The Se en Wonde s o he Wo ld
Lec u e no es on 21s -cen u y physics
P.G.L. Po a Mana
Wo king d a e sion 0.4, upda ed 11 Oc obe 2025
pglpm.gi hub.io/7wonde s
Licence
P.G.L. Po a Mana <pgl po amana.o g>
h ps://o cid.o g/0000-0002-6070-0784
Typese wi h L
A
TEX.
No la ge language models we e used in he p epa a ion o his documen .
Co e : he Pale Blue Do image o he Ea h,
aken by Voyage 1 om igh ou side Plu o’s o bi .
h ps://science.nasa.go / esou ce/ oyage -1s-pale-blue-do /
Con en s
P e ace o he s uden 9
Thanks ................................. 10
0 O e iew 13
0.1 The plan o hese no es .................... 13
0.2 P e equisi es .......................... 14
0.3 Fea u es o he ex ....................... 15
0.4 Code o nume ical simula ions ............... 16
0.5 Addi ional exe cises ...................... 18
0.6 No a ion and e minology .................. 18
URLs o chap e 0 .......................... 21
1 Physics, quan i ies, uni s 23
1.1 Physics? ............................. 23
1.2 Wha is “ undamen al” physics? ............... 24
1.3 Physical laws .......................... 27
1.4 Physical quan i ies ....................... 29
1.5 Physical dimensions and uni s ................ 32
1.6 Ma hema ics wi h quan i ies and uni s ........... 34
URLs o chap e 1 .......................... 37
2 Time and space 39
2.1 Time and p ope ime ..................... 39
2.2 Coo dina e ime ........................ 45
2.3 Space, leng h, dis ance ..................... 46
2.4 Rada dis ance ......................... 48
2.5 Coo dina e sys ems ...................... 50
2.6 Spa ial coo dina e dis ance and leng h ........... 53
2.7 Coo dina e posi ion ...................... 55
3
Con en s
2.8 Coo dina e eloci y and accele a ion ............ 56
2.9 Angles .............................. 59
URLs o chap e 2 .......................... 60
3 Main physical quan i ies 61
3.1 Se en p imi i e quan i ies .................. 61
3.2 Two basic p ope ies ...................... 62
3.3 Ma e .............................. 64
3.4 Elec ic cha ge ......................... 67
3.5 Magne ic lux .......................... 67
3.6 Ene gy-mass .......................... 68
3.7 Momen um ........................... 73
3.8 Angula momen um ...................... 75
3.9
Ene gy-mass, momen um, angula momen um a e
coo dina e-dependen ..................... 77
3.10 En opy ............................. 78
3.11 Auxilia y quan i ies ...................... 79
3.12 Me ic .............................. 81
URLs o chap e 3 .......................... 83
4 Volume con en s, luxes, supplies 85
4.1 Con en , lux, supply ..................... 85
4.2 Symbols, no a ion, and ex ensi i y .............. 87
4.3 Con ol olumes and con ol su aces ............ 90
4.4 Choices o con ol olumes and su aces .......... 93
4.5 Volume con en ......................... 94
4.6 Flux o scala quan i ies .................... 96
4.7 Rep esen a ion o scala luxes ................ 99
4.8 Flux o ec o quan i ies and i s ep esen a ion ...... 101
4.9 Fluxes h ough di e en su aces .............. 104
4.10 Flux and supply o momen um: o ce ............ 106
4.11 Flux o momen um is su ace o ce ............. 107
4.12 P essu e, ension, shea o ce ................. 110
4.13 Closed con ol su aces, in luxes, e luxes .......... 112
4.14 Time-in eg a ed luxes and supplies ............. 114
4.15 The ela ion be ween luxes and eloci ies ......... 116
URLs o chap e 4 .......................... 118
5 Physical laws 119
4
Con en s
5.1 Some classi ica ions o physical laws ............ 119
5.2 Uni e sal laws s cons i u i e ela ions ........... 120
5.3 Balance laws and conse a ion laws ............. 122
5.4 Conse a ion laws ....................... 125
5.5 Balance laws .......................... 129
5.6 Examples ............................ 132
5.7 Balance laws: di e en ial exp ession ............. 137
5.8 Se en uni e sal balance laws ................. 142
5.9 Cons i u i e ela ions ..................... 146
5.10
Summa y o di e ences be ween he se en balance laws
and cons i u i e ela ions ................... 150
5.11 New on’s laws ......................... 151
URLs o chap e 5 .......................... 155
6 In e ence, p edic ion, simula ion 157
6.1 Gene al in e ence o physical quan i ies ........... 157
6.2 P edic ion and o ecas .................... 159
6.3 Nume ical ime in eg a ion and simula ions ........ 161
6.4 I e a ion: he special ole o balance laws .......... 165
6.5
I e a ion: he oles o bounda y condi ions and cons i u i e
ela ions ............................. 166
6.6 Compu e code o nume ical ime in eg a ion ....... 169
6.7 Nume ical ime in eg a ion o posi ion ........... 170
6.8 Applicabili y o nume ical ime in eg a ion ......... 173
URLs o chap e 6 .......................... 174
7 Conse a ion & balance o ma e 175
7.1 Fo mula ion and gene ali ies ................. 175
7.2 Examples o cons i u i e ela ions .............. 176
7.3 Examples o applica ions ................... 179
URLs o chap e 7 .......................... 183
8 Conse a ion o elec ic cha ge 185
8.1 Fo mula ion and gene ali ies ................. 185
9 Conse a ion o magne ic lux 187
9.1 Fo mula ion and gene ali ies ................. 187
10 Balance o momen um 189
10.1 Fo mula ion and gene ali ies ................. 189
5
Con en s
10.2 Momen um supply: g a i y and ine ial o ces ....... 190
10.3
New on’s app oxima e ela ion be ween momen um and
ma e .............................. 193
10.4 Analy ical ime in eg a ion o posi ion and momen um . . 194
10.5 Nume ical ime in eg a ion o posi ion and momen um . 198
10.6
Cons i u i e ela ions o momen um lux: ma e ials science
201
10.7 Consequences o simpli ying assump ions ......... 203
10.8 Hookean ma e ials ....................... 206
10.9 Applica ion: Hookean sp ing and ha monic oscilla o . . 208
10.10
The concep o physical s a e. A s a egy o nume ical ime
in eg a ion ........................... 213
10.11 Non-Hookean and mo e gene al ma e ials ......... 223
10.12 Beyond he no-mass simpli ica ion .............. 226
10.13 Many-body sys ems ...................... 227
10.14 F ic ion and no mal o ces .................. 228
10.15 Applica ion: s a ics ....................... 232
10.16 Applica ion: A mosphe ic p essu e ............. 234
10.17 Applica ion: Ai bo ne ligh ................. 236
10.18 Applica ion: Rocke s ...................... 238
10.19 Applica ion: S a ics again and cable ca s .......... 238
10.20 Applica ion: Momen um luxes in a gas ........... 242
10.21 Choice o con ol su aces and olumes ........... 244
URLs o chap e 10 .......................... 246
11 Balance o ene gy-mass 247
11.1 Fo mula ion and gene ali ies ................. 247
11.2 De ini ions o o al ene gy ................... 248
11.3 Is ene gy conse ed? ...................... 251
11.4 Fo ms o ene gy ........................ 252
11.5 In e nal, kine ic, g a i a ional po en ial ene gy ...... 253
11.6 In e nal and kine ic ene gy: obse a ion scales ....... 255
11.7
The exchange be ween in e nal, kine ic, g a i a ional po-
en ial ene gies ......................... 255
11.8 Hea lux and mechanical powe ............... 257
11.9 Hea and powe : obse a ion scales ............. 262
11.10
Summa y: cons i u i e ela ions o ene gy in he p esence
o ma e ............................ 265
11.11
Ene gy con en , lux, supply depend on he coo dina e
sys em .............................. 268
6
Con en s
11.12 Rigid mo ion and igid bodies ................ 272
11.13 Cons i u i e ela ions o ideal gases ............ 275
11.14 Example applica ions: ideal gas and pis on ......... 282
11.15 Su aces o discon inui y ................... 294
URLs o chap e 11 .......................... 299
12 Balance o angula momen um 300
12.1 Fo mula ion and gene ali ies ................. 300
12.2 De ini ions o angula momen um .............. 302
12.3 Angula momen um as a wis ed ec o ........... 302
12.4 Cons i u i e ela ions o angula momen um ....... 304
12.5
P elimina y ema ks on cons i u i e ela ions o igid
mo ion .............................. 308
12.6 Cen e o mass-ene gy ..................... 308
12.7 Angula eloci y ........................ 310
12.8 Tenso o ine ia ........................ 311
12.9 Rigid mo ion II ......................... 313
12.10 Cons i u i e ela ions o a igid body ............ 314
12.11 Nume ical ime in eg a ion o igid body ......... 316
12.12 Cons i u i e ela ions o o ces and o ques in a igid body318
URLs o chap e 12 .......................... 320
13 Balance o boos momen um 321
13.1 Fo mula ion and gene ali ies ................. 321
14 Rema ks on momen um and ene gy 323
14.1
Common misunde s andings on momen um, ene gy, an-
gula momen um ........................ 323
15 Balance o en opy 325
15.1 Fo mula ion and gene ali ies ................. 325
15.2 The physical ole o he balance o en opy ......... 328
15.3 I e e sibili y .......................... 329
15.4 En opy balance as a me a-law ................ 331
15.5 En opy depends on a e e ence s a e ............ 332
15.6 Examples o cons i u i e ela ions .............. 333
15.7 The mal engines ........................ 335
15.8 One- empe a u e he mal engines: hea e s ......... 337
15.9
Two- empe a u e he mal engines: bounds on mechanical
wo k ............................... 339
7
Con en s
15.10 Example applica ion: ic ion coe icien ........... 343
URLs o chap e 15 .......................... 344
Pos ace o he eache 345
Validi y o he ma hema ical o m o he balance laws in Gene al
Rela i i y ............................ 349
URLs o chap e Pos ace o he eache ............... 351
Bibliog aphy 353
8
P e ace o he s uden
I don’ know wha ’s he ma e wi h people: hey don’
lea n by unde s anding; hey lea n by some o he way – by
o e, o some hing. Thei knowledge is so agile!
Finally, I said ha I couldn’ see how anyone could be
educa ed by his sel -p opaga ing sys em in which people
pass exams, and each o he s o pass exams, bu nobody
knows any hing.
R. P. Feynman 1989
These no es a e aimed a unde g adua e s uden s in Physics and in
Enginee ing p og ammes, bu g adua e s uden s may also ind hem
use ul.
You a e p obably awa e o he a ie y o physics b anches, such as
mechanics, he modynamics, chemis y, elec omagne ics, luid mechanics,
s a ics, nuclea physics, and many o he s. Maybe you a e acquain ed wi h
some o hem. P obably you a e also awa e o he exis ence o di e en
physical heo ies, like New onian Mechanics, Gene al Rela i i y, Quan um
Theo y, which gi e di e en explana ions and o mulae o he same
physical phenomena. Some physical heo ies a e said o be mo e exac o
mo e app oxima e han o he s; New onian mechanics, o ins ance, is an
app oxima ion o Gene al Rela i i y. Does his wild a ie y o b anches
and heo ies also mean a wild a ie y o p inciples, me hods, ma hema ical
o mulae – which you’ll ha e o lea n i you wan o need o s udy any
pa icula b anch o heo y?
Yes, i does.
Bu he e is also a co e o e y ew p inciples ha apply uni e sally
o e e y b anch and e e y heo y known oday, be i mechanics o he -
modynamics, New onian Mechanics o Gene al Rela i i y. I you lea n
hese p inciples, you’ll be able o immedia ely wo k wi h, and unde s and,
9
0. O e iew 0.4 Code o nume ical simula ions
Hype links and bibliog aphy
Some pieces o ex a e hype links, like his one abou One Punch Man
1
.
You ecognize hem om hei di e en colou and om he li le oo no e
numbe ha ollows hem. The links’ URLs a e also lis ed a he end o
each chap e , in case you’ e eading a p in ed copy o hese no es.
The ex gi es bibliog aphic e e ences, like “Eins ein 1905a”, o sci-
en i ic li e a u e. The e e ences a e lis ed in he inal Bibliog aphy on
page 353. Bibliog aphic e e ences a e gi en o wo easons:
•Fo you own cu iosi y. “Belie e no hing, O monks,
me ely because you ha e been
old i , o because i is
adi ional, o because you
you sel es ha e imagined i . Do
no belie e wha you eache
ells you me ely ou o espec
o he eache .”
(a ibu ed o Gau ama
Buddha)
•
To back up wha ’s w i en in he ex . In science you should no belie e
some hing jus because you’ e ead i somewhe e. You should, as
much as possible, go and check o you sel he logic and he expe imen al
e idence behind he s a emen .
0.4 Code o nume ical simula ions
Why code?
As we lea n abou physical laws and equa ions, we shall explo e and apply
hem bo h analy ically, ha is wi h pen & pape , and nume ically, ha is by
w i ing and unning code in a p og amming language. Why using code?
Fo h ee di e en easons.
Fi s , a la ge and la ge pa o physics and enginee ing esea ch
and applica ions equi es he nume ical solu ion o physics equa ions, by
means o code in di e en p og amming languages. Code is used e en
o p oblems ha could be sol ed wi h pen & pape , because doing so
minimizes he possibili y o human calcula ion e o s. So i ’s good i you
ge amilia wi h “ ansla ing” physics p oblems in o code and sol ing
hem by unning code.
Second, many – we can say he majo i y – o physics phenomena
canno be ully s udied analy ically. Bu many o hem a e easy o simula e
and s udy nume ically. The e o e using code allows you o explo e many
addi ional in e es ing physical phenomena.
Thi d, w i ing code allows us o see mo e clea ly he di e en oles
ha di e en physical laws ha e in physical phenomena, as well as he
in e ela ionship among hem.
16
0. O e iew 0.4 Code o nume ical simula ions
The hi d eason abo e is he mos impo an o hese no es. This is
also why ou goal is no o w i e e icien code, bu o w i e code ha allows
us o unde s and he physics.
Oc a e
The sc ip s o nume ical simula ions p esen ed in he ex a e w i en in
Oc a e2. Why Oc a e o hese no es? Se e al easons:
•I ’s ee and open sou ce.
•I uns on all majo ope a ing sys ems.
•
I includes a g aphical use in e ace and de elopmen en i onmen ,
whe e you can edi you code and see esul ing plo s.
•
You can un i immedia ely, wi hou needing o compile i i s . This
means ha i ’s slowe han some o he p og amming languages; bu
as s a ed abo e, ou goal is no e icien code, bu code ha makes us
unde s and physics.
•
I ssyn axis e yclose ophysics’sma hema icalno a ion.Fo ins ance,
a sum o wo ec o s exp essed ma hema ically as
[1,2,3] + [5,6,7]
is w i en in Oc a e like his: [1, 2, 3] + [5, 6, 7] .
Compa e he same exp ession in o he p og amming languages:
–Py hon: np.a ay([1, 2, 3]) + np.a ay([5, 6, 7])
(you mus load he numpy lib a y i s ).
–Ja aSc ip : ma h.add([1, 2, 3], [5, 6, 7])
(you mus load he ma hjs lib a y i s ).
–R: c(1, 2, 3) + c(5, 6, 7) .
–
Rus :
A ay1:: om( ec![1, 2, 3]) + A ay1:: om( ec![5, 6, 7])
(you mus load nda ay).
–C, Visual Basic, Pe l, Go: be ween 5 and 10 lines o code.
•You can use i di ec ly, wi hou loading ex a lib a ies.
•
You don’ need o wo y abou “en i onmen s” o e sion dependen-
cies.
•
I s syn ax is e y simila o ma lab’s, which is a language used in
some o he enginee ing cou ses; ypically you can un Oc a e’s code
in ma lab and ice e sa.
17
0. O e iew 0.5 Addi ional exe cises
Keep in mind ha we a e no saying ha Oc a e is he bes p og amming
language o physics; wha ’s bes depends on he speci ic physics applica-
ion. We’ e jus saying ha Oc a e is pa icula ly con enien o he goals
we ha e in hese no es.
You can ind many in oduc ions o Oc a e online. Check ou i s own
documen a ion3 o example.
Code colou s
The sc ip s usually p esen wo colou s:
•Blue lines a e s ic ly ela ed o he physical simula ion.
•
G ey lines ake ca e o auxilia y ope a ions, like plo ing he esul s
o sa ing hem o ile.
O he p og amming languages
Feel ee o ansla e he Oc a e sc ip s used in hese no es in o you
a ou i e p og amming language; ha ’s a g ea exe cise! Please do ge
in ouch i you wan o make hem a ailable in hese no es. Fo example,
some Py hon
4
e sions a e also gi en o some o he sc ip s, linked on he
ma gin. No e ha hey need he NumPy
5
and Ma plo lib
6
packages. The
Py hon sc ip s should wo k on ma plo lib.online7.
0.5 Addi ional exe cises
Addi ional exe cises, and accompanying solu ions o some o hem, a e
a ailable a
h ps://pglpm.gi hub.io/7wonde s/
0.6 No a ion and e minology
Ma hema ical no a ion ollows as much as possible he s anda ds gi en
by he In e na ional Sys em o Uni s (SI)
8
, lis ed o example in iso 2009
and iso 2019. One sligh excep ion a e physical dimensions
§1.5 p.32
, whose
no a ion will be explained in ha sec ion.
In physics, enginee ing, and mos o he scien i ic ields he e is no
uni e sal s anda d o symbols. I ’s he e o e impo an ha you ocus on he
18
0. O e iew 0.6 No a ion and e minology
concep ha a symbol s ands o , a he han he symbol i sel . O he wise
you’ll ace many di icul ies eading li e a u e and o he scien is s’ epo s.
The p esen no es y o use as much as possible he symbols ecommen-
ded by he In e na ional O ganiza ion o S anda diza ion (ISO)
9
, which
a e o en also adop ed by o he in e na ional bodies like he In e na ional
Bu eau o Weigh s and Measu es (BIPM)
10
, he Ins i u e o Elec ical and
Elec onics Enginee s (IEEE)
11
, o he In e na ional Union o Pu e and
Applied Chemis y (IUPAC)
12
. Some changes a e una oidable, because
di e en disciplines o en use he same symbol o di e en quan i ies. Fo
ins ance, he symbol ‘
𝐸
’ is used in many ields o ‘ene gy’, in mechanics
o he ‘Young modulus’ (a mechanical cha ac e is ic o elas ic ma e ials),
in elec omagne ics o he ‘magni ude o he elec ic ield’.
The mos common symbols used in he p esen no es a e lis ed in
able 0.1 on page 20.
19
0. O e iew 0.6 No a ion and e minology
Table 0.1 Lis o common symbols
(G eek symbols, wi h names)
𝜅(kappa) g a i a ional cons an
𝜇(mu) iscosi y coe icien
𝜇kcoe icien o kine ic ic ion
𝜇scoe icien o s a ic ic ion
𝛱(capi al pi) en opy lux
𝜌( ho) mola mass
𝛷(capi al phi) ene gy-mass lux
𝜴(capi al omega) angula - eloci y ma ix
𝝎(omega) angula - eloci y ec o
(La in symbols)
𝐴usually: a ea; in some con ex s: jus a coe icien
𝒜ma e supply
ℬmagne ic lux
𝐶usually: mola hea capaci y; in some con ex s: jus a coe icien
𝑐usually: speed o ligh ; in some con ex s: jus a coe icien
𝐸ene gy-mass con en
ℰelec ic ension (line in eg al o elec ic ield s eng h)
𝑭momen um lux (su ace o ce)
𝑮momen um supply ( olume o ce)
𝒈accele a ion ec o o ee all
ℎsu ace coe icien o hea ans e
𝑰c enso o ine ia (wi h espec o cen e o mass-ene gy)
ℐelec ic cu en
𝐽ma e lux
𝑘elas ic cons an
𝑳angula -momen um con en
𝑴angula -momen um lux (su ace o que)
𝑚mass-ene gy con en , ypically a es
𝑁ma e con en (amoun o subs ance)
𝑷momen um con en
𝑝p essu e
𝑄hea lux
𝒬elec ic-cha ge con en
𝑅usually: mola gas cons an ; in some con ex s: adius
ℛene gy-mass supply
𝒓coo dina e-posi ion ec o
𝑆en opy con en
𝑇 he modynamic empe a u e (in kel ins)
𝓣angula -momen um supply ( olume o que)
𝑡coo dina e ime
𝑈in e nal ene gy
𝑉 olume
𝒗coo dina e- eloci y ec o
𝑥posi ion coo dina e, ypically ho izon al
𝑦posi ion coo dina e, ypically ho izon al
𝑧posi ion coo dina e, ypically e ical
20
0. O e iew 0.6 No a ion and e minology
URLs o chap e 0
1. h ps://onepunchman. andom.com
2. h ps://oc a e.o g
3. h ps://docs.oc a e.o g/la es /
4. h ps://www.py hon.o g
5. h ps://numpy.o g
6. h ps://ma plo lib.o g
7. h ps://ma plo lib.online
8. h ps://www.nis .go /pml/special-publica ion-811
9. h ps://www.iso.o g
10. h ps://www.bipm.o g
11. h ps://www.ieee.o g
12. h ps://iupac.o g
21
0. O e iew 0.6 No a ion and e minology
22
1
Physics, quan i ies, uni s
Philosophy is w i en in his g and book, he uni e se,
which s ands con inually open o ou gaze. Bu he book
canno be unde s ood unless one i s lea ns o
comp ehend he language and ead he le e s in which i is
composed. I is w i en in he language o ma hema ics,
and i s cha ac e s a e iangles, ci cles, and o he geome ic
igu es wi hou which i is humanly impossible o
unde s and a single wo d o i .
G. Galilei 1623
1.1 Physics?
I you hink abou i , many hings we o dina ily do e e y day a e some
so o magic. Think o how you can ins an aneously see and speak wi h a
pe son li ing on ano he con inen , in eal ime, using jus a small widge
in he palm o you hand. Think o how you can ins an aneously see whe e
you a e on he Ea h, using he same widge . Think o how as you can go
o ano he coun y, by lying in a huge me al hing. Think o how you can
command and in e ac wi h a pu ely ic i ious anima ed wo ld when you
play on you compu e . The lis can go on o e e . O he hings a e luckily
less o dina y, bu s ill inspi e a lo o awe: hink o he de as a ing powe
unleashed by some hing oughly as small as a ennis ball, in an a omic
bomb.
We can do hese as onishing hings hanks o ou unde s anding o
how he wo ld wo ks. Tha ’s Physics.
Many hings can be said and ha e been said abou science and physics.
Ra he han epea ing wha ’s been al eady w i en in many excellen books,
I in i e you o ake a b eak he e and go ead hei in oduc ions. Choose as
you please; compa e wha hey say; don’ limi you sel o popula books.
23
1. Physics, quan i ies, uni s 1.2 Wha is “ undamen al” physics?
1.2 Wha is “ undamen al” physics?
Bu wha ’s he “ul ima e” goal o physics? Wha ’s “ undamen al” physics?
The answe o his ques ion is again subjec i e – also in his case physics
le s you exp ess you p ocli i ies and pe sonali y. In he his o y o physics
one can p obably iden i y wo main concep ions o “ undamen al” physics.
Fo some physicis s i is abou inding he ul ima e building blocks, so
ha one day we can say “. . . and hese a e he cons i uen s, and hey obey
hese equa ions”. The his o y o physics seems o show ha his goal is
o e u ned e e y ew gene a ions. And ye e e y gene a ion says “Now
we almos ha e he comple e pic u e – i ’s igh behind he co ne . I ’s ue
ha p e ious gene a ions hough hey almos had i , and u ned ou o
be w ong. Bu his ime is di e en , his ime we ha e he eal deal!”. The
heo e ical and pa icle physicis Ge ey Chew
1
depic ed his si ua ion as
in ig.1.1. Fo his eason some physicis s a e a li le scep ical abou his
goal; maybe i ’s a ne e -ending s uc u e, wi h su p ises a e e y deepe
look.
So o o he physicis s undamen al physics is abou inding some
poin o iew o ma hema ical s uc u e ha is ich enough o make use ul
p edic ions, and ye lexible enough o accommoda e any new pa e ns o
objec s ha we migh disco e . In a manne o speaking, i is abou inding
“pa e ns o pa e ns” o “laws abou physical laws”.
The wo concep ions abo e a e no mu ually exclusi e, and bo h a e
always pu sued, e en i ime-changing ashions may emphasize he one
o he o he .
In hese no es we ake a poin o iew sligh ly close o he second
concep ion. This will also be e lec ed in he main di ision be ween wo
kinds o physical laws ha we’ll d aw in chap e 5.
24
1. Physics, quan i ies, uni s 1.2 Wha is “ undamen al” physics?
Figu e 1.1 (Con inues on p. 26)The p og ess o “ undamen al” physics, om Chew 1970 as
ep oduced in T uesdell 1987
25
1. Physics, quan i ies, uni s 1.5 Physical dimensions and uni s
Time and space a e aken as p imi i e quan i ies – o ob ious easons –
in almos all cu en physical heo ies. We shall discuss hem in chap e 2.
The o he se en physical quan i ies, which we shall discuss in chap e 3,
a e chosen because o se e al ad an ages:
✓
They can be unde s ood physically, and ea ed ma hema ically, in
a simila way. The e o e as we ge amilia in hinking abou and
handling any one o hem, we au oma ically ge also amilia wi h all
he o he s.
✓
The way in which hey can be unde s ood and handled is in ui i e
and lends i sel o men al isualiza ion.
✓
They lead o physical laws ha ha e an almos iden ical exp ession.
And, as men ioned be o e, his exp ession ep esen a so o “budge ”
and is he e o e in ui i e.
✓
These laws a e common o all ou physical heo ies, and hey can be
exp essed in a ma hema ical o m ha ’s he same in any heo y.
The p ice o pay o he ad an ages abo e is ha some o hese quan i ies
may be less amilia han o he s; bu he ad an ages seem o ou weigh his
disad an age.
We shall also use o he quan i ies, some o which a e e y amilia , like
empe a u e o p essu e. Bu he quan i ies lis ed abo e a e ou undamen al
building blocks.
1.5 Physical dimensions and uni s
Measu emen is he p ocess by which we de e mine he alue o a physical
quan i y. Measu emen s can be ex emely complex, and can ex emely
di e en e en i hey a e abou he same quan i y. Conside he ways we
can measu e he mass-ene gy o a oo ball, compa ed o he ways we can
measu e he mass-ene gy o he Sun.
To each quan i y we associa e a physical dimension. The e m ‘dimen-
sion’ he e has no hing o do wi h physical ex ension, as in “ he dimensions
o his box”; be ca e ul no o con use he wo. Usually i ’s clea which one
is mean om he con ex .
Physical dimensions e lec he ma hema ical ela ionships ha exis
be ween physical quan i ies. Fo ins ance, we ake dis ance o ha e dimen-
sion leng h, and ime in e al o ha e dimension ime; i we now de ine speed
as dis ance di ided by a ime in e al, hen he physical dimension o speed
32
1. Physics, quan i ies, uni s 1.5 Physical dimensions and uni s
is leng h
/
ime. Physical dimensions help us o a oid doing ma hema ical
ope a ions ha don’ make sense wi h some quan i ies. Fo example, i
doesn’ make sense o sum up he olume o a glass o wa e wi h i s em-
pe a u e. The In e na ional Sys em o Uni s (SI)
6
ep esen s he physical
dimensions o se e al base quan i ies by single capi al le e s; o ins ance
‘T’ s ands o he physical dimension ime. In he p esen no es we use
he di e en , non-s anda d con en ion o deno ing a physical dimension
using his on , jus o a oid ambigui y and con usion wi h o he one-le e
symbols.
Wi h each physical dimension we can associa e a uni o measu emen ,
which as men ioned in §1.4 exp esses a basic s anda d o compa ing he
measu emen esul s o quan i ies ha ing ha physical dimension. Uni s
a e e y impo an and mus always be w i en, o se e al easons. Fi s ,
a numbe wi hou uni s doesn’ ell us any hing. I I ell you “ he place is
a a dis ance 100 om he e”, you ha e no idea how a he place is. “100”
wha ? 100 me es? 100 kilome es? These a e comple ely di e en dis ances.
Second,uni sgi eususe ul in o ma ionabou physicalquan i iesand hei
ela ionships and measu emen . I you see he exp ession “
3m/s
”, hen
he e’s a s ong possibili y ha ha ’s a eloci y. I you see he exp ession
“
5J/m2
”, hen you ha e a hin ha i could be measu ed by measu ing an
ene gy-mass and an a ea, and hen di iding hem. Thi d, keeping ack o
uni s o en allows us o quickly ca ch e o s in sol ing a physical p oblem.
I a physical quan i y is de ined in e ms o o he quan i ies, hen i s
uni is usually gi en in e ms o he de ining quan i ies. Fo example,
i we de ine speed as leng h di ided by ime, hen i s uni is ‘me es pe
second’, ‘
m/s
’. Some combina ions o uni s ecei e special uni names.
Fo ins ance, powe is de ined as ene gy-mass di ided by ime; i s uni
is he e o e ‘
J/s
’ (‘joules pe second’). Bu his compound uni is usually
called ‘wa ’, symbol ‘
W
’. In o he wo ds, ‘one wa ’ and ‘one joule pe
second’ a e he same:
1W ≡1J/s.
The opics o measu emen , physical dimensions, and uni s, which a e
s udied in me ology and in dimensional analysis, could occupy an en i e
cou se by hemsel es! I assume ha you will look in he documen s by he
SI o mo e complica ed de ails. We shall speak u he abou uni s in he
nex chap e s. The main quan i ies and uni s used in he p esen no es a e
summa ized in able 3.1 on page 82 and able 4.1 on page 88.
33
1. Physics, quan i ies, uni s 1.6 Ma hema ics wi h quan i ies and uni s
-How o p onounce he quo ien and he p oduc o uni s
Acco ding o he ules o he SI, he quo ien o wo uni s is p onounced
‘pe ’ in English; o ins ance ‘
m/s
’ is p onounced ‘me es pe second’.
The p oduc o wo uni s is simply p onounced by conca ena ing he
names o he uni s; o ins ance ‘
N·s
’ is p onounced ‘new on seconds’.
Fo o he ules abou p in ing and epo ing uni s ake a look a he
NIST Check Lis 7.
1.6 Ma hema ics wi h quan i ies and uni s
Va iables and unc ions
When a physical quan i y is deno ed by a symbol o a iable, keep in mind
ha a uni is “con ained” in he symbol, so o speak. Fo example i he
a iable
𝑡
deno es a ime, hen i includes some ime uni , say seconds. This
becomes appa en when we w i e he alue o he symbol, o ins ance
“
𝑡=120s
”. The uni is no p ede e mined, bu i mus co espond o
he dimension o ha quan i y. We could o ins ance w i e “
𝑡=2min
”
ins ead; he wo exp essions a e comple ely equi alen .
This ac mus be kep in mind when combining symbols. Fo example,
i
𝑑
is a dis ance and
𝑡
is a ime, hen w i ing
𝑣=𝑑/𝑡
ells us ha
𝑣
is a
eloci y, and i has app op ia e uni s ha come om
𝑑
and
𝑡
, o ins ance
m/s.
Uni s o he wise beha e jus like li e al cons an s o all ma hema ical
pu poses, jus like he le e ‘
𝑎
’ in he exp ession ‘
𝑎 𝑥
’. This is why hey can
be simpli ied; o ins ance:
3mol/s·5s =3mol
s·5s=15mol .
Pa icula ca e mus be aken wi h igonome ic and exponen ial
unc ions, like
sin()
,
cos()
,
an()
,
exp()
,
log()
; hese unc ions only admi a
dimensionless a gumen (which o he igonome ic ones co esponds
o adians). So he e canno be uni s like ‘
s
’ o ‘
m
’ wi hin hese unc ions:
we mus make su e ha any uni s p esen wi hin cancel ou .
34
1. Physics, quan i ies, uni s 1.6 Ma hema ics wi h quan i ies and uni s
This makes sense, because we wouldn’ know how o in e p e he
a gumen o he wise. Suppose you ead “
cos(60s)
” somewhe e: how much
is ha ? I we say “jus disca d he uni ”, we would ha e
cos(60s)?
=cos(60) ≈ −0.95
bu wai :
60s ≡1min
, so we could equi alen ly w i e “
cos(1min)
”. Then,
acco ding o he hypo he ical ule “jus disca d he uni ”, we would ha e
cos(60s) ≡ cos(1min)?
=cos(1) ≈ +0.54
a comple ely di e en esul !
Fo his eason an exp ession like ‘
cos(𝑡)
’, wi h
𝑡
deno ing ime, doesn’
make sense: he e’s a ime uni in he a gumen o
cos()
. I we wan o
exp ess an oscilla ion wi h ime, we mus w i e ins ead some hing like
cos𝑡
𝑇
whe e
𝑇
is he pe iod o he oscilla ion, a symbol which also includes a
ime uni , which simpli ies wi h he one in
𝑡
. I he pe iod o he oscilla ion
is 𝑇=1s hen we can also simply w i e
cos(𝑡/s)
This exp ession is now unambiguous: suppose ha
𝑡=60s ≡1min
, hen
cos(𝑡/s)=cos(60s/s)=cos(60) ≈ −0.95
=cos(1min/s)=cos(1·60 s/s=cos(60) ≈ −0.95
Also emembe ha he esul s o igonome ic and exponen ial
unc ions a e dimensionless numbe s as well, so an exp ession like
‘
3 cos(...)
’ deno es a pu e numbe , wi h no uni s. I you wan o exp ess
ha he esul is a leng h, he app op ia e uni s mus appea . We can o
ins ance w i e
𝐿cos(...)
whe e
𝐿
is a leng h, and he e o e includes some kind o leng h uni such
as ‘m’. I his leng h is, say, 𝐿=2m we can also simply w i e
2 cos(...)m
35
1. Physics, quan i ies, uni s 1.6 Ma hema ics wi h quan i ies and uni s
De i a i es
When we ollow he ules abo e, all o he ma hema ical ope a ions au o-
ma ically ake ca e o e e y hing. The de i a i e, o ins ance, is calcula ed
in he usual way, ea ing any isible uni s as li e al cons an s. Le ’s see a
conc e e example. This exp ession
𝑥(𝑡)=2 cos(𝑡/s)m
says ha he posi ion o some objec oscilla es wi h ime, be ween he
alues
−2m
and
+2m
. When
𝑡=0s
, he posi ion is
𝑥=+2m
. The posi ion
𝑥=−2m is eached when he a gumen o cos() is π, ha is
𝑡/s= π ⇒𝑡≈3.14s .
The eloci y
§2.8 p.56
o he objec is gi en by he de i a i e o his
exp ession wi h espec o
𝑡
. Le ’s calcula e i ea ing all uni symbols as
li e al cons an s:
d𝑥(𝑡)
d𝑡=d
d𝑡2 cos(𝑡/s)m=2h−sin(𝑡/s) · 1
s
chain ule im=−2 sin(𝑡/s)m/s
and you see ha he co ec uni s o eloci y ha e au oma ically appea ed.
36
1. Physics, quan i ies, uni s 1.6 Ma hema ics wi h quan i ies and uni s
URLs o chap e 1
1. h ps://www.physics.lbl.go / emembe inggeo eychew/
2.
h ps://www.pa keha ison.com/bodies-o -wo k/a chi ec -s-b o he /ea
h-elegies
3. Join Commi ee o Guides in Me ology (JCGM)
4. h ps://jcgm.bipm.o g/ im/en/1.1.h ml
5. h ps://www.you ube.com/wa ch? =nYg6jzo iAc& =893s
6. h ps://www.nis .go /pml/special-publica ion-811
7.
h ps://www.nis .go /pml/special-publica ion-811/nis -guide-si-check
-lis - e iewing-manusc ip s
37
1. Physics, quan i ies, uni s 1.6 Ma hema ics wi h quan i ies and uni s
38
2
Time and space
I we wan o desc ibe he mo ion o a ma e ial poin , we
gi e he alues o i s coo dina es as a unc ion o ime.
Howe e , we should keep in mind ha o such a
ma hema ical desc ip ion o ha e physical meaning, we
i s ha e o cla i y wha is o be unde s ood he e by “ ime”.
We ha e o bea in mind ha all ou p oposi ions in ol ing
ime a e always p oposi ions abou simul aneous e en s. I ,
o example, I say ha “ he ain a i es he e a 7 o’clock”,
ha means, mo e o less, “ he poin ing o he small hand o
my clock o 7 and he a i al o he ain a e simul aneous
e en s”.
A. Eins ein 1905a
2.1 Time and p ope ime
Time is a p imi i e quan i y. We unde s and he no ion o ime in ui i ely,
e en i i is di icul o explain – ha ’s why i is aken as p imi i e. In
1905, wi h he Theo y o Rela i i y, pa o ou e e yday in ui ion abou
he no ion o ime was se iously shaken. Fo many yea s a e wa ds ou
old in ui ion could s ill be used in p ac ice and in applica ions. Bu he
new, co ec in ui ion is becoming mo e and mo e impo an in e e yday
li e and echnologies. GPS na iga ion, o example – which we use e e y
day in leisu e ac i i ies like hiking o sigh seeing, as well as in mo e
c i ical ones like ae oplane landing – c ucially depends on he co ec
no ion, in ui ion, and measu emen o ime. Luckily he new in ui ion o
ime is also becoming mo e and mo e widesp ead hanks o ilms and
mass-media; hink o mo ies like In e s ella 1.
Le ’s see, by means o a hough expe imen , how ou adi ional
in ui ion abou ime goes as ay. He e’s Alice, Bob, and Cha lie. They ha e
ex emely p ecise clocks, buil in exac ly he same way. They synch onize
39
2. Time and space 2.1 Time and p ope ime
hei clocks and s ay e y close o one ano he . Keeping close, hey go
a ound, maybe on an ae oplane o on a space ship, and hey cons an ly
compa e hei h ee clocks. They no ice ha hei clocks s ay pe ec ly
synch onized all he ime, no ma e whe e hey go and wha hey do.
A some poin hey sepa a e, and each one goes a ound independen ly.
One o hem migh s ay in place, ano he migh ake a helicop e , and
ano he migh go o a ip o Ma s and back.
Alice and Bob a some poin mee again, and compa e hei clocks.
They see ha hei clocks a e no synch onized anymo e; he di e ence
could be as small as mic oseconds, o as la ge as yea s. In ac , i his
ime disc epancy is la ge, hey no ice ha hey ha e also aged di e en ly:
he ime di e ence is no only appa en in hei clocks, bu also in hei
bodies. Le ’s say o conc e eness ha Alice’s clock is ahead o Bob’s, o
equi alen ly ha Bob’s clock is behind Alice’s. No e he ollowing aspec s:
Fi s , bo h Alice and Bob can say “my clock has been wo king i e, so i
should be co ec ”: nei he has no iced any hing s ange abou he passage
o ime.
Second, i hey now s ay oge he , hey see ha hei clocks emain
exac ly synch onized, besides he ime disc epancy hey no iced when
hey me again. This disc epancy doesn’ inc ease o dec ease. They migh
e en e ace oge he Alice’s and Bob’s p e ious ips; hei clocks will s ill
emain synch onized.
Thi d, hey migh wonde wha he ime is on Cha lie’s clock. Bu
Cha lie is a some dis ance away. They could decide o con ac Cha lie,
say ia adio, and ask “wha does you clock show, igh now?”. Bu hey
would no ice ha he e’s a delay, e en i ex emely small, in he adio
ansmission; so i ’s unclea o wha ime Cha lie’s answe would apply.
I we say “le ’s accoun o he adio-signal speed”, we see ha he e’s a
logical p oblem: speed is dis ance di ided by ime, and he e we ha e a
p oblem in exac ly de e mining wha ’s he “co ec ” ime! So we would
be easoning in ci cles. Bu e en neglec ing hese di icul ies, Cha lie’s
answe could e eal a ime ha is comple ely di e en om Alice’s and
om Bob’s – i could be yea s ahead o behind bo h o hei s!
The expe ience jus desc ibed will occu again any ime Alice, Bob,
Cha lie, o any wo o hem, mee . The e could be a hund ed obse e s
like Alice, Bob, Cha lie, ini ially a he same place and wi h synch onized
clocks. Whene e wo o mo e o hem mee a e ha ing been sepa a ed,
hey will no ice disc epancies in hei clocks (and in he ageing o hei
40
2. Time and space 2.1 Time and p ope ime
Figu e 2.1 A space ime diag am ha illus a es o he expe iences o Alice (dashed
),
Bob (solid
), Cha lie (do -dashed
) wi h ime. The a ea wi hin he g ey do ed line
ep esen s a wo-dimensional space ime. Space is mo e ho izon al han e ical; ime is
mo e e ical han ho izon al, and lows upwa d. No e ha we can’ p ecisely say, o
ins ance, “ he e ical di ec ion is ime”, because ou p oblem is o see whe he we can
eally sepa a e ime and space a all.
Lowe pa : Alice, Bob, Cha lie s ay close and obse e hei clocks a e pe ec ly synch on-
ized om 12:00 o 12:10. Then hey sepa a e.
Righ : Cha lie isi s a egion nea a s ong mass-ene gy sou ce. Upon mee ing again
wi h Bob, he wo no ice hei clocks di e : 16:00 o Bob, 12:30 o Cha lie. Ye his clock
di e ence s ays he same while hey a el oge he o 10min.
Le : Alice wande s a ound, a elling a high speed wi h espec o he ixed s a s. When
he clock displays 16:00, she wonde s wha ’s he ime “ igh now” o Bob and Cha lie.
Bu his ques ion doesn’ make sense, because (1) when Bob and Cha lie a e oge he
hei clocks di e – impossible o say wha ’s “ he” ime a hei posi ion; (2) i is no
clea which ins an in Bob & Cha lie’s ajec o y should be conside ed as “now” o Alice
(yellow dashed lines).
Uppe pa : When Alice, Bob, Cha lie mee again, hei clocks ha e comple ely di e en
eadings; and hey ha e aged di e en ly. Bu hei clocks un again a he same a e as
long as hey s ay close again.
41
2. Time and space 2.4 Rada dis ance
measu ed, which a e gene ally no equi alen o one ano he . Cosmology,
o example, uses a ple ho a o di e en dis ances
15
. One mus he e o e
be ca e ul abou which de ini ion o dis ance is being used. In he nex
sec ions we shall ocus on wo o hem: ada dis ance and coo dina e dis ance.
2.4 Rada dis ance
The de ini ion o dis ance ha ’s ega ded as he mos “physical” is ada
dis ance, de ined as ollows.
Conside again a si ua ion in which an objec B is mo ing wi h espec
o you. You s and B’s mo ions in space ime, a ound he ime when you
clock displays ime
𝑡
, a e illus a ed in he nex side igu e. You wo ldline
in space ime is he blue one on he le ; he wo ldline o objec B is he g een
one on he igh . A you p ope ime
𝑡0
you send a ligh pulse owa ds
objec B. The pulse a els in emp y space, and upon hi ing objec B i
immedia ely bounces back o you. I eaches you a you p ope ime
𝑡1
.
The wo ldline o he ligh pulse is shown in dashed yellow in he igu e.
A p ope ime
Δ𝑡=𝑡1−𝑡0
has elapsed o you be ween emission and
ecep ion o he ligh pulse, and he ime exac ly in be ween emission and
ecep ion is
𝑡=(𝑡1+𝑡0)/2
. The ada dis ance
𝑑
o objec B om you a
ime 𝑡is de ined as
𝑑:=1
2𝑐Δ𝑡(2.1)
whe e 𝑐is he speed o ligh in acuum, a uni e sal physical cons an :
𝑐=299792458m/s(exac ly). (2.2)
The SI uni o leng h, he me e
16
, is based on he measu ing p ocedu e
abo e. Common lase dis ance me e s also wo k by he same p ocedu e,
and he e o e yield ada dis ance. As he name indica es, his is also he
dis ance measu ed by ada s.
A lase dis ance me e ( he
ligh beam is no isible in
eali y).
In using ada dis ance, howe e , we mus be wa y o he ollowing
peculia i ies:
•
Rada dis ance makes sense only i he ime lapse
Δ𝑡
is small enough
compa ed o a ia ions in he ela i e mo ion o he obse e and he
objec , so ha his mo ion is app oxima ely uni o m. Fo his eason his
dis ance canno be used i he objec is oo a away: he a he away i is,
he longe i akes o a ligh beam o a el o and o. Rada dis ance can
be used be ween he Ea h and o he Sola Sys em plane s; bu i canno
be used o galaxies o o he dis an cosmological objec s.
48
2. Time and space 2.4 Rada dis ance
•
Rada dis ance is no symme ic: he ada dis ance o B om A a
A’s ime
𝑡
is gene ally di e en om he ada dis ance o A om B a B’s
ime 𝑡.
•
The alue o ada dis ance depends on he ela i e mo ion be ween
A and B. Imagine ha a iend o you s is loca ed e y close o you a
ime
𝑡
, bu is mo ing wi h espec o you. Upon measu ing objec B’s
ada dis ance, you iend will gene ally ind a alue di e en om you s.
The disc epancy be ween you and you iend’s measu ed alues will be
he la ge , he highe is he ela i e eloci y be ween you wo. Se e al
obse e s in mo ion wi h espec o one ano he will gene ally disag ee on
he dimensions o an app oxima ely igid objec s in hei icini y.
How a s ee in Tübingen
would look like (excep o
colou and some o he ea-
u es) i we a elled h ough
i a a ound
240000000m/s
( om Rela i i y isualized17)
The dependence on ela i e mo ion also a ec s, a high speeds, how
we see objec s, which appea s mo e and mo e de o med. You can ind
beau i ul isualiza ions, bo h s a ic and anima ed, a Rela i i y isualized
18
.
Luckily, o ela i e speeds ha a e no oo high compa ed o he speed
o ligh , he ada dis ances measu ed by di e en obse e s di e by
amoun s ha a e negligible in e e yday ci cums ances. As an example,
conside a ca mo ing on a oad a
100km/h
, ha is a ound
28m/s
. The
ca ’s d i e measu es he leng h o he ca o be
4m
by ada dis ance.
A pedes ian ha sees he ca passing by ins ead measu es i s leng h o
be
3.99999999999998m
by ada dis ance. This is ob iously a negligible
di e ence.
Leng h and dis ance ha e SI dimension leng h, and we shall usually
measu e hem using he uni me e, symbol ‘m’.
«Exe cise 2.2
Imagine ha you and a iend o you s a e measu ing you dis ance
om a wall, using a lase dis ance me e each. You and he wall a e
s a ic wi h espec o Ea h’s su ace. You iend is mo ing wi h a speed
𝑣
owa ds he wall, and is igh beside you a he exac momen o he
measu emen .
In his speci ic si ua ion, i
𝑑you
is he dis ance measu ed by you, and
𝑑 iend
he dis ance measu ed by you iend, he wo a e ela ed by he
o mula
𝑑 iend =𝑑you ·p1−𝑣2/𝑐2,
whe e
𝑐
is he speed o ligh , gi en in eq.
(2.2)
. No e ha his o mula
49
2. Time and space 2.5 Coo dina e sys ems
is also alid i you iend is mo ing away om he wall, a he han
owa ds i .
1.
Suppose you ind ha he dis ance o he wall om you is
200m
.
You iend’s speed is
300m/s
. How much is he dis ance om he
wall o you iend, who’s igh beside you, as measu ed by you
iend? (You’ll need a high-p ecision calcula o and 18 signi ican
digi s.)
2.
Now ins ead suppose you ind ha he dis ance o he wall om you
is
500m
. You iend measu ed (when igh beside you) a dis ance
o 499m. How as was you iend mo ing?
2.5 Coo dina e sys ems
“Hence o h, space by i sel , and
ime by i sel , a e doomed o
ade away in o me e shadows,
and only a kind o union o he
wo will p ese e an independ-
en eali y.”
Minkowski 1908
F om ou discussion abou ime and space we conclude ha physical
e en s happen in space ime, and he e is no unique way o a ibu e a
uni e sal ime o a uni e sal posi ion in space o a physical e en .
In he p e ious sec ions we used he wo d ‘e en ’, in o mally aking
i s meaning o g an ed. Le ’s be mo e p ecise now. We call e en o
space ime poin a e y small egion o space ha las s o a e y sho
lapse o ime, so ha i can be conside ed as a poin in a ou -dimensional
space. When we say ‘small egion’ o ‘sho ime lapse’, i doesn’ ma e
which de ini ion o dis ance o ime lapse we’ e using.
The wo d ‘e en ’ is used because ypically we iden i y such a space ime
poin by means o a physical phenomenon o limi ed spa ial ex ension
and du a ion. How “limi ed” should hese ex ension and du a ion be? I
depends on he kind o physical phenomenon we’ e in e es ed in. The
sudden bu s o a soap bubble can be conside ed as an e en in compa ison
o geological dis ances and imes; bu i canno be conside ed as an e en
i we’ e s udying suba omic pa icles.
The peculia i ies o space ime can make i di icul o communica e he
posi ionso objec s and e en sby elyingondis ances.In gi ing indica ions
abou he loca ion o a shop we can say “i ’s 200 me es down he oad”
wi hou ambigui y. Bu in si ua ions whe e much highe p ecision is
needed and ex eme mo ions o g a i a ional ields a e in ol ed, we would
need o know he eloci y o he pe son we’ e alking o, because he
dis ances measu ed by us and by ha pe son could be e y di e en .
50
2. Time and space 2.5 Coo dina e sys ems
In he case o ime, we bypassed he p oblem ha ime lapse depends
on he obse e ’s mo ion by in oducing a coo dina e ime
§2.2 p.45
. This
way each e en ge s an a bi a y bu ag eed-upon ime label. We can
bypass he analogous p oblem ha leng h and dis ance depend on he
obse e ’s mo ion, by in oducing a se o a bi a y spa ial coo dina es o
each coo dina e ime.
All hese coo dina es oge he o m a coo dina e sys em, also called
e e ence sys em:
]Coo dina e sys em o e e ence sys em
Acoo dina e sys em, also called e e ence sys em, is he assignmen ,
by ag eemen , o ou nume ical labels o e e y poin in space ime:
one coo dina e ime and h ee spa ial coo dina es. We use symbols such
as
(𝑡, 𝑥, 𝑦, 𝑧)
, o
(𝑡, 𝑟, 𝜃,𝜙)
, o o he s, o hese coo dina es. These ou
coo dina es a e ob iously he same o all obse e s, because hey a e
decided by ag eemen .
A coo dina e sys em is usually de ined on a limi ed egion o space ime.
The loca ion whe e all spa ial coo dina es ha e alue ze o is called he
o igin o he spa ial coo dina es.
O en he coo dina es ha e physical meaning – like he p ope ime
elapsed o a speci ic clock, o he dis ance om some e en as measu ed
by a speci ic obse e – bu hey don’ need o. Typically we use h ee
spa ial coo dina es. In special si ua ions, such as loca ing poin s on he
Ea h’s su ace neglec ing al i ude, only wo o e en one spa ial coo dina e
can be enough.
A coo dina e sys em can be isualized as a g id made by a se o
lines o planes, one se o each coo dina e, which allow us o ead he
coo dina es o any poin . The side igu e shows an example wi h spa ial
coo dina es
(𝑥, 𝑦)
in wo dimensions, o a speci ic coo dina e ime. I is o
cou se assumed ha he g id can be e ined as much as needed. We a e
all amilia wi h he coo dina e sys em
(𝜆,𝜙)
o la i ude and longi ude o
iden i y loca ions on Ea h’s su ace, and used in e nally by he loca ion
sys ems o mobile phones.
Since a coo dina e sys em is a bi a y, we o en choose one adap ed
o he physical phenomenon unde s udy. I ’s e y common o choose
a coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
ha has ime coo dina e
𝑡=0s
a he
51
2. Time and space 2.5 Coo dina e sys ems
beginning o ou obse a ion o he phenomenon, and spa ial coo dina es
(𝑥, 𝑦, 𝑧)=(0,0,0)m
a a loca ion close o whe e he phenomenon happens.
The spa ial coo dina es may be chosen so as o ha e pa icula physical
p ope ies,whichin u n may lead osimple exp essions o some physical
laws as we shall see la e
§5.9 p.146
. Fo ins ance, he coo dina e lines o
migh be s aigh lines (geodesics), in which case we speak o ec ilinea
coo dina es. I hey a e no , hen we call hem cu ilinea coo dina es; he
coo dina es in he p e ious side igu e a e cu ilinea .
I he spa ial coo dina e lines a e o hogonal o one ano he a e e y
poin , ha is, hey in e sec a
𝜋/2 ad
, hen we call hem o hogonal
coo dina es. This is a e y use ul p ope y, which simpli ies many physics
o mulae. In hese no es we shall always use o hogonal coo dina es. No e
ha a coo dina e sys em can be cu ilinea and o hogonal.
In as onomy, in space and sa elli e communica ion, and in geodesy
19
( he science o accu a ely measu ing and unde s anding he Ea h’s geo-
me ic shape, o ien a ion in space, and g a i y ield), se e al impo an
coo dina e sys ems20 a e used. Fo example:
Map o some dis an as o-
nomicalobjec sused ode ine
he In e na ional Celes ial
Re e ence F ame ( om The
ICRF21).
•
The In e na ional Celes ial Re e ence Sys em (ICRS)
22
is a coo dina e
sys em wi h o hogonal spa ial coo dina es, which a e almos ec i-
linea . Thei o igin is close o he cen e o he Sun. Se e al dis an
cosmological objec s, like quasa s
23
, ha e ixed spa ial coo dina es
in his e e ence sys em. I s coo dina e ime is called Ba ycen ic
Coo dina e Time (TCB).
•
The Geocen ic Celes ial Re e ence Sys em (CGRS) is a coo dina e sys em
wi h o hogonal spa ial coo dina es, which a e almos ec ilinea .
Thei o igin is close o he cen e o he Ea h. Dis an cosmological
objec s ha e almos cons an spa ial coo dina es in his e e ence
sys em; his means ha he Ea h o a es wi h espec o i . I s
coo dina e ime is called Geocen ic Coo dina e Time (TGB), which
is e y simila o he TAI.
•
The In e na ional Te es ial Re e ence Sys em (ITRS)
24
is a coo dina e
sys em simila o he CGRS, bu wi h he impo an di e ence ha
he Ea h is app oxima ely s a ic in his e e ence sys em; his means
ha dis an cosmological objec s o a e wi h espec o i . I has he
same coo dina e ime as he ICRS.
52
2. Time and space 2.6 Spa ial coo dina e dis ance and leng h
•
The Wo ld Geode ic Sys em 1984 (WGS 84)
25
is e y simila o he ITRS,
besides a disc epancy o some cen ime es. I is used by he sys em
o GPS sa elli es.
Bu how do we de e mine he posi ion and ime o an e en in hese
coo dina e sys ems? The p ocedu e can be ex emely complica ed in ac .
The s a ing poin is he assignmen o some p ede ined coo dina es o
objec s ha seem o ha e ixed spa ial coo dina es in he coo dina e sys em;
o ins ance dis an cosmological objec s like quasa s
26
in he case o he
ICRS and CGRS, o pa icula e e ence s a ions
27
on Ea h’s su ace in
he case o he ITRS and WGS 84. The coo dina e o o he e en s a e hen
calcula ed om measu emen s o p ope imes, ada dis ances, and angles
om he e e ence objec s, using o mulae om gene al ela i i y. As we
men ioned in §2.1, e en you mobile phone pa icipa es in hese complex
calcula ions.
The assignmen o coo din-
a es on and a ound Ea h
depends on nominal alues
assigned o dis an as o-
nomical objec s ( om Capi-
aine 2010)
«Exe cise 2.3
On Ea h’s su ace we o en use he sys em o wo coo dina es called
la i ude
𝜆
and longi ude
𝜙
, measu ed in deg ees. Coo dina e lines o
cons an la i ude a e called pa allels; hose o cons an longi ude a e
called me idians.
1. Find wha ’s a hese h ee coo dina e pai s:
(𝜆,𝜙)=(60.369002◦,5.350336◦)
(𝜆,𝜙)=(35.658587◦,139.745424◦)
(𝜆,𝜙)=(−13.163069◦,−72.545265◦)
2.
A e la i ude and longi ude o hogonal coo dina es? Explain why o
why no .
2.6 Spa ial coo dina e dis ance and leng h
I we ha e chosen a coo dina e sys em, we can de ine a no ion o dis ance
called coo dina e dis ance be ween wo poin s a any coo dina e ime
𝑡
. The
idea is simple: we measu e he leng h o he sho es pa h in space ime
joining he wo poin s, and all in e media e poin s on he pa h mus ha e
53
2. Time and space 2.6 Spa ial coo dina e dis ance and leng h
he same coo dina e ime
𝑡
. O cou se i can happen ha he e is mo e
han one pa h ha ing sho es leng h.
Coo dina e dis ance is di e en om ada dis ance. I has wo ad-
an ages: i doesn’ depend on ela i e mo ion, and can be de ined also
be ween objec s ha a e e y a apa . In egions o space ime wi h
low cu a u e and slow ela i e mo ions, howe e , ada dis ance and
coo dina e dis ance based on s anda d coo dina e imes a e app oxima ely
he same, and app oxima ely independen o any mo ions.
No e also ha he speed o ligh de ined wi h espec o coo dina e
dis ance need no ha e he alue
𝑐
, o e en be cons an ! You migh ha e
hea d o ead abou dis an cosmological objec s, like quasa s, ha a e said
o be eceding om us a speeds as e han ligh ’s. How is ha possible?
The eason is ha he ‘speeds’ hey’ e alking abou a e de ined wi h
espec o coo dina e dis ance, no ada dis ance.
]Ca esian coo dina es
We call Ca esian coo dina es a se o spa ial coo dina es
(𝑥, 𝑦, 𝑧)
wi h a
e y special p ope y: he coo dina e dis ance be ween wo loca ions A
and B ha ing spa ial coo dina es
(𝑥A, 𝑦A, 𝑧A)
and
(𝑥B, 𝑦B, 𝑧B)
is simply
gi en by
𝑑AB =q(𝑥B−𝑥A)2+ (𝑦B−𝑦A)2+ (𝑧B−𝑧A)2.(2.3)
Pe ec Ca esian coo dina es do no exis , because space ime is cu ed.
Bu i is possible o choose coo dina es ha a e app oxima ely Ca esian in
limi ed egions o space ime. The coo dina e sys ems ICRS, GCRS, ITRS
discussed in §2.5 a e no Ca esian: hey include he e ec s o cu a u e
gene a ed by he Ea h and o he bodies in he Sola Sys em.
In mos o hese no es we will no ha e o wo y abou he disc epancies
be ween ada dis ance and coo dina e dis ance, and abou he mo ion o
he obse e o ins umen ha is measu ing a dis ance. So we shall use he
e m ‘dis ance’ wi hou speci ica ions. And we shall o en use Ca esian
coo dina es.
«Exe cise 2.4
In he Geocen ic Celes ial Re e ence Sys em, de ined by he In e na-
ionalAs onomicalUnion, hedis ancebe ween wo e ycloseloca ions
54
2. Time and space 2.7 Coo dina e posi ion
A and B ha ing spa ial coo dina es
(𝑥A, 𝑦A, 𝑧A)
and
(𝑥B, 𝑦B, 𝑧B)
close o
Ea h’s su ace is app oxima ely gi en by
𝑑GCRS,AB =1+𝑔𝑅/𝑐2q(𝑥B−𝑥A)2+ (𝑦B−𝑦A)2+ (𝑧B−𝑧A)2(2.4)
whe e
𝑔≈9.8m/s2
is he g a i a ional accela a ion,
𝑅≈6371×103m
is
Ea h’s adius, and
𝑐
is he speed o ligh . This is also he ada dis ance
o B om A, i A is no mo ing wi h espec o he coo dina e sys em.
1.
Ve i y ha he o mula abo e is dimensionally co ec : bo h le and
igh side should ha e dimension leng h.
2.
Assume ha you coo dina es a e
(𝑥A, 𝑦A, 𝑧A)
, and conside an
objec B a a
𝑥
-coo dina e di e ence o
100m
om you, ha is, wi h
coo dina es
𝑥B=𝑥A+100m , 𝑦B=𝑦A, 𝑧B=𝑧A.
How much is he di e ence be ween he dis ances
𝑑AB
and
𝑑GCRS,AB
,
calcula ed be ween you and he objec ?
(The inaccu acy in he speci ica ion o
𝑔
and
𝑅
is ac ually much la ge han
he di e ence you jus ound.)
2.7 Coo dina e posi ion
Le us ag ee on some no a ion and e minology ha will be used in hese
no es.
We shall o en deno e he ou coo dina es o a coo dina e sys em by
he le e s
(𝑡, 𝑥, 𝑦, 𝑧).
Unless s a ed o he wise, he coo dina e ime
𝑡
will be aken o be UTC
§2.2 p.45
, and he spa ial coo dina es
(𝑥, 𝑦, 𝑟)
will be aken o be Ca esian
§2.6 p.54
. As men ioned in he p e ious sec ion, he de ini ion o he spa ial
coo dina es is usually di e en om p oblem o p oblem; so i is always
impo an o speci y how he coo dina e sys em you’ e using is de ined.
55
2. Time and space 2.8 Coo dina e eloci y and accele a ion
]Posi ion o loca ion ec o
The iple o spa ial coo dina es is called he posi ion o loca ion ec o
and is o en deno ed by 𝒓:
𝒓:=(𝑥, 𝑦, 𝑧)o 𝒓:=[𝑥, 𝑦, 𝑧]o 𝒓:=
𝑥
𝑦
𝑧
use ound b acke s ‘
()
’ o squa e b acke s ‘
[]
’, and ho izon al o e ical
no a ion as you p e e .
Whene e we speak o a “ egion o space” o o a “su ace in space”,
we mean a 3D o 2D egion a some speci ic coo dina e ime 𝑡.
Some physical phenomena happen app oxima ely along a line, in one
dimension; hink o ins ance o a small alling objec . O he phenomena
happen app oxima ely on a su ace, in wo dimensions; hink o ins ance
o a swinging pendulum. In hese cases we can omi wo o one o he
spa ial coo dina es, and simply assume ha he omi ed ones ha e some
cons an , unimpo an alues. In such cases we can simply w i e, o
ins ance, (𝑡, 𝑥)o (𝑡, 𝑥, 𝑦)as ou coo dina es.
2.8 Coo dina e eloci y and accele a ion
In some si ua ions he spa ial coo dina es
𝒓=(𝑥, 𝑦, 𝑧)
may be unc ions
o he ime coo dina e
𝑡
. A ypical example is when we desc ibe how he
spa ial posi ion o a small olume o body changes wi h coo dina e ime.
We can w i e his unc ional dependence in di e en ways, o ins ance
𝒓(𝑡)o 𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡).
So
𝒓
is a ec o unc ion o ime, which simply means ha we ha e a
collec ion o h ee unc ions o ime.
I we ake he de i a i e o each coo dina e wi h espec o he ime
𝑡
,
we ob ain he coo dina e eloci y:
56
2. Time and space 2.8 Coo dina e eloci y and accele a ion
]Coo dina e eloci y
The coo dina e eloci y is a ec o de ined as he de i a i e o he
posi ion ec o 𝒓(𝑡)wi h espec o coo dina e ime 𝑡:
𝒗(𝑡):=d
d𝑡𝒓(𝑡)o
𝑣𝑥(𝑡)
𝑣𝑦(𝑡)
𝑣𝑧(𝑡)
:=
d
d𝑡𝑥(𝑡)
d
d𝑡𝑦(𝑡)
d
d𝑡𝑧(𝑡)
.
The wo d speed means he magni ude o he eloci y:
|𝒗| ≡ q𝑣𝑥2+𝑣𝑦2+𝑣𝑧2.
The de i a i e o some quan i y wi h espec o coo dina e ime is o en
deno ed by a do o e he quan i y. So we can also w i e
𝒗(𝑡)=¤𝒓(𝑡)=¤
𝑥(𝑡),¤
𝑦(𝑡),¤
𝑧(𝑡)
The coo dina e eloci y is usually di e en om he physical eloci y, which
anobse e would measu e using p ope imeand space, o ins anceusing
bouncing ligh ays. In many e e yday si ua ions he di e ence be ween
coo dina e and physical eloci y is so small ha i can be neglec ed, so we
shall simply use he wo d ‘ eloci y’. Bu in si ua ions in ol ing suba omic
pa icles a high speed, o example, one mus ake in o accoun ha he
wo eloci ies a e di e en .
Taking hede i a i eo he eloci ywi h espec ocoo dina e ime, ha
is, he second de i a i e o posi ion, we ob ain he coo dina e accele a ion:
]Coo dina e accele a ion
The coo dina e accele a ion is a ec o de ined as he de i a i e o he
eloci y ec o 𝒗(𝑡)wi h espec o coo dina e ime 𝑡:
𝒂(𝑡):=d
d𝑡𝒗(𝑡)=d2
d𝑡2𝒓(𝑡)=d2
d𝑡2𝑥(𝑡),d2
d𝑡2𝑦(𝑡),d2
d𝑡2𝑧(𝑡).
57
3. Main physical quan i ies 3.3 Ma e
heo y. Ye , hese se en quan i ies a e uni e sal in ou p esen way o
doing physics and o desc ibing and unde s anding physical phenomena
all a ound and wi hin us.
Le us make a i s acquain ance wi h hese se en quan i ies. The
discussion ha ollows is mean as an in oduc ion. We shall epea and
say mo e abou each quan i y in la e chap e s. Bu emembe
§1.4 p.29
ha
i is e y di icul , i no impossible, o answe ques ions like “wha is eally
he quan i y. . . ?”.
3.3 Ma e
One mole o di e en sub-
s ances (image: NIST1).
]Ma e : uni s and no a ion
Ma e , which includes amoun o subs ance
2
in chemis y, is a scala
quan i y. The uni o he amoun o ma e is he mole
3mol
; he uni o
ma e lux and supply is mole pe second
mol/s
. In s a is ical mechanics
and pa icle physics, ma e is o en simply coun ed and hus measu ed
in dimensionless uni s, a he han in moles.
The amoun o ma e in a olume is usually deno ed
𝑁
. The lux o
ma e h ough a su ace is deno ed
𝐽
; and supply o ma e in a olume,
𝒜
. In chemis y we usually speci y wha kind o ma e we a e speaking
abou , w i ing o ins ance
𝑁Ca =5.3mol
, o indica e an amoun o
5.3mol o calcium4a oms.
Ma e is p obably he easies quan i y o g asp in ui i ely: i is wha
we o dina ily call “s u ”. I is usually classi ied in o se e al kinds. The
classi ica ion depends on he physical phenomena and heo y one wo ks
wi h. A building enginee , o ins ance, could classi y “ma e ” in o di -
e en kinds o ma e ials – such as wood, conc e e, s eel, sand, plas ic, and
so on – keeping ack o he amoun o each ma e ial in di e en egions
o space, i s mo emen , i s a e o p oduc ion and ans o ma ion. Each
ma e ial has di e en physical p ope ies.
A chemis could classi y ma e in o di e en subs ances – such as wa e ,
hyd ogen, oxygen, ca bon dioxide, and so on – again keeping ack o
hei amoun s, mo emen s, p oduc ion. Acco ding o his classi ica ion,
he “ma e ials” o he building enginee would be mix u es o he di -
e en subs ances. Bu no e ha he e is no clea bounda y be ween one
classi ica ion and he o he .
64
3. Main physical quan i ies 3.3 Ma e
A chemis could also classi y ma e in o di e en kinds o a oms – such
as hyd ogen
5
,helium
6
,li hium
7
, and he o he kinds ha appea in he
pe iodic able8– and seeing subs ances and ma e ials as combina ions o
hese di e en a omic kinds o ma e . This classi ica ion is special because
hese di e en kinds ha e, a leas app oxima ely, he p ope y o being
conse ed
§5.3 p.122
: hei amoun s in a con aine o in a egion o space
can only change i hese kinds a e en e ing h ough an opening in he
con aine o h ough he bounda y o he egion o space. In o he wo ds,
hey canno be c ea ed o des oyed. This conse a ion p ope y is only
app oxima e, howe e . Radioac i e a oms
9
can ansmu e om one kind
o ano he . This possibili y is c ucial and mus be aken in o accoun in
phenomena in ol ing adioac i i y10 and nuclea ene gy-mass11.
A chemis o a pa icle physicis may classi y ma e in o ewe di -
e en kinds: p o ons, neu ons, elec ons, an i-p o ons, an i-neu ons,
an i-elec ons (also called posi ons), seeing he di e en a omic kinds as
being made o hese six basic ones. These kinds may be conse ed e en
when kinds o a oms a e no .
Bu a nuclea o pa icle physicis knows ha he conse a ion p op-
e ies o he six kinds abo e is also only app oxima e, and he e a e also
o he kinds, p oduced only in special ci cums ances.
We he e o e go down in o mo e and mo e sub le classi ica ions. This
kind o esea ch is s ill open, bu i seems ha he o al amoun o ba yonic
12
,
(includingp o onsandneu ons)andlep onic
13
(includingelec ons)ma e
is always conse ed.
Acco ding o he de ini ion o ma e ha we’ e adop ing, he o al
amoun o some kind o ma e in a egion can in p inciple be nega i e. A
nega i e amoun simply deno es he p esence o an i-ma e
14
. An i-ma e
appea s in small amoun s in e e yday li e, o example in connec ion wi h
common adioac i i y p ocesses. I is also c ea ed and used in medicine,
in posi on-emission omog aphy (PET)
15
scans. In o dina y chemical
applica ions, howe e , all amoun s o ma e wi hin a egion a e usually
posi i e o ze o.
In posi on-emission omo-
g aphy he e is c ea ion o
amoun s o ma e (lep onic
ma e ) ha can be con-
side ed nega i e: hei lep on
numbe is nega i e (image:
Helse Be gen16).
Why do we need o wo y abou how ma e ge s classi ied depending
on he applica ion? Because o desc ibing aphysical sys emandp edic ing
i s beha iou we usually ha e o use a leas one physical law o each kind
o ma e . So he mo e kinds o ma e we ha e o keep ack o , he mo e
equa ions we will ha e.
65
3. Main physical quan i ies 3.3 Ma e
-Ambigui y o he e m ‘ma e ’
In hese no es we use he e m ‘ma e ’ in he gene ic sense discussed
abo e. Bu be awa e ha in some disciplines his e m may ha e a much
mo e specialized and sligh ly di e en meaning. I may no e en be
used a all. In chemical applica ions, o ins ance, one ypically speaks
speci ically o ‘compounds’, ‘mix u es’, ‘subs ances’, ‘elemen s’, a he
han ‘ma e ’. A pa icle physicis s speaks o ma e and an i-ma e , bu
in he p esen no es he e m ‘ma e ’ e e s o bo h.
-Ma e is di e en om mass-ene gy
I is impo an o clea ly dis inguish ma e om mass-ene gy. Mass-
ene gy can be conside ed a p ope y o ma e , bu he wo a e di e en .
In nuclea eac ions, o ins ance, he mass-ene gy o some amoun o
ma e may change, while he amoun o ma e s ays he same.
As a as we know, he o al amoun o mass-ene gy associa ed wi h an
amoun o ma e is always posi i e, whe he he amoun o ma e is
posi i e o nega i e (an ima e ). This is he eason why an ima e “ alls
downwa d” jus like posi i e ma e , a ac ha has been expe imen ally
con i med: see Ande son, Bake , e al. 2023.
How many posi ons do ba-
nanas p oduce?
«Exe cise 3.1
Acco ding o s a emen s on symme ymagazine.o g
17
and quan umdi-
a ies.o g18,
The a e age banana ( ich in po assium) p oduces a posi on
oughly once e e y 75 minu es.
Un o una ely he o iginal si e whe e he his s a emen was discussed,
and he co esponding calcula ion made, seems no o exis anymo e.
1. Do a li le esea ch and ind ou whe he his s a emen is ue.
2.
F om you esea ch, app oxima ely quan i y he lux o posi ons
a ound an o dina y banana, exp essing i in pa icles/s.
66
3. Main physical quan i ies 3.4 Elec ic cha ge
3.4 Elec ic cha ge
]Elec ic cha ge: uni s and no a ion
Elec ic cha ge is a scala quan i y. The uni o he amoun o elec ic-
cha ge is he coulomb
19 C
. The lux o elec ic cha ge is called elec ic
cu en , i s uni is he ampe e20 A=C/s.
{To be comple ed in a la e e sion
3.5 Magne ic lux
“magne ic- lux lines eme ging
om he su ace o a Type II su-
pe conduc o ”
Essmann & T äuble 1971
]Magne ic lux: uni s and no a ion
Magne ic luxisascala quan i y. The uni o magne ic lux is he webe
21
Wb
. The “ lux” o magne ic lux is called elec ic po en ial di e ence o
elec ic ension; i s uni is he ol 22 V=Wb/s.
The magne ic lux is usually calcula ed by means o a ec o quan i y
called magne ic lux densi y; i s uni is he esla23 T=Wb/m2.
The elec ic po en ial di e ence is usually calcula ed by means o a
ec o quan i y called elec ic ield s eng h; i s uni is he ol pe me e
(V/m).
As we shall see in mo e de ail in chap e 9, magne ic lux di e s om
he o he six main quan i ies in ha i answe s he wo “how much?”
ques ions in one lowe dimension: “How much magne ic lux is in his
su ace?” and “How much magne ic lux c osses his line in he uni o
ime?”. I also equi es a sligh ly di e en no ion o o ien a ion o a su ace.
The “ lux o magne ic lux”, o elec ic po en ial di e ence, is he e o e a low
connec ed o a line.
“ske ch o he magne ic lines o
o ce in a magne ic ilamen ex-
ending up h ough he pho o-
sphe e.” Pa ke 1974a
The magne ic lux densi y and he elec ic ield s eng h oge he
a e usually called “elec omagne ic ield”, which is he e o e commonly
ep esen ed by wo ec o s associa ed o each poin in space. Bu i can
also be in e p e ed and isualized as a collec ion o spinning magne ic
ubes o lines, ei he closed o ex ending inde ini ely, which mo e a ound.
This isualiza ion is somewha analogous o how we isualize ma e and
cha ge, as mo ing blobs o poin s, bu wi h one mo e dimension. This
in e p e a ion goes back o Fa aday (1846), Maxwell (1855), and la e Di ac
(1955) among o he s, and oday is con enien ly used in some ields such
67
3. Main physical quan i ies 3.6 Ene gy-mass
as sola physics
24
, o example o s udy sunspo s
25
(see Ryu o a 2018). In
pa icula si ua ions, o example in some supe conduc o s subjec ed o
ex e nal magne ic ields, he magne ic- lux lines can li e ally be seen and
e en acked as hey mo e a ound (see p e ious side igu e).
{To be comple ed in a la e e sion
3.6 Ene gy-mass
Thechemical ene gy-masscon-
en in an o dina y AA ba -
e y is a ound
10000J
. The
o al ene gy-mass con en , in-
cluding es ene gy-mass, is
a ound 1015 J.
]Ene gy-mass: uni s and no a ion
Ene gy is a scala quan i y. The uni o he amoun o ene gy is he joule
J
; he uni o ene gy lux and supply is joule pe second
J/s
, also called
wa W=J/s.
Equi alen ly we can speak o mass. The uni o he amoun o mass
is he kilog am
26 kg
; he uni o mass lux and supply is kilog am pe
second kg/s.
The amoun o ene gy in a olume is usually deno ed
𝐸
, o
𝑚
i we
desc ibe i as mass. The lux o o al ene gy h ough a su ace is deno ed
𝛷; and supply in a olume, ℛ.
The no ion o ene gy is ex emely impo an oday, and cen al in many
wo ld-wide discussions and wo ies – hink o oday’s “ene gy c isis”, he
need o “ enewable ene gy”, and so on. I is somewha unny ha despi e
i s impo ance i ’s ac ually di icul o answe ‘wha is ene gy, eally?’.
O en we speak abou ene gy as some hing ha “ lows”, is “ anspo ed”,
“con e ed”, “s o ed”, and simila isualiza ions. This in ui ion will be
enough in hese no es. The no ion o mass is also e y in ui i e in ou
e e yday li e; we associa e i wi h he “ esis ance” we eel when se ing
objec s in o mo ion, o wi h he weigh o objec s.
F om Rela i i y Theo y – and expe imen ally – we know ha ene gy
and mass a e he same quan i y, and in hese no es we shall emphasize his
expe imen al ac .
Ene gy is mass, mass is ene gy
Le ’s see some examples o why i is impossible o make a dis inc ion
be ween ene gy and mass. The ollowing examples ha e been simpli ied
in some o hei aspec s, bu hei main poin is alid.
68
3. Main physical quan i ies 3.6 Ene gy-mass
Hea ed gas. Imagine we ha e a box wi h a gi en amoun o gas, say
1mol
o oxygen molecules. Using an ex emely p ecise weighing scale, we
obse e ha he mass o he gas is, say, exac ly
0.031999540000000000kg .
Now we hea he gas, p o iding
60J
o ene gy, while making su e ha
no a single molecule o oxygen ge s in o ou o he box. The empe a u e
o he gas inc eases by a ound
3K
. We ac ually obse e ha he weigh
measu ed by he scale inc eases while we hea he gas, eaching he new
alue
0.031999540000000668kg .
Clea ly he mass has inc eased, bu no molecules we e added! The addi-
ional mass is he
60J
o ene gy ha we p o ided o he gas by hea ing.
Ene gy has weigh , ene gy is mass.
When we s e ch a ub-
be band, i s mass inc eases
sligh ly – e en i he amoun
o ubbe emains exac ly he
same.
S e ched o mo ing ubbe band. Take a common ubbe band, and
imagine again ha we ha e an ex emely p ecise weighing scale. The
ubbe band, uns e ched, has a mass o exac ly
0.000500000000000000000kg .
Now we s e ch he band a li le. By doing so we gi e ene gy o he band,
which is said o acqui e ‘elas ic ene gy’. Le ’s say we ha e gi en
0.3J
o he
band in his way. Now we weigh he ubbe band again, while s e ched.
We obse e a mass o app oxima ely
0.000500000000000003338kg .
The ex emely small di e ence o a ound
3×10−18 kg
om he ini ial mass
is exac ly he elas ic ene gy ha we p o ided by s e ching. Ene gy has
weigh ; ene gy is mass.
Now se he uns e ched band in mo ion. Owing o he mo ion, he
band is said o ha e acqui ed ‘kine ic ene gy’; le ’s say an amoun
0.3J
.
I we could weigh he band while in mo ion (bu wi hou mo ing he
weighing scale), we would obse e again a mass o app oxima ely
0.000500000000000003338kg .
The small di e ence om he ini ial mass is he addi ional kine ic ene gy
o he band. Ene gy has weigh ; ene gy is mass.
69
3. Main physical quan i ies 3.6 Ene gy-mass
Hyd ogen Bomb Tes , 1954
(Na ionalMuseumo Nuclea
Science & His o y27)
Fission and a omic bombs. The a omic bomb
28
is a da k example o
he ac ha mass is ene gy. In phenomena o nuclea ission, we no ice
a dec ease in he weigh , measu ed a es , o nuclea ma e ial be o e
and a e he phenomenon o ission. Bu we also obse e ha a g ea
amoun o (kine ic) ene gy is eleased. This amoun is exac ly equal o he
appa en ly missing weigh .
Elec ic hea e . As a inal example conside a
1000W
elec ic hea e ,
which is adia ing
1000J
in one second. The hea e is also losing a ound
0.00000000000001kg
o mass e e y second owing o his hea adia ion –
al hough i ’s also acqui ing he same amoun o mass as elec omagne ic
ene gy.
The p ac ical use o he wo ds ‘mass’ and ‘ene gy’
F om he examples abo e i becomes clea ha ene gy and mass a e wo
names o he same hing. The equi alence be ween ene gy and mass
is gi en by he amous o mula
𝐸=𝑚𝑐2
, whe e
𝑐
is he speed o ligh ,
eq. (2.2). In hei espec i e uni s his gi es
1kg =89875517873681764J (exac ly)
1J ≈0.0000000000000000111265kg
To g asp hese numbe s, conside ha he mass o he ubbe band in he
“we a e led o he mo e gene al
conclusion: The mass o a body
is a measu e o i s ene gy con-
en ; i he ene gy changes by
𝐿
,
he mass changes in he same
sense by
𝐿/9·1020
, i he ene gy
is measu ed in e gs and he mass
in g ams. ” Eins ein 1905b
example abo e,
0.5g
, is compa able o he ene gy eleased by he a omic
bomb o e Hi oshima29.
Bu i also becomes clea ha in ou daily expe ience we deal wi h
ene gy-mass in wo di e en ways:
On he one hand, we deal wi h huge (a om-bomb-like) amoun s o
ene gy-mass packed in e y small olumes: he huge amoun s o ene gy-
mass ha go oge he wi h objec s and s u like pens, keys, bicycles, ca s,
houses, wa e , and so on. We mo e, push, pull hese huge ene gy-mass
amoun s om one place o ano he , and e en pu hem in ou pocke s.
We ou sel es a e huge bundles o ene gy-mass mo ing a ound. These
amoun s o ene gy-mass change a li le, all he ime, as in he examples
wi h he ubbe band abo e. Bu hese changes a e so small as o be o en
unde ec able wi h o dina y weigh scales, and negligible o p ac ical
pu poses. We use he wo d ‘mass’ o any such huge amoun o ene gy-
mass,andmeasu ei wi hauni –
kg
– ha doesn’ lead o idiculouslyla ge
70
3. Main physical quan i ies 3.6 Ene gy-mass
numbe s. And we also ag ee o neglec he imp ecision and luc ua ion in
i s measu emen , say any imp ecision unde
0.00001%30
. So we say “ he
ubbe band has a mass o
0.0005kg
”, a he han “ he ubbe band has
an ene gy-mass o 45000000000000J”.
On he o he hand, we also deal wi h he small ene gy changes and
exchanges in all hese objec s. These ene gy exchanges ha a e e y
impo an o ou daily li e: hey keep us wa m, keep ou cells ac i e, make
ou lap ops wo k. In dealing wi h hese ene gy exchanges, we don’ ca e
abou he huge ene gy ese oi s hey come om. So we ag ee o measu e
hem wi h a uni –
J
– ha doesn’ lead o idiculously small numbe s.
And we also ag ee no o be p ecise abou he o al amoun in he ese oi
om which hese ene gy bi s come om.
As an analogy, hink o when we speak abou he amoun o people in
di e en coun ies. We can say ha in No way he e a e
5
millions, and in
India
1500
millions, so in India he e a e
300
imes mo e people. By his we
don’ mean ha in No way he e a e exac ly
5000000
people and ha India
has exac ly
300
imes mo e people. These numbe s a e changing sligh ly all
he ime, bu we don’ ca e abou di e ences o 10 o e en
10000
people.
A he same ime, i we ha e h ee dea iends o ela i es isi ing us om
ab oad, hen he amoun o
3
people is now o us e y impo an – e en i
i is a e y small amoun compa ed o he o al popula ion o a coun y.
The dis inc ion abo e is o cou se no clea -cu . In dealing wi h some
physical phenomena, o example wi h ew molecules o wi h suba omic
pa icles, he ic i ious bu p agma ic dis inc ion be ween mass and ene gy
becomes oo blu y and no use ul anymo e. In discussing hese phenom-
ena, indeed, one o en uses he e ms ‘mass’ and ‘ene gy’ in e changeably,
as well as a common uni o bo h, such as he elec on ol 31.
In hese no es we shall o en use he exp essions ‘ene gy-mass’ and
‘mass-ene gy’ o emind ou sel es ha hese wo wo ds deno e he same
physical hing.
Diffe en ‘ o ms’ o ene gy-mass
Weo en speak o di e en o ms o ene gy-mass. Themos impo an o ms
o us will be in e nal ene gy-mass,kine ic ene gy-mass,g a i a ional
po en ial ene gy-mass,elec omagne ic ene gy-mass o be discussed
la e .
71
3. Main physical quan i ies 3.6 Ene gy-mass
The di e ences among hese o ms o ene gy-mass a ise om he
way hey a e calcula ed om o he quan i ies, as we shall see la e . Fo
example, i in a olume he e’s an amoun o a pa icula kind o ma e ,
hen in ha olume he e mus also be an amoun o ene gy-mass, gi en
by a o mula ha in ol es he amoun o ma e . And i ha ma e is
mo ing, hen we ha e o add an ex a amoun o ene gy-mass gi en by
ano he o mula which in ol es he eloci y. And i in ha olume he e’s
a g a i a ional ield ( ha is, a pa icula kind o space ime cu a u e),
hen ano he ex a amoun o ene gy-mass mus be added, gi en by ye
ano he o mula in ol ing he g a i a ional ield. Simila ly i we know
ha an elec omagne ic ield is in ha olume.
We also speak o di e en o ms o lux o ene gy-mass. The mos
impo an o us will be hea and mechanical powe . The di e ence is
again in how hese luxes a e calcula ed depending on whe he he e a e
also luxes o ma e and o o he quan i ies.
Hyd oca bon uel pa icles
32
.
The small blobs ha e size o
a ound 2×10−8m.
The dis inc ions be ween di e en o ms o ene gy-mass also depend
on he obse a ion scale and he heo y used. Take, o ins ance, he wa e
in a glass es ing on able. We can obse e and desc ibe ha wa e on a
mac oscopic scale o cen ime es, seeing i as a s ill, uni o m luid. On
his mac oscopic scale we say ha he wa e has in e nal ene gy-mass, o
ha he e is in e nal ene gy-mass in he glass. Bu we can also obse e
and model ha same wa e as a collec ion o molecules, on a mic oscopic
scale o nanome es (
10−9m
). On his mic oscopic scale, we speak o
in e nal ene gy-mass and kine ic ene gy-mass, because he molecules
a e in cons an mo ion. The o al amoun o ene gy-mass is he same on
he cen ime e-scale and on he mic oscopic scale, bu i s pa i ion in o
di e en “ o ms” depends on he scale: only in e nal on he mac oscopic
scale, and in e nal plus kine ic on he mic oscopic scale.
The same is ue o lux o ene gy-mass. Wha we call ‘hea ’ on one
obse a ion scale appea s as a lux o ene gy-mass no associa ed wi h
he mo ion o ma e . Bu on a ine scale i is ins ead called ‘wo k’, and i
appea s as an ene gy-mass lux associa ed wi h he mic oscopic mo ion o
ma e .
«Exe cise 3.2
In an hou , 14 people exi h ough a doo . Taking he a e age human
weigh o be
62kg
(Walpole, P ie o-Me ino, e al. 2012), wha ’s he
a e age lux o ene gy-mass , in J/s, h ough ha doo ?
72
3. Main physical quan i ies 3.7 Momen um
3.7 Momen um
A walking pe son has, wi h
espec o he g ound, an
amoun o ho izon al mo-
men um o a ound
70Ns
(im-
age: An onio Romei33).
A
10cm
beam o ligh om a
60W
o ch has an amoun o
momen um a ound
10−16 Ns
.
]Momen um: uni s and no a ion
Momen um, also called linea momen um o ansla ional momen um o
dis inguish i om angula momen um, is a ec o quan i y. The amoun
o momen um can be exp essed in se e al equi alen uni s; we shall
keep in mind especially hese h ee:
new on second
N·s≡kilog am me e pe second
kg ·m/s≡joule second pe me e
J·s/m
Flux o momen um is also called con ac o ce o su ace o ce. Supply o
momen um is also called body o ce o olume o ce. They can be exp essed
in se e al equi alen uni s:
new on
N≡kilog am me e pe squa ed second
kg ·m/s2≡joule pe me e
J/m
Since momen um and momen um lux a e ec o quan i ies, hey a e
usually exp essed wi h h ee numbe s, ypically hei
𝑥
-,
𝑦
-, and
𝑧
-
componen s.
The amoun o momen um in a olume is usually deno ed
𝑷
. The lux
o momen um (su ace o ce) is deno ed
𝑭
; and supply o momen um
( olume o ce), 𝑮.
Momen um is a sub le quan i y, e en sub le han ene gy-mass. Tex -
books ha ocus on New onian mechanics de ine i as he p oduc o he
mass and he eloci y o a body, usually w i en “
𝒑=𝑚𝒗
”. This ela ion,
howe e , is only alid in special ci cums ances, and canno be used in many
e e yday echnological applica ions, especially when elec omagne ism o
high speeds a e in ol ed. And ha ela ion is ac ually only an app oxim-
a ion e en in he ci cums ances whe e i ’s used. Fo example, in a small
egion whe e he e a e ma e and elec omagne ic ields, momen um
is app oxima ely gi en by
𝑷=𝑚𝒗+𝜖0𝑬×𝑩
, whe e he quan i ies
𝑬
(elec ic ield s eng h) and
𝑩
(magne ic lux densi y) a e ela ed o he
elec omagne ic p ope ies o ma e . And a high speeds, he momen um
o ma e is be e app oxima ed by
𝑷=(𝑚+𝐸/𝑐2)𝒗
, whe e
𝑚
is he
es -ene gy-mass and 𝐸is he emaining ene gy-mass o ma e .
I is he e o e bene icial o sepa a e ou idea o momen um om he
73
3. Main physical quan i ies 3.11 Auxilia y quan i ies
magne iza ion. Some auxilia y quan i ies a e no ex ensi e; o ins ance we
canno ask “wha ’s he o al amoun o empe a u e in his egion?”.
The dimensions, uni s, and scala o ec o cha ac e o all quan i ies
men ioned so a a e summa ized in able 3.1 on page 82.
Tempe a u e
We ha e an in ui i e unde s anding o he no ion o ho ness and coldness.
Tempe a u e quan i ies hese no ions.The physical bases andmeasu emen
p ocedu es o his quan i ica ion a e a om i ial, bu we shall ake
hem o g an ed in he p esen no es.
Fo some physical phenomena, especially hose in ol ing gases, we
know ha empe a u e is ela ed o he in isible mo ion o mic oscopic
pa s o ma e , such as molecules. Bu he e a e also physical phenomena
o which ou mic oscopic unde s anding o empe a u e is mo e complex,
and in some cases s ill unclea .
Tempe a u e is a scala quan i y. The e a e se e al de ini ions and scales
o measu emen o empe a u e. O special impo ance is he modynamic
empe a u e
40
, also called absolu e empe a u e, which is measu ed in kel ins
(
K
). The modynamic empe a u e has he special p ope y o being always
posi i e in mos physical phenomena ( he e a e excep ions, especially in
some phenomena whe e s a is ical mechanics41 becomes ele an ).
In hese no es we shall use he modynamic empe a u e, deno ing i
𝑇
. I s ela ion wi h Celsius empe a u e
𝑇C
, which is measu ed in deg ees
Celsius ◦C, is gi en by
𝑇C=𝑇−273.15K (3.2)
ha is, a ede ini ion o he “ze o” alue; o ins ance
25.00◦C=298.15K
.
No e ha empe a u e di e ences a e he same o he wo empe a u e
scales: Δ𝑇C= Δ𝑇, because he cons an ze o- alue cancels ou .
Tempe a u e is use ul because i en e s in many physical laws ha
in ol e ene gy-mass, bu i ’s easie o measu e han ene gy-mass. Tempe -
a u e gene ally depends on he ime and place, so i can be a unc ion o
he coo dina es: 𝑇(𝑡, 𝑥, 𝑦, 𝑧).
£Wha is empe a u e?
In en ing Tempe a u e by H. Chang (2004) gi es a b illian accoun o he his o y o
in en ion o empe a u e, as well as an in e es ing po ai o how scien i ic concep s
a e bo n and de elop.
80
3. Main physical quan i ies 3.12 Me ic
Is The e a Tempe a u e? by T.S. Bi ó (2011) discusses ascina ing physical phenomena
o which ou mic oscopic unde s anding o empe a u e is s ill incomple e.
3.12 Me ic
A e y impo an quan i y is a undamen al building block o all ou phys-
ical heo ies: he me ic. I is qui e di e en om he se en undamen al
quan i ies, om a physical and also om a geome ical poin o iew.
The me ic cha ac e izes ou measu emen s o space and ime. I ’s he
objec ha allows us o calcula e how much physical ime has elapsed, he
physical dis ance be ween wo objec s, he olume (say, in cubic me es) o
a h ee-dimensional egion o space, and he a ea (say, in squa e me es)
o a su ace. In Gene al Rela i i y he me ic allows us o calcula e he
cu a u e o space ime.
The me ic i sel , in he echnical sense o he so-called “me ic enso ”,
is no an ex ensi e quan i y. We can’ ask “wha ’s he o al amoun o me ic
in his egion?”; ha ’s a meaningless ques ion. The e a e, howe e , o he
quan i ies which can be de i ed om he me ic and which a e ex ensi e.
Impo an examples a e he so-called “ olume densi y” and “a ea densi y”,
which a e he ones ha allow us o calcula e ex ended olumes and a eas.
In he New onian app oxima ion, ha is, o speeds smalle han he
speed o ligh and low ene gy-mass densi ies (hence weak g a i a ional
ields and small space ime cu a u e), he me ic is jus a s a ic, uni o m
objec , he same e e ywhe e in space ime; and space ime is la , ha is,
i has no cu a u e. This is why we can speak o an ‘absolu e ime’ and
‘absolu e dis ances’ in his app oxima ion. In hese no es we shall o he
mos pa use his New onian app oxima ion.
In Gene al Rela i i y he me ic is a dynamic objec ins ead: i can
change wi h coo dina e ime, and can a y om one poin in space o
ano he . These changes a e de e mined by he se en main quan i ies, and
he me ic, in u n, de e mines changes in he se en quan i ies.
81
3. Main physical quan i ies 3.12 Me ic
Quan i y SI Dimension Uni
Time ime second s
Leng h leng h me e m
Ma e amoun o subs ance mole mol
Elec ic cha ge cu en · ime coulomb C
Magne ic lux mass ·leng h2/(cu en · ime2)webe Wb
Ene gy-mass mass ·leng h2/ ime2,
mass
joule J,
kilog am kg
Momen um mass ·leng h/ ime N·s,
kg ·m/s,
J·s/m
Angula momen um mass ·leng h2/ ime N·m·s,
kg ·m2/s,
J·s
En opy mass ·leng h2/( ime2· empe a u e)J/K
Tempe a u e empe a u e kel in K
Table 3.1 SI dimensions and uni s o he main physical quan i ies used in hese no es.
Thei luxes ha e he dimensions di ided by ime, and he e o e uni s di ided by seconds.
Quan i ies in bold ace a e ec o s, he o he s a e scala s.
82
3. Main physical quan i ies 3.12 Me ic
URLs o chap e 3
1. h ps://www.nis .go /image/moleedi 2jpg
2. h ps://doi.o g/10.1351/goldbook.A00297
3. h ps://www.nis .go /si- ede ini ion/ ede ining-mole
4. h ps://pubchem.ncbi.nlm.nih.go /elemen /Calcium
5. h ps://pubchem.ncbi.nlm.nih.go /elemen /Hyd ogen
6. h ps://pubchem.ncbi.nlm.nih.go /elemen /Helium
7. h ps://pubchem.ncbi.nlm.nih.go /elemen /Li hium
8. h ps://iupac.o g/wha -we-do/pe iodic- able-o -elemen s/
9. h ps://www.ciaaw.o g/ adioac i e-elemen s.h m
10. h ps://www.iaea.o g/newscen e /news/wha -a e- adioac i e-sou ces
11.
h ps://www.iaea.o g/newscen e /news/wha -is-nuclea -ene gy- he-sci
ence-o -nuclea -powe
12. h p://hype physics.phy-as .gsu.edu/hbase/Pa icles/had on.h ml#c6
13. h p://hype physics.phy-as .gsu.edu/hbase/Pa icles/lep on.h ml#c1
14. h ps://www.b i annica.com/science/an ima e
15. h ps://www.b i annica.com/ opic/posi on-emission- omog aphy
16.
h ps://www.helse-be gen.no/a delinge / adiologisk-a deling/sen e
- o -nuklee medisin-og-pe / il ising- il-pe
17. h ps://www.symme ymagazine.o g/2009/07/23/an ima e - om-bananas
18. h ps://www.quan umdia ies.o g/2009/07/21/posi ons- om-bananas/
19. h ps://doi.o g/10.1351/goldbook.C01365
20. h ps://www.nis .go /si- ede ini ion/ampe e-in oduc ion
21. h ps://doi.o g/10.1351/goldbook.W06666
22. h ps://doi.o g/10.1351/goldbook.V06634
23. h ps://doi.o g/10.1351/goldbook.T06283
24. h ps://doi.o g/10.1093/ac e o e/9780190871994.013.21
25. h ps://spaceplace.nasa.go /sola -ac i i y/
26. h ps://doi.o g/10.1351/goldbook.K03391
27.
h ps://nuclea museum.pas pe ec online.com/A chi e/716477C1-5E7A-4
85C-8BE1-857919471563
28. h ps://www.b i annica.com/science/nuclea - ission
29. h ps://www.b i annica.com/s o y/a omic-bombing-o -hi oshima
30.
h ps://www.nis .go /si- ede ini ion/kilog am-dissemina ing-new-kil
og am
31. h ps://home.ce n/ ags/13- e
32. h ps://doi.o g/10.4209/aaq .2019.04.0177
33. h ps://www. lick .com/pho os/33486695@N06/13566555795
34. h ps://www. g-dance.com/ icha dals ondancecompany/
35. h ps://www.b i annica.com/science/polyme
36. h ps://nssdc.gs c.nasa.go /plane a y/ ac shee /ea h ac .h ml
37. h ps://doi.o g/10.1351/goldbook.K03374
38. h ps://doi.o g/10.1351/goldbook.B00695
39. h ps://www.b i annica.com/science/liquid-c ys al
40. h ps://www.iso.o g/obp/ui/#iso:s d:iso:80000:-5:ed-2: 1:en: ab:1
41. h ps://www.b i annica.com/science/s a is ical-mechanics
83
3. Main physical quan i ies 3.12 Me ic
84
4
Volume con en s, luxes, supplies
When we ega d ene gy as esiding in insically in a body,
we may measu e i s in ensi y by he amoun con ained in
uni o olume. [. . .][T]he only way we ha e o de ining he
mo ion o he luid is by conside ing i as a lux [. . .]. This
dis inc ion is s ill mo e necessa y when we come o hea
and elec ici y. The lux o hea o o elec ici y canno be
e en hough o in any way excep as he quan i y which
lows h ough a gi en a ea in a gi en ime.
J. Cle k Maxwell 1869
4.1 Con en , lux, supply
The concep o physical quan i y allows us o look a he wo d in ways ha
can be quan i ied and exp essed wi h numbe s and ma hema ics. This is
how we can o mula e physical laws
§1.3 p.27
. In he p e ious chap e we
madeou acquain ancewi hse enphysicalquan i ies.Theywe echosenas
ou building blocks because hei quan i ica ion is easy o g asp in ui i ely
and o isualize. In he p esen chap e we s udy his quan i ica ion mo e
igo ously, and s a de eloping he necessa y ma hema ics.
Fo each o he se en p imi i e quan i ies, excep magne ic lux, we can
measu e h ee kinds o amoun : olume con en , lux, and supply:
]Volume con en
Volume con en o olume in eg al is he amoun o quan i y con ained
wi hin a h ee-dimensional egion, a a speci ic ime ins an .
The olume con en o a quan i y does no depend on how he olume
is mo ing. The olume con en has he same physical dimension as he
quan i y.
85
4. Volume con en s, luxes, supplies 4.1 Con en , lux, supply
]Flux
Flux o cu en o low a e is he amoun o quan i y lowing h ough
a wo-dimensional su ace in a gi en di ec ion, pe ime, a a pa icula
ime ins an .
The lux o a quan i y h ough a su ace depends on how ha su ace
is mo ing and de o ming. The lux has he physical dimension o ha
quan i y di ided by ime.
]Supply
Supply o sou ce is he amoun o quan i y being p oduced o des oyed
wi hin a h ee-dimensional egion pe ime, a a pa icula ime ins an .
The supply o a quan i y in a egion depends on how ha egion is
mo ing and de o ming. The supply has he physical dimension o ha
quan i y di ided by ime.
Fo he magne ic lux we shall in oduce analogous kinds o amoun ,
bu in one less dimension.
The olume con en , lux, and supply o he ou scala quan i ies –
ma e , elec ic cha ge, ene gy-mass, en opy – a e also scala s; ha is, each
is exp essed by one numbe and a uni . The olume con en , lux, and
supply o he wo ec o quan i ies – momen um and angula momen um
– a e also ec o s; ha is, each is usually exp essed by h ee numbe s and
uni s.
-Di e en e minology ac oss disciplines and li e a u e
Keep in mind ha many disciplines and hei li e a u e use e ms
di e en om ‘con en ’, ‘ lux’, ‘supply’, o indica e he same no ions. I ’s
he e o e impo an ha you unde s and he concep s hese e ms s and
o , wi hou ge ing oo a ached o he e ms hemsel es.
The e m con en is e y a ely used. In mos disciplines, one simply
speaks o he quan i y in he con ol olume, wi hou men ioning
any ‘con en ’. Fo ins ance, ins ead o saying “ he ene gy con en in he
con ol olume is
2J
”, one simply says “ he ene gy in he con ol olume
is
2J
”. In hese no es we use he wo d ‘con en ’ o clea ly dis inguish i s
86
4. Volume con en s, luxes, supplies 4.2 Symbols, no a ion, and ex ensi i y
no ion om hose o lux and supply.
The e m lux is o en used in disciplines like luid mechanics. In solid
s a e physics and some o he disciplines he e m cu en is used ins ead;
he mos e iden example is elec ic cu en , which is exac ly he same
as lux o elec ic cha ge. In physical chemis y and o he disciplines, he
e m low a e is used ins ead. You may e en mee o he echnical e ms
o he same concep . In hese no es ‘ lux’ is chosen because i ’s sho
and unlikely o be misin e p e ed in a quali a i e sense.
One mo e con using aspec abou he e ms lux and cu en is ha some
disciplines use hem o deno e he no ions we mean in hese no es, bu
di ided by a ea. This means ha hei uni s will also be di e en in hose
disciplines. The In e na ional O ganiza ion o S anda diza ion (ISO)
ecommends
1
ha he e ms ‘a eic lux’ o ‘ lux densi y’, and simila ly
wi h ‘cu en ’, be used ins ead o deno e he di ision o a lux by he
a ea o he su ace h ough which i occu s. Ou e minology ag ees
wi h he ISO.
The e m supply is o en used in disciplines like luid mechanics. In solid
s a e physics and some o he disciplines he e m sou ce is used ins ead,
and sink o deno e a supply ha ing a nega i e alue. Some disciplines
use supply and sou ce o deno e hese no ions bu di ided by olume. The
ISO ecommends ins ead o use he e ms ‘ olumic supply’ o ‘supply
densi y’ in his case.
4.2 Symbols, no a ion, and ex ensi i y
The h ee kinds o amoun , o each quan i y, a e deno ed by special
symbols. Table 4.1 on page 88 summa izes he symbols, uni s, and scala o
ec o cha ac e o he olume con en , lux, supply o he se en quan i ies,
as used in hese no es. Le ’s see some examples.
Conside abo leona able.Youmeasu e heamoun o wa e molecules
in i a a gi en ime, and ind ha i ’s
51.3mol
. We can exp ess his as
ollows:
𝑁H2O=51.3mol .
The symbol ‘
𝑁
’ is used o he olume con en o ma e . The kind o ma e
in his case is wa e , so we append i s chemical o mula
H2O
as a subsc ip ;
bu we could also ha e used he wo d ‘wa e ’ o he le e ‘w’. We could
87
4. Volume con en s, luxes, supplies 4.2 Symbols, no a ion, and ex ensi i y
Quan i y Vol. con en [uni ] Flux [uni ] Supply
ma e 𝑁[mol]𝐽[mol/s]𝒜
elec ic cha ge 𝒬[C]ℐ[C/so A]
ene gy-mass 𝐸[J]
𝑚[kg]𝛷[J/so W]
[kg/s]ℛ
momen um 𝑷[Ns]𝑭[N]𝑮
angula
momen um
𝑳[Nms]𝑴[Nm]𝓣
en opy 𝑆[J/K]𝛱[J/(Ks)]
Table 4.1 Symbols and uni s o olume con en , lux, supply o six main quan i ies.
Vec o quan i ies a e in bold ace. Supplies ha e he same uni s as luxes.
Quan i y Flux [uni ] Ci cui a ion [uni ]
magne ic lux ℬ[Wb]−ℰ [Wb/so V]
Table 4.2 Symbols and uni s o lux and ci cui a ion o magne ic lux.
88
4. Volume con en s, luxes, supplies 4.2 Symbols, no a ion, and ex ensi i y
also append he wo d ‘bo le’ o indica e ha we’ e speaking abou he
olume con en o he bo le.
As a second example, conside wo o dina y ba e ies inside some
de ice; one is on he le , he o he on he igh in he ba e y compa men .
You measu e he amoun o chemical ene gy-mass in each ba e y a a
gi en ime, and ind
9873J
o he ba e y on he le , and
4221J
o he
one on he igh . We can exp ess his as ollows:
𝐸L=9873J , 𝐸R=4221J .
The symbol ‘
𝐸
’ is used o he olume con en o ene gy-mass. In his
case we’ e speaking abou chemical ene gy-mass, and we assume his is
clea om he con ex , so we don’ indica e his in he symbols. Bu we
mus dis inguish he olume con en s o he wo ba e ies, so we use he
subsc ip s ‘L’ and ‘R’; bu we could ha e used ‘l’ and ‘ ’, o ‘le ’ and ‘ igh ’.
As a hi d example, conside a comb ha has acqui ed elec ici y, say
om ubbing wi h hai . You measu e he ne olume con en o elec ic
cha ge in he comb when you s opwa ch shows
5s
, and ind
−0.0000004C
(no e ha he cha ge is mo e concen a ed a he bounda y o he olume).
Then you measu e i again 10 seconds la e and ind ha he e is no ne
cha ge. We can exp ess his as ollows:
𝑡0=5s 𝒬(𝑡0)=−0.0000004C ,
𝑡1=15s 𝒬(𝑡1)=0.0000000C .
The symbol ‘
𝒬
’ is used o he olume con en o elec ic cha ge. We assume
ha i ’s clea om he con ex ha we’ e speaking abou he cha ge o he
comb, so we don’ indica e his in he symbols. The olume con en
𝒬
o
he cha ge is changing wi h ime, ha is, i is a unc ion o ime. Thus we
indica e i s alue a di e en imes by explici ly w i ing i s a gumen in
ound b acke s.
The examples abo e show ha i depends on he con ex wha kinds o
subsc ip o addi ional signs ha we use oge he wi h he symbols o
olume con en , lux, supply. We ha e some eedom in which addi ional
signs o use; wha ’s impo an is o make he message unambiguous.
Volume con en , lux, and supply always e e o some pa icula
ime, so s ic ly speaking hey a e unc ions o ime and would need a
ime a gumen like ‘
(𝑡)
’. Bu in si ua ions whe e we a e conside ing one
pa icula ime only, o whe e he quan i ies a e cons an in ime, we can
o b e i y omi he ime a gumen .
89
4. Volume con en s, luxes, supplies 4.6 Flux o scala quan i ies
«Exe cise 4.2
Recall ex ensi i y, he second p ope y o ou se en p imi i e quan i ies:
he amoun in a olume consis ing o sepa a e olumes is equal o he
o al o he sepa a e amoun s.
We ha e a egion consis ing o wo sub egions; he amoun s o mo-
men um in each sub egion a e shown below.
y
x
1.
W i e he o al momen um in each sub egion in componen o m,
(𝑃𝑥, 𝑃𝑦), acco ding o he coo dina e sys em shown.
2.
Calcula e hemomen umin hewhole egion; ep esen i g aphically
as ec o and w i e i in componen o m.
£Adding ec o s in Gene al Rela i i y
We a e used o he idea o adding ec o s placed a di e en poin s in space: we only
ha e o mo e each ec o , keeping i pa allel o i sel , o a common poin ; and hen
add hem all a ha poin wi h he usual ule.
This ope a ion canno be done so simply in Gene al Rela i i y: he no ion o pa allelism
doesn’ apply anymo e in a simple way, owing o he cu a u e o space ime. The
addi ion would lead o di e en esul s depending on how we anspo ed he ec o s.
Bu i is s ill possible o add he momen um o wo di e en spa ial egions, simply
because momen um is de ined wi h espec o a coo dina e sys em. This coo dina e sys em
selec s, so o speak, a unique way o anspo he momen um ec o s o a common
poin . We a e again eminded o he ac ha momen um is a coo dina e-dependen
quan i y §3.9 p.77.
In Gene al Rela i i y momen um is no eally a ec o , bu jus a special iple o
quan i ies.
4.6 Flux o scala quan i ies
To ge an in ui i e g ip o he no ion o lux, conside a low o people
h ough an open doo . The doo is ou con ol su ace. We can ask how
96
4. Volume con en s, luxes, supplies 4.6 Flux o scala quan i ies
many people c ossed he doo du ing a minu e. Bu one mo e de ail abou
his low is impo an : in which di ec ion did he people c oss he doo ? This
de ail is impo an because, o example, he doo leads o a class oom
and we need o keep ack o how many sea s a e ee. We he e o e need
o know whe he each pe son who c ossed he doo was ac ually en e ing
o lea ing he class oom.
Acco ding o he c ossing
di ec ion indica ed by he
blue wiggly a ow, he pe -
son c ossing he doo coun s
as ‘+1’.
Acco ding o he c ossing
di ec ion indica ed by he
blue wiggly a ow, he pe -
son c ossing he doo coun s
as ‘−1’.
In o de o do his we can p oceed as ollows:
1.
Assign a c ossing di ec ion o he doo , o ins ance he di ec ion om
ou side o inside he class oom.
2.
Coun as ‘posi i e’ each pe son who c osses he doo in he chosen
c ossing di ec ion, and as ‘nega i e’ each pe son who c osses he
doo in he opposi e di ec ion.
The o al o his coun ing ells us he ne numbe o people who c ossed
he doo in he chosen di ec ion. I we chose a c ossing di ec ion om
ou side o inside he class oom, hen his o al is he ne numbe o people
who en e ed he class oom. I we chose a c ossing di ec ion om inside o
ou side he class oom, hen his o al is he ne numbe o people who le
he class oom.
The e o e a lux ep esen s only a ne amoun c ossing a con ol
su ace. No e also ha his ne amoun can be nega i e. Fo ins ance i we
chose a c ossing di ec ion om ou side o inside he class oom, and he
ne amoun is
−3
, hen i means ha mo e people go ou han in: 9 pe sons
may ha e en e ed he oom du ing ha minu e, and 12 pe sons le . O
maybe no pe son en e ed he oom, and 3 pe sons le . In ei he case, he
inal si ua ion is ha hose who go ou du ing ha minu e we e h ee
mo e han hose who go in.
One impo an aspec o his example and e minology is he ollowing
symme y:
•I is comple ely a bi a y which c ossing di ec ion we choose.
•
I we choose he o he c ossing di ec ion, hen he ne amoun will
swi ch sign.
The physical si ua ion is o cou se he same. The sen ences
“+5pe sons en e ed he oom”
and
“−5pe sons le he oom”
97
4. Volume con en s, luxes, supplies 4.6 Flux o scala quan i ies
a e saying exac ly he same hing.
Now conside a simila example, bu ins ead o people, hink o a
quan i y ha can o dina ily also be nega i e, such as elec ic cha ge. Le ’s
choose he doo -c ossing di ec ion om ou side o inside he oom. I
we’ e old ha a ne amoun
−5C
o cha ge c ossed he doo in he chosen
di ec ion in one minu e, hen his could ha e happened in se e al ways:
•a cha ge o −5C was b ough in o he oom
•a cha ge o +5C was b ough ou o he oom
•
a cha ge o
−2C
was b ough in o he oom du ing he i s
30s
, and
a cha ge o +3C was b ough ou in he emaining 30s
•
a cha ge o
−2C
was b ough in o he oom du ing he whole minu e,
and a cha ge o +3C was b ough ou a he same ime
•. . . and many o he possible combina ions.
So he s a emen ha “ he lux o elec ic cha ge in o he oom was
−5C
in
one minu e” does no ell us which o he si ua ions abo e occu ed.
“Fechne [in 1845] supposed
e e y cu en o consis in a
s eaming o elec ic cha ges, he
i eous cha ges a elling in
one di ec ion, and he esinous
cha ges, equal o hem in mag-
ni ude and numbe , a elling in
he opposi e di ec ion wi h equal
eloci y.” Whi ake 1951
In ac , o dina y elec ici y in wi es was hough o some ime o be
associa ed wi h mo emen s o nega i e and posi i e cha ges in opposi e
di ec ions. Today we know ha i consis s in he mo emen o nega i e
cha ges only.
The pu pose o he p e ious examples is o make you awa e o some
undamen al aspec s o wha we call “ lux”. These aspec s a e i ial bu
impo an when conside ing luxes o physical quan i ies:
]Fundamen al aspec s and symme y o lux
•
A lux in a pa icula su ace-c ossing di ec ion only ells us he
ne amoun o subs ance ha c osses he su ace in ha di ec ion
pe ime.
•A lux can be nega i e.
•
Symme y o lux: A lux in a pa icula su ace-c ossing di ec ion
is equi alen o a lux o opposi e sign in he opposi e c ossing
di ec ion.
This is called Cauchy’s unda-
men al lemma in he echnical
li e a u e.
-Wha a lux alue does no ell
•
A lux alue does no ell us he amoun ha c ossed du ing sho e
imes o h ough di e en pa s o he su ace.
98
4. Volume con en s, luxes, supplies 4.7 Rep esen a ion o scala luxes
•
A lux alue does no ell us whe he he ans e o he quan i y
h ough he su ace occu s because he quan i y is lowing, o
because he su ace is mo ing, o bo h.
4.7 Rep esen a ion o scala luxes
How can we g aphically ep esen he lux o a quan i y, in a way ha
akes ca e o all i s undamen al aspec s?
Fi s le ’s conside a su ace h ough which we’ e measu ing a lux, a
a pa icula ins an o ime. He e i is ep esen ed as line, emo ing one
spa ial dimension o simpli y he d awing:
Keep in mind ha he su ace could ha e any o he shape – as long as i
can be gi en a clea c ossing di ec ion – and could also be in mo ion.
Le ’s indica e a c ossing di ec ion h ough he su ace by one o mo e
wiggly a ows:
Keepinmind ha hesea owsa e no ec o s!Theydon’ ha ea‘magni ude’
o ‘componen s’. They simply indica e a sense in which we imagine he
su ace o be c ossed. We could also ha e used only one wiggly a ow o
h ee ins ead o wo.
Le ’s ake a scala quan i y such as ene gy. A lux o ene gy
+5J/s
h ough he su ace, in he i s c ossing di ec ion, can hen be depic ed as
ollows:
99
4. Volume con en s, luxes, supplies 4.7 Rep esen a ion o scala luxes
This pic u e says ha a ne amoun o
5J
is c ossing he su ace, pe second,
om he le side o he igh side. This also means ha pe second a
ne amoun o
5
J is “disappea ing” om he le side o he su ace and
“appea ing” on he igh side.
Now conside he opposi e c ossing di ec ion, depic ed like his:
Because o he symme y o lux
§4.6 p.98
, we can say ha he lux o ene gy
equals −5J/sin his opposi e di ec ion. We depic his as ollows:
This pic u e says ha a ne amoun o
−5
has c ossed he su ace om he
igh o he le side, in a uni o ime. This also means ha a ne amoun
o
−5
has “disappea ed” om he le side o he su ace and “appea ed”
on he igh side, in a uni o ime.
Bu his is indeed exac ly he same si ua ion as be o e. Bo h pic u es
he e o e ep esen he same lux:
is exac ly he same as
I is ex emely impo an ha you emembe ha he wo kinds o
pic u e abo e a e comple ely equi alen . You can always men ally swi ch
om one o he o he . A lux in one c ossing di ec ion is exac ly he same
as a lux wi h opposi e sign in he opposi e di ec ion.
The wo equi alen pic u es do no say ha a gi en amoun is only
mo ing om le o igh , o ice e sa. We ha e seen ha in gene al we
don’ know his. Bo h pic u es say ha on he le side he amoun o
quan i y is changing a a a e o
−5J
pe second, and on he igh side by
+5J
pe second. In hese no es we shall usually display only one o hese
wo equi alen ep esen a ions.
100
4. Volume con en s, luxes, supplies 4.8 Flux o ec o quan i ies and i s ep esen a ion
«Exe cise 4.3
Fo each ques ion, answe in an unambiguous way and ske ch a pic u e
ep esen ing he lux.
1.
The wo sides o a pa icula su ace a e called ‘up’ and ‘down’.
Du ing
0.2s
, an ene gy-mass o
+3J
lows om he up-side o he
down-side, and an ene gy-mass o
−4J
lows om he down-side
o he up-side. How much is he lux o ene gy-mass h ough he
su ace?
2.
Th ough he same su ace, a a la e ime,
2mol
o neu ons low
om he up- o he down- side in
0.01s
, and
2mol
o neu ons low
om he down- o he up-side du ing he same ime. How much is
he lux o ma e h ough he su ace?
3.
The wo sides o a su ace a e called ‘in’ and ‘ou ’. Du ing
0.01s
he e is a low o
1000
elec ons om he in-side o he ou -side, and
also a low o
1000
posi ons (an i-elec ons) in he same di ec ion.
How much is he lux o ma e h ough he su ace?
4.
The side igu e shows a con ol su ace mo ing om le o igh a
a (cons an ) eloci y o
1m/s
. The space o i s igh has wo s a ic
egions wi h some amoun o ene gy-mass as shown ( he e’s no
ene gy-mass behind o he le o he su ace). How much is he lux
o ene gy-mass h ough he su ace in 1s?
1 m
1 m/s
4.8 Flux o ec o quan i ies and i s
ep esen a ion
The lux o a ec o quan i y is also a ec o , because i is gi en by an
amoun o ha quan i y, which is a ec o , di ided by ime, which is a
scala . We can hink o i as h ee luxes o h ee scala quan i ies.
The in ui ion and men al ep esen a ion o he lux o a ec o quan i y
h ough a con ol su ace may be less s aigh o wa d han o a scala
quan i y. Think again o he p e ious examples wi h people o elec ic
cha ges c ossing a doo o con ol su ace. In he case o lux o a ec o
quan i y, we may imagine ha wha ’s c ossing he con ol su ace a e
ec o s. We a e going o discuss some possible g aphical ep esen a ions.
101
4. Volume con en s, luxes, supplies 4.8 Flux o ec o quan i ies and i s ep esen a ion
The ema ks abou he choice o c ossing di ec ion and abou he
symme y o lux, which we made o scala quan i ies, also apply in
analogous ways o he lux o ec o quan i ies. Fo ins ance, i he lux o
momen um h ough a su ace in a pa icula c ossing di ec ion is
[−3,4,0]N,
hen i we choose he opposi e c ossing di ec ion he lux is
−[−3,4,0]N=[+3,−4,0]N.
I we hink o ec o s as a ows, we mus only emembe ha a minus sign
changes hei o ien a ion:
↖=−1· ↘
We can de ise a g aphical ep esen a ion o he lux o a ec o quan i y
simila o ha o he lux o a scala quan i y §4.7 p.99.
Fi s , i ’s impo an o indica e he c ossing di ec ion, and we can do
ha again wi h one o mo e wiggly a ows; o ins ance:
Now we ha e o indica e how much is he lux. In he case o a scala
quan i y we simply epo ed he alue, including he uni . Fo he lux o a
ec o quan i y we ha e h ee alues, so one possibili y is o simply epo
hem. Suppose we a e speaking abou momen um and he lux in he
gi en c ossing di ec ion is [3,−4,0]N; we can hen w i e his explici ly:
Ano he al e na i e is o d aw a ec o ep esen ing hese componen s:
102
4. Volume con en s, luxes, supplies 4.8 Flux o ec o quan i ies and i s ep esen a ion
This pic u e says ha a ne ec o amoun o momen um is c ossing,
pe second, he su ace om he le side o he igh side. To help ou
in ui ion we can imagine he ec o “mo ing” ac oss he con ol su ace in
he di ec ion indica ed by he wiggly a ow; an anima ed ep esen a ion
o his can be ound a his link6.
In he opposi e c ossing di ec ion he lux ge s a minus sign, because
o he symme y o luxes. The co esponding g aphical ep esen a ion is
o
no e how he componen s ha e lipped sign, and how he ec o has
lipped di ec ion keeping he same magni ude. The pic u es abo e say ha
a ne ec o amoun o momen um
[3,−4,0]Ns
, o g aphically , is
c ossing, pe second, he su ace om he igh side o he le side.
Bu his is exac ly he same lux as be o e, because
[3,−4,0]Ns =−[−3,4,0]Ns =−
In o he wo ds, he ollowing wo pic u es ep esen he same ec o lux:
is exac ly he same as
is exac ly he same as
An impo an aspec o ec o luxes which we mus y no o ge
con used abou is he applica ion poin o he ec o , ha is, he base poin
o he a ow. Jus as o ec o olume con en s
§4.5 p.95
, he applica ion
103
4. Volume con en s, luxes, supplies 4.9 Fluxes h ough diffe en su aces
poin o he ec o ep esen ing he lux is unimpo an ; he ec o e e s
o one side o he su ace as a whole. G aphically:
hese ou pic u es ep esen exac ly he same lux
«Exe cise 4.4
A ho izon al su ace is gi en, and he e is a lux o a ec o quan i y
h ough i ; o he momen we neglec uni s:
1.
I we ake he downwa d c ossing di ec ion as ‘posi i e’, he lux
𝑥𝑦𝑧
-componen s a e
[5,5,0]
. Rep esen his lux g aphically, in he
way discussed in he p esen sec ion. Use he coo dina e sys em
y
x
we e 𝑦poin s upwa d.
2.
Taking he same c ossing di ec ion, ep esen g aphically he lux
[0,−2,0]ins ead.
3.
Taking he samedownwa dc ossingdi ec ion,wea e now old ha
he e is a lux wi h componen s
[1,−2,3]
. Wha a e he componen s
o his lux i we ake he upwa d c ossing di ec ion as posi i e?
4.9 Fluxes h ough diffe en su aces
Wha happens o he alue o a lux h ough a su ace, i we conside a
di e en su ace, maybe in e sec ing he o iginal one? I ’s impo an o keep
in mind ha
•A lux e e s only o a speci ic su ace.
•
The luxes h ough wo dis inc su aces can be e y di e en , e en i
he wo su aces a e qui e close.
•
The lux depends on he mo ion o he su ace. So i we conside he
same su ace bu wi h a di e en ins an aneous mo ion, hen he lux
may be e y di e en .
104
4. Volume con en s, luxes, supplies 4.9 Fluxes h ough diffe en su aces
Conside o ins ance he pic u e on he side. I depic s wo in e sec ing
+5 J/s
–1 J/s
P
su aces (as usual simpli ied by emo ing one dimension) and wo chosen
c ossing di ec ions on hem. The c ossing di ec ions a e bo h oughly
igh wa d. Ye he ene gy-mass lux h ough he solid blue su ace is
+5J/s
,
whe eas he ene gy-mass lux h ough he dashed ed su ace is −1J/s.
The e is, howe e , a ela ion be ween he luxes h ough su aces
ha sha e a common poin . I we know he lux h ough h ee di e en ,
small, s a ic su aces ha ing a common poin , hen we can ind he lux
h ough any o he small, s a ic su ace passing h ough ha same poin .
This possibili y leads o he ep esen a ion o lux h ough a small su ace
asa ec o , called ‘ lux-densi y ec o ’. In hese no es we shall no conside
lux densi ies and hei ec o ep esen a ion.
We saw ha he lux o a scala quan i y can be e y di e en i we ake
a sligh ly di e en su ace, o he same su ace wi h a di e en mo ion.
The same is ue o he lux o a ec o quan i y: in pa icula , he ec o s
ep esen ing he luxes h ough wo sligh ly di e en su aces can poin
in comple ely di e en di ec ions.
He e is an example. Take a ixed poin
𝑃
. Now ake a small e ical
su ace passing h ough
𝑃
, and choose a c ossing di ec ion om le o
igh . The lux o a ec o quan i y (momen um o example) h ough his
quan i y could be as in his pic u e:
(9, –6, 0)
P
his lux has componen s (9,−6,0), wi h magni ude a ound 10.8.
Now o ge abou ha su ace, and ake ins ead a small ho izon al
su ace passing h ough he same poin
𝑃
, and upwa d c ossing di ec ion.
The lux o he same quan i y h ough his new su ace could be as in his
pic u e:
(–6, –3, 0)
P
i has componen s (−6,−3,0), wi h magni ude a ound 6.7.
105
4. Volume con en s, luxes, supplies 4.13 Closed con ol su aces, in luxes, effluxes
Shea o ce is he kind o momen um lux ha we expe ience unde ou
ee when we walk o un, and ha occu s be ween a ca ’s wheels and he
g ound.
In gene al, a momen um lux won’ ha e any o he h ee special
di ec ions abo e, bu a he a combina ion o hem.
«Exe cise 4.6
Using you in ui ion, y o iden i y he a ious momen um luxes ha
occu in he di e en beams o a owe c ane. Which momen um luxes
a e app oxima ely comp essi e, ensile, and shea ing?
4.13 Closed con ol su aces, in luxes, effluxes
We shall o en conside closed con ol su aces, ha is, con ol su aces
ha don’ ha e a im o bo de o holes, like he su ace o a sphe e o
o a cube. A closed su ace delimi s a speci ic h ee-dimensional olume,
and we can he e o e speak o i s in e io and i s ex e io . An example
(simpli ied by emo ing one dimension as usual) is he su ace in he side
pic u e.
The wo c ossing di ec ions o a closed su ace he e o e ake on special
names: inwa d, om ex e io o in e io ; and ou wa d, om in e io o
ex e io . A lux h ough he su ace is usually called in lux i we a e
conside ing he inwa d c ossing di ec ion, and e lux o ou lux i we a e
conside ing he ou wa d c ossing di ec ion. Ob iously, by he symme y
o lux,
in lux ≡ −e lux e lux ≡ −in lux
The in lux and e lux a e luxes h ough he whole su ace. Conside o
ins ance hese in luxes o ene gy-mass and o momen um:
112
4. Volume con en s, luxes, supplies 4.13 Closed con ol su aces, in luxes, effluxes
In he le pic u e we ha e a ne in lux o
5J/s
on he whole inne side o
he con ol su ace. Remembe ha we don’ know whe he hese
5J/s
a e
being e enly dis ibu ed o e he su ace, o jus a pa icula spo s o i ,
o whe he hey a e he ne esul o posi i e and nega i e amoun s on
di e en pa s o he su ace. In he igh pic u e we ha e a ec o in lux o
8N
, he ec o poin ing app oxima ely igh wa d. Again we don’ know
wha a e he lux ec o s o pa s o he su ace: he ec o in he pic u e
is jus he g and o al.
Le ’s see ano he example o his ac . The pic u e on he le below
shows he ou wa d luxes h ough h ee pa s (do ed yellow,dashed
g een,do -dashed ed) o a closed con ol su ace. The pic u e on he igh
shows he o al e lux h ough he same su ace:
The indi idual luxes and he o al e lux a e consis en because
[2,−1,0]N+[−2,−1,0]N+[0,4,0]N=[0,2,0]N
«Exe cise 4.7
1.
A e he ou pa ial in luxes shown on he le (wi h di e en colou s
and line s yles) consis en wi h he o al in lux shown on he igh ?
Why o why no ?
113
4. Volume con en s, luxes, supplies 4.14 Time-in eg a ed luxes and supplies
2.
Take an imagina y cylind ical su ace enclosing one con ol od
11
in a nuclea - ission eac o
12
(see side igu e). Le ’s say ha in a
eac o he e a e 20 such ods. App oxima ely
5×1019
neu ons a e
libe a ed in a second in he whole eac o by he ission uel, bu 2/3
o hese a e abso bed by he con ol ods.
How much, on a e age, is he e lux o neu ons (ma e ) h ough
he su ace o one con ol od?
Exp ess he esul i s in
neu ons/s
, and hen in
mol/s
, using he
A ogad o cons an
𝑁A=6.02214076 ×1023 pa icles/mol .
Be ca e ul abou he signs!
4.14 Time-in eg a ed luxes and supplies
A lux is de ined as he amoun o a quan i y c ossing a con ol su ace
in a sho ime lapse
Δ𝑡
, di ided by ha ime lapse. Deno ing he lux by,
say,
𝐽
, his de ini ion also means ha he amoun o quan i y c ossing he
su ace in a sho ime Δ𝑡is app oxima ely equal o 𝐽Δ𝑡.
Now conside a con ol su ace be ween wo ime ins an s
𝑡0
and
𝑡1
;
du ing his ime lapse i could be mo ing and changing shape. Choose a
c ossing di ec ion h ough he su ace. A each in e media e ime ins an
𝑡
we can measu e he lux o a quan i y c ossing he su ace in ha di ec ion,
a ha ins an ; deno e his lux by 𝐽(𝑡).
The o al amoun o quan i y ha c osses he su ace be ween imes
𝑡0
and
𝑡1
can be ound by in eg a ing
𝐽(𝑡)
. Tha is, we di ide he ime in e al
in o e y sho ime lapses o leng h
Δ𝑡
; o each sho ime lapse we know
ha he amoun ha c osses he su ace is
𝐽(𝑡)Δ𝑡
; he o al is hen ob ained
by adding hese small amoun s. As we conside sho e and sho e
Δ𝑡
,
his sum is by de ini ion an in eg al:
]Time-in eg a ed lux
The o al amoun o quan i y lowing h ough a con ol su ace in
a speci ied c ossing di ec ion, be ween imes
𝑡0
and
𝑡1
, is called he
114
4. Volume con en s, luxes, supplies 4.14 Time-in eg a ed luxes and supplies
ime-in eg a ed lux and is gi en by
∫𝑡1
𝑡0
𝐽(𝑡)d𝑡 , (4.1)
whe e 𝐽(𝑡)is he lux o he quan i y a ime 𝑡.
The meaning o he in eg al abo e should be clea o any scala
quan i y, o which he lux is also a scala . In he case o a ec o quan i y,
o ins ance momen um, he lux is also a ec o , ep esen ed by h ee
componen s. The in eg al o a ec o is ob ained by calcula ing he in eg al
o each componen , ob aining h ee esul s, which a e he componen s o
a new ec o . Geome ically his co esponds o summing a la ge numbe
o e y sho ec o s.
Take he case o momen um, whose lux ( o ce) we deno e
𝑭=
[𝐹𝑥, 𝐹𝑦, 𝐹𝑧]. The ime in eg al o his lux is hen
∫𝑡1
𝑡0
𝑭(𝑡)d𝑡:="∫𝑡1
𝑡0
𝐹𝑥(𝑡)d𝑡 , ∫𝑡1
𝑡0
𝐹𝑦(𝑡)d𝑡 , ∫𝑡1
𝑡0
𝐹𝑧(𝑡)d𝑡#.(4.2)
An analogous discussion can be made abou he supply o a quan i y
in a con ol olume:
]Time-in eg a ed supply
The ne amoun o quan i y c ea ed in a a con ol olume, be ween
imes 𝑡0and 𝑡1, is called he ime-in eg a ed supply and is gi en by
∫𝑡1
𝑡0
𝒜(𝑡)d𝑡 , (4.3)
whe e
𝒜(𝑡)
is he supply o he quan i y a ime
𝑡
. I he in eg al is
nega i e, i means ha a ne amoun o quan i y has been des oyed.
-
Time-in eg a ed quan i ies can be ze o e en wi h non-ze o lux o
supply
The esul o he in eg al de ining a ime-in eg a ed lux o supply can
be ze o. This means ha no ne amoun o quan i y lowed h ough
he su ace, o was c ea ed in he olume, be ween
𝑡0
and
𝑡1
. Ye he
115
4. Volume con en s, luxes, supplies 4.15 The ela ion be ween luxes and eloci ies
lux o supply can be non-ze o, e en a all imes; o cou se i needs o
be posi i e a some imes, and nega i e a o he s, in o de o he ime
in eg al o be ze o.
«Exe cise 4.8
1.
Wha a e he physical dimensions o he ime-in eg a ed lux and
he ime-in eg a ed supply o a quan i y?
2.
Suppose ha we calcula e he ime in eg al o a pa icula con ol
su ace in he case o ma e , inding a o al o
∫𝑡1
𝑡0𝐽(𝑡)d𝑡=7mol
.
Now we change ou mind and choose he opposi e c ossing di ec ion
o ha su ace. How does he esul abo e change?
4.15 The ela ion be ween luxes and eloci ies
The idea o lux e okes he idea o mo emen , and he e o e o eloci y. Is
he e a ela ionship be ween lux and eloci y?
The answe is yes: he eloci y o a quan i y is essen ially de ined i s
lux and i s olume con en . Conside o example how we measu e he
eloci y o an objec : we a e ac ually keeping ack o a low o ma e – he
ma e ha makes up he objec – om a egion o space o ano he .
The igo ous de ini ion o eloci y om lux is somewha in ol ed,
so he e we’ll jus see a simpli ied and app oxima e example o how his
de ini ion wo ks.
Take a scala ex ensi e quan i y like ma e , elec ic cha ge, ene gy-
mass, o en opy. Fo conc e eness le ’s ake ma e . Choose a coo dina e
sys em
(𝑡, 𝑥, 𝑦, 𝑧)
and conside a poin in space a a speci ic coo dina e
ime. A ound his poin , choose a e y small s a ic cuboid egion, as in he
a ea A
x
N
olume V
x
y
z
xΔ
Jx
This con ol olume is small,
s a ic, and wi h sides pa allel
o he coo dina e axes.
side igu e, delimi ed by six small ec angula s a ic su aces: wo pa allel
o he
𝑦𝑧
-coo dina e plane, wo o he
𝑧𝑥
one, and wo o he
𝑥𝑦
one. The
cuboid has olume
𝑉
, and he olume con en o ma e in i is
𝑁
. The
wo
𝑦𝑧
su aces (one o hem is in da k ed in he pic u e) ha e a ea
𝐴
,
app oxima ely he same o bo h; and he e is a lux o ma e
𝐽𝑥
c ossing
ei he o hese wo su aces in he posi i e-
𝑥
di ec ion. These wo pa allel
su aces ha e app oxima ely he same a ea and he same lux because he
cuboid egion is e y small.
116
4. Volume con en s, luxes, supplies 4.15 The ela ion be ween luxes and eloci ies
The
𝑥
-componen o he coo dina e eloci y o ma e in his egion is
hen de ined as
𝑣𝑥:=𝐽𝑥/𝐴
𝑁/𝑉(4.4)
wi h analogous de ini ions o he 𝑦- and 𝑧-componen s.
The eloci y
𝒗=[𝑣𝑥, 𝑣𝑦, 𝑣𝑧]
so de ined has he ollowing in ui i e
p ope y. I you choose any e y small su ace cen ed a his poin , and
mo e i wi h eloci y
𝒗
, in he di ec ion speci ied by he eloci y, hen he
ma e lux h ough i is ze o. This e lec s he in ui i e unde s anding
ha i a su ace is mo ing oge he wi h he ma e , a he same speed,
hen we shouldn’ obse e any lux h ough i .
«Exe cise 4.9
1.
T y o p o e he o mula
(4.4)
ela ing lux and eloci y in an in ui i e
way, e e ing o he pic u e abo e. As a s a ing poin , conside his
ques ion: i he amoun o ma e
𝑁
in he olume
𝑉
is mo ing wi h
eloci y
𝑣𝑥
in he posi i e-
𝑥
di ec ion, how much o i will c oss he
a ea 𝐴du ing ime Δ𝑡?
2.
A small cuboid egion has a olume o
1×10−9m3
, and i s sides
pa allel o he
𝑥𝑦
axes ha e each a ea
1×10−3m3
. Th ough each o
hese sides he e is a lux o ene gy-mass
𝛷𝑧=3J/s
in he posi i e
𝑧
-di ec ion. The cuboid egion con ains
𝐸=0.5J
o ene gy-mass.
How much is he eloci y o ene gy-mass in he 𝑧-di ec ion?
£Veloci ies o quan i ies in Gene al Rela i i y
One consequence o he ela ionship be ween eloci ies and luxes is ha we can
de ine such a eloci y o any ex ensi e quan i y. So we ha e a eloci y ma e om
he lux o ma e , bu also a eloci y o ene gy-mass om he lux o ene gy-mass.
In New onian app oxima ion hese wo eloci ies a e app oxima ely equal, so we do
no need o dis inguish hem. Bu in si ua ions whe e he New onian app oxima ion
is no alid, we ha e o ake in o accoun he eloci y o ma e and he eloci y
o ene gy-mass sepa a ely. This di e ence is impo an o ins ance in he s udy o
plasma in s a s and in nume ical Gene al Rela i i y.
The e is an ongoing discussion as o which o he wo eloci ies is mo e con enien o
use; see o ins ance Kandus & Tsagas 2008, especially he sec ion Ecka ame e sus
Landau ame, which e e s o he choice be ween hese wo eloci ies.
117
4. Volume con en s, luxes, supplies 4.15 The ela ion be ween luxes and eloci ies
URLs o chap e 4
1. h ps://doi.o g/10.6028/NIST.SP.811e2008
2. h ps://www. u bosquid.com/3d-models/class oom-1726208
3. h p://www.009.cd2.com
4.
h ps://www.g c.nasa.go /WWW/k-12/Vi ualAe o/Bo leRocke /ai plane/
lap.h ml
5. h p://www.you ube.com/wa ch? =M5xnAdVPbgQ
6. h ps://pglpm.gi hub.io/7wonde s/media/ ec o luxanim2.webp
7. h ps://www.imdb.com/ i le/ 0013099/
8. h ps://pglpm.gi hub.io/7wonde s/media/p essu e.webp
9. h ps://pglpm.gi hub.io/7wonde s/media/ ension.webp
10. h ps://pglpm.gi hub.io/7wonde s/media/shea o ce.webp
11. h ps://ene gyeduca ion.ca/encyclopedia/Con ol_ od
12. h ps://www.b i annica.com/ echnology/nuclea - eac o
118
5
Physical laws
E e y b anch o physical science is based on wo se s o
undamen al equa ions. The i s se is ha o basic laws o
physics, which a e pos ula ed o hold alid o all bodies
unde all concei able ci cums ances [. . .]. The second se o
undamen al equa ions a e he cons i u i e equa ions: hese
a e ela ionships which a e no supposed o hold o all
bodies, bu only o desc ibe he beha io o some es ic ed
class o bodies, o possibly o a la ge class o bodies o a
mo e es ic ed class o phenomena.
G. As a i a 1990
5.1 Some classi ica ions o physical laws
Physical laws, e y gene ally speaking, a e ma hema ical ela ions be ween
physicalquan i ies.Gi enin o ma ionabou somequan i ies,physicallaws
allow us o deduce in o ma ion abou o he quan i ies, o abou he same
quan i ies a o he imes o spa ial egions. As p e iously discussed
§1.3 p.27
,
we use many physical laws e e y day, in a quali a i e and app oxima e
way, wi hou e en hinking abou hem.
Physical laws can be classi ied o ca ego ized in many di e en ways;
o ins ance by he quan i ies hey in ol e, o by he kind o ma hema ics
hey use. So we speak o ‘laws o mechanics’ and ‘elec omagne ic laws’;
o o ‘di e en ial laws’ and ‘in eg al laws’; and so on.
One classi ica ion dis inguishes be ween undamen al s de i ed physical
laws. This dis inc ion is simila o he one be ween p imi i e and de i ed
quan i ies
§1.4 p.29
.Aphysicallawis‘ undamen al’i i is akenasempi ically
alid, and as he s a ing poin o make p edic ions o calcula e o he kinds
o consequences. A physical law is ‘de i ed’ i i can be deduced om
o he undamen al laws: in a manne o speaking, i doesn’ eally say
119
5. Physical laws 5.2 Uni e sal laws s cons i u i e ela ions
any hing new ha wasn’ al eady a consequence o he undamen al laws.
Bu i may s ill be a e y use ul sho cu .
He e is an ex emely simple imagina y example. Suppose we ha e
h ee di e en physical quan i ies deno ed by
𝑎
,
𝑏
,
𝑐
. One undamen al
physical law s a es ha
𝑎
and
𝑏
a e equal; ano he undamen al law s a es
ha 𝑏and 𝑐a e equal oo:
𝑎=𝑏 , 𝑏 =𝑐( undamen al laws)
Combining hese wo undamen al laws we ob ain a u he law:
𝑎
is equal
o 𝑐:
𝑎=𝑏 , 𝑏 =𝑐=⇒𝑎=𝑐(de i ed law)
This la e physical law doesn’ ell us any hing new ha wasn’ al eady
implici in he i s wo oge he . Bu in some si ua ions i is use ul o
simply emembe di ec ly ha 𝑎=𝑐.
h ps://xkcd.com/1489
The dis inc ion be ween undamen al and de i ed physical laws is no
objec i e, bu mos ly a ma e o con enience and e en o pe sonal as e.
We can o en p omo e a de i ed law o undamen al law, demo ing some
o he undamen al law o a de i ed-law s a us. In he example abo e, we
could ake ‘
𝑎=𝑏
’ and ‘
𝑎=𝑐
’ as undamen al laws; hen ‘
𝑏=𝑐
’ becomes a
de i ed law, because i can be ob ained om he o he wo. I is impo an
o be awa e o his lexibili y in wha ’s undamen al and wha ’s de i ed.
You’ll ind physics and enginee ing ex s ha p esen a physical heo y
as consis ing in a pa icula collec ion o undamen al laws, and o he
ex s ha p esen he same physical heo y as consis ing in a collec ion
o sligh ly di e en undamen al laws. The e is no con adic ion he e: i
means ha some laws aken as de i ed in some ex s, bu as undamen al
in o he s, and ice e sa.
Ye , he choice undamen al laws is no wi hou consequence. A pa icu-
la choiceo undamen al laws, a he han o he s,mayco e mo ephysical
si ua ions. I may also sugges new physical ideas and gene aliza ions,
leading o he disco e y o new physical phenomena.
5.2 Uni e sal laws s cons i u i e ela ions
Ano he impo an dis inc ion can be made be ween uni e sal physical
laws and cons i u i e physical ela ions. This dis inc ion is de e mined
no by con enience and pe sonal as e, bu by expe imen :
120
5. Physical laws 5.2 Uni e sal laws s cons i u i e ela ions
]Uni e sal laws s cons i u i e ela ions
•
Uni e sal laws ep esen uni e sal physical pa e ns ha we ob-
se e in all possible physical phenomena we manage o in es iga e
•
Cons i u i e ela ions ep esen physical pa e ns and physical
p ope ies ha a e peculia o – and he e o e a e only alid o –
speci ic phenomena, o speci ic scales o ime and space, o speci ic
kinds o con ol olumes and su aces, o speci ic physical heo ies.
Cons i u i e ela ions a e also called cons i u i e equa ions o closu e
equa ions.
One o he meanings o
he wo d cons i u i e is ‘ ha
makes a hing wha i is’ (Ox-
o d English Dic iona y 2009).
The dis inc ion abo e is di e en om he one be ween ‘ undamen al’
and ‘de i ed’. Le ’s y o unde s and his di e ence by means o ou
p e ious simple example.
Suppose we obse e ha he law ‘
𝑎=𝑏
’ is always alid in all physical
phenomena we explo e, unde all possible ex eme condi ions, ci cum-
s ances, egions o space, and ime. Also suppose we obse e ha ‘
𝑏=𝑐
’ is
ins ead only alid in speci ic physical phenomena and condi ions, bu no
in o he s. Fo ins ance, we may obse e ha i is only alid when we make
expe imen s wi h gases and low speeds, bu no wi h solids o high speeds.
The conclusion (un il we ind a disp oo ) is ha he physical law ‘
𝑎=𝑏
’ is
uni e sal, whe eas he physical law ‘
𝑏=𝑐
’ is cons i u i e: cons i u i e o
gases and low speeds. Ye bo h laws a e undamen al, because we canno
deduce ‘
𝑏=𝑐
’ om ‘
𝑎=𝑏
’: he law abou
𝑏
and
𝑐
says some hing new,
al hough some hing ha is ue only in some ci cums ances. In conc e e
applica ions we may he e o e need o use bo h ‘𝑎=𝑏’ and ‘𝑏=𝑐’.
Bu a e he e eally uni e sal physical laws, which can be applied o
e e y physical phenomenon, wi hou exclusions o excep ions?
The answe is yes.We shall soon mee hem
§5.8 p.142
, and we shall
see ha hey a e igh ly connec ed wi h he se en p imi i e quan i ies
discussed in chap e 3. Ou s udy o physics will indeed comple ely hinge
on hese uni e sal laws. They a e applied wi h ex eme con idence o e e y
new phenomenon we obse e, and hey o en allow us o make p edic ions
o a leas a quali a i e cha ac e wi hou he need o cons i u i e ela ions.
We would modi y hese uni e sal laws only as a las eso ; so a his
has a ely o ne e been necessa y. On he o he hand, we ha e a la ge
eedom in modi ying cons i u i e ela ions, and in p oposing new ones
o accoun o newly obse ed physical phenomena.
121
5. Physical laws 5.4 Conse a ion laws
and his is he ne amoun o ice ha en e ed he con ol su ace du ing
he 0.5s.
No e ha wha we ound is he ime-in eg a ed in lux h ough he
whole closed con ol su ace. The conse a ion law doesn’ ell us any hing
abou he in eg a ed in lux h ough pa s o he su ace. In he p esen case
we can di ide he su ace in o h ee pa s: a ci cula su ace a he bo om,
a side su ace, and a ci cula su ace a he op. I we p o ide he ex a
knowledge ha he luxes h ough he side and bo om su aces a e ze o,
hen by he ex ensi i y p ope y §3.2 p.63 we ha e
∫𝑡1
𝑡0
𝐽(𝑡)d𝑡=∫𝑡1
𝑡0𝐽 op(𝑡) + 𝐽side(𝑡) + 𝐽bo (𝑡)d𝑡
53.4mol =∫𝑡1
𝑡0𝐽 op(𝑡)+0mol/s+0mol/sd𝑡
om which we ind, as was in ui i ely clea , ha he ne amoun o ice
ha c ossed he op su ace in a downwa d di ec ion is 53.4mol.
«Exe cise 5.2
Sol e he ollowing exe cise no jus by using in ui ion, bu by explaining
s ep-by-s ep how you use a conse a ion law o ob ain he esul :
•Wha is he ele an ime in e al?
•
How do you de ine he closed con ol su ace and and i s mo emen , as
well as any subdi isions o he su ace?
•
Wha a e he alues o olume con en and o ime-in eg a ed lux known
o you? Which ones do you wan o ind?
Fo he p esen exe cise we assume ha ene gy-mass sa is ies a conse -
a ion law.
An apa men ’s oom has wo iden ical elec ic hea e s along a wall.
An elec ic hea e can be conside ed as a piece o su ace ac oss which
ene gy-mass lows in o he oom: he ene gy-mass is en e ing a ound
he elec ic wi es in he o m o elec omagne ic ene gy-mass, and is
con e ed in o in e nal ene gy-mass (mainly o he oom’s ai ) by means
o he hea e . Suppose ha each hea e co esponds o an in lux o
200J/s.
The oom also has a window, which is he only o he pa o he oom’s
bounda y whe e ene gy-mass can low in o ou .
128
5. Physical laws 5.5 Balance laws
In one hou we measu e ha he o al amoun o ene gy-mass in he
oom has no changed. How much is he in eg a ed ene gy-mass in lux
h ough he window du ing ha ime?
5.5 Balance laws
Ou in ui i e unde s anding o a conse ed quan i y is ha i canno
be “c ea ed” o “des oyed”; i can only “mo e a ound”. Bu he e a e
quan i ies ha a e no conse ed. Such quan i ies sa is y a balance law. They
could be p oduced o disappea in a con ol olume; ecall ha we do ha e
a e m and measu emen o his kind o p oduc ion o disappea ance: he
supply §3.2 p.62.
To o mula e a gene al balance law we use he same se up
§5.3 p.123
as o
a conse a ion law. We ake a closed con ol su ace, delimi ing a con ol
olume, be ween coo dina e imes
𝑡0
and
𝑡1
. Ha ing chosen one o he
se en quan i ies excep he magne ic lux, we measu e i s olume con en a
ime
𝑡0
, i s ime-in eg a ed in lux be ween imes
𝑡0
and
𝑡1
, i s ime-in eg a ed
supply be ween imes 𝑡0and 𝑡1, and i s olume con en a ime 𝑡1.
]Balance law
A quan i y is said o sa is y a balance law,o obebalanced, i he
ollowing equali y holds o any closed con ol su ace and olume, and
any coo dina e imes 𝑡0and 𝑡1:
olume con en (𝑡1)= olume con en (𝑡0)+∫𝑡1
𝑡0
in lux(𝑡)d𝑡+∫𝑡1
𝑡0
supply(𝑡)d𝑡(5.2)
Fo example, in he case o ene gy-mass wi h olume con en
𝐸
, in lux
𝛷
,
supply ℛ, he balance law would be
𝐸(𝑡1)=𝐸(𝑡0) + ∫𝑡1
𝑡0
𝛷(𝑡)d𝑡+∫𝑡1
𝑡0
ℛ(𝑡)d𝑡 .
The meaning o a balance law is in ui i e: he amoun o quan i y in
he inal con ol olume:
𝐸(𝑡1)
, mus be equal o he amoun in he ini ial
con ol olume:
𝐸(𝑡0)
, plus he o al amoun ha lowed in h ough he
su ace be ween hose imes:
∫𝑡1
𝑡0𝛷(𝑡)d𝑡
, plus he o al amoun ha was
c ea ed in he olume: ∫𝑡1
𝑡0ℛ(𝑡)d𝑡.
129
5. Physical laws 5.5 Balance laws
We see ha a conse a ion law is a special, powe ul case o a balance
law. Le ’s make his connec ion explici :
]Connec ion be ween balance and conse a ion laws
A quan i y is said o sa is y a conse a ion law i i sa is ies a balance
law and i s supply is always ze o, o any closed con ol su ace and
olume and any coo dina e imes 𝑡0and 𝑡1.
A conse a ion law is powe ul because i means ha he supply is always
known in ad ance and has an ex emely simple alue: ze o. The impo an
consequence o his ac is ha i allows us o p edic he amoun o
quan i y in a inal olume by knowing wha happens only on he bounda y
o he olume du ing he ime lapse. A gene al balance law, ins ead, equi es
us o know also wha happens a e e y poin wi hin he olume du ing
he ime lapse: we need o know whe he some amoun o quan i y was
c ea ed o des oyed wi hin he olume.
A conse a ion law in ol es
only knowledge abou he ini-
ial and inal con ol olumes
( ep esen ed by he ed
ha ched disks), and abou he
closed con ol su ace du ing
he ime lapse (cu ed ed
con ou s); bu no abou he
con ol olume du ing he
ime lapse. A balance law in-
s ead in ol es also his addi-
ional in o ma ion.
Fo hese easons a balance law is in some espec s mo e i ial han a
conse a ion law. I we measu e ha
𝐸(𝑡0) − 𝐸(𝑡1) − ∫𝑡1
𝑡0𝛷(𝑡)d𝑡
is no ze o
– so he e’s no conse a ion law – we can always say ha some amoun o
quan i y mus ha e been c ea ed o des oyed wi hin he con ol olume
be ween
𝑡0
and
𝑡1
. An ex ensi e quan i y can he e o e always be said o
sa is y a balance law. A balance is no i ial, howe e , i we ha e some
o he physical law ha ells us in ad ance how he amoun c ea ed o
des oyed a each ins an , ℛ(𝑡), can be calcula ed.
-Supplies a e e y di e en om luxes
One could objec : “why don’ you jus pu he lux
𝛷(𝑡)
and he supply
ℛ(𝑡)
oge he , adding he wo in eg als in equa ion
(5.2)
? Wouldn’ you
ge
𝐸(𝑡1)=𝐸(𝑡0) + ∫𝑡1
𝑡0𝛷(𝑡) + ℛ(𝑡)d𝑡=0
which looks like a conse a ion law?”
Un o una ely he ma hema ical exp ession abo e wouldn’ be a conse -
a ion law, despi e i s appea ance. The poin is his: he lux
𝛷
in ol es
only wha ’s happening on he con ol su ace; he supply
ℛ
, ins ead,
in ol es wha ’s happening wi hin he con ol olume. A conse a ion
130
5. Physical laws 5.5 Balance laws
law does no equi e us o know wha ’s happening wi hin he con ol
olumes, excep a he ini ial and inal ime.
Remembe , mo eo e , ha luxes always sa is y a symme y p inciple
§4.6 p.98
, bu supplies in gene al do no sa is y any analogous symme y
p inciple.
Balance law o ec o quan i ies
Le usnowconside an ex ensi e ec o quan i ylikemomen um,which will
be e y impo an in ou u u e in es iga ions. I s ini ial olume con en
is deno ed
𝑷
; i s in lux, also called su ace o ce,
𝑭(𝑡)
; and i s supply,
also called body o ce,
𝑮
. They a e ime-dependen ec o s, de ined in a
coo dina e sys em (𝑡, 𝑥, 𝑦, 𝑧):
𝑷(𝑡)=
𝑃𝑥(𝑡)
𝑃𝑦(𝑡)
𝑃𝑧(𝑡)
𝑭(𝑡)=
𝐹𝑥(𝑡)
𝐹𝑦(𝑡)
𝐹𝑧(𝑡)
𝑮(𝑡)=
𝐺𝑥(𝑡)
𝐺𝑦(𝑡)
𝐺𝑧(𝑡)
.
The balance law o a ec o quan i y like momen um has exac ly he
same exp ession we al eady know:
𝑷(𝑡1)=𝑷(𝑡0) + ∫𝑡1
𝑡0
𝑭(𝑡)d𝑡+∫𝑡1
𝑡0
𝑮(𝑡)d𝑡(5.3)
wi h he only di e ence ha he quan i ies in ol ed a e ec o s. So i
co esponds o a sys em o h ee balance equa ions, one pe componen :
𝑃𝑥(𝑡1)=𝑃𝑥(𝑡0) + ∫𝑡1
𝑡0
𝐹𝑥(𝑡)d𝑡+∫𝑡1
𝑡0
𝐺𝑥(𝑡)d𝑡
𝑃𝑦(𝑡1)=𝑃𝑦(𝑡0) + ∫𝑡1
𝑡0
𝐹𝑦(𝑡)d𝑡+∫𝑡1
𝑡0
𝐺𝑦(𝑡)d𝑡
𝑃𝑧(𝑡1)=𝑃𝑧(𝑡0) + ∫𝑡1
𝑡0
𝐹𝑧(𝑡)d𝑡+∫𝑡1
𝑡0
𝐺𝑧(𝑡)d𝑡
(5.4)
131
5. Physical laws 5.6 Examples
£Balance laws in Gene al Rela i i y
Conse a ion and balance laws appea simple om he poin o iew o Rela i i y
Theo y. F om a ou -dimensional space ime pe spec i e, a 3D olume is a egion,
called hype su ace, ha ing one less dimension han space ime. Bu a mo ing 2D
su ace ollowed h ough ime is also a space ime egion ha has one less dimension
han space ime: wo spa ial dimensions and one empo al one. Thus he dis inc ions
among he 3D egion a
𝑡0
, he mo ing 2D su ace be ween
𝑡0
and
𝑡1
, and he 3D egion
a
𝑡1
, disappea : hey a e seen o be jus di e en pa s o he same h ee-dimensional
hype su ace. We pe cei e some pa s o his hype su ace as belonging “ o he same
ime”, showing hei h ee dimensions all a once; and o he pa s o i as ex ending
h ough ime, showing only wo dimensions a any ime. In ac , di e en obse e s
make his di ision in di e en ways.
And om a space ime pe spec i e, he amoun o a quan i y
𝐸(𝑡0)
o
𝐸(𝑡1)
wi hin a
3D egion is seen as a lux h ough ime; so i s appa en di e ence om he lux
𝛷
also disappea .
Space ime ep esen a ion o
he e olu ion o a closed
con ol su ace, con aining
some poin like objec s (adap-
ed om Misne , Tho ne, e
al. 2017)
£Wha abou he magne ic lux?
In he case o magne ic lux, he idea o a conse a ion law is analogous, bu is
o mula ed wi h one less spa ial dimension, and wi h a di e en no ion o o ien a ion: we
conside a con ol su ace ha exis s be ween imes
𝑡0
and
𝑡1
; his su ace has a closed
con ol cu e as bounda y. The magne ic lux u ns ou o be a quan i y o which i ’s
possible o ask how much o i is in e sec ing a su ace, and how much o i is c ossing
a closed cu e. One way o unde s and his is o imagine magne ic lux as a bundle o
ubes o lines ha a e ei he closed o ex end o in ini y. I will be discussed in dep h
in chap e 9.
5.6 Examples
Le us now s udy a couple o example applica ions o balance laws.
In o de o apply a balance law we mus choose one o mo e closed
con ol su aces and co esponding con ol olumes. Fix a coo dina e
sys em
(𝑡, 𝑥, 𝑦, 𝑧)
. We p e iously discussed ha wo ypical choices o
con ol olumes and su aces
§4.4 p.93
wi h espec o a coo dina e sys em
a e: (a) a mo ing one, usually “w apping” some solid objec ; (b) a s a ic
one. Le ’s see how a balance law is applied in hese wo cases. Fo he
quan i y o s udy we choose momen um, so ha we can ge immedia ely
ge acquain ed wi h handling ec o equa ions.
132
5. Physical laws 5.6 Examples
Mo ing con ol su ace
Le ’s say ha coo dina es
𝑥
,
𝑦
ha e ho izon al di ec ions, and
𝑧
an upwa d
di ec ion.Conside a lying ennisball.Choosean imagina y, closedcon ol
su ace ha pe ec ly w aps he ennis ball and mo es wi h i .
A a pa icula ime ins an
𝑡0
he ennis ball has momen um
[0,1.70,0.98]Ns
; by his we mean ha he con ol olume co esponding
o he ball con ains ha amoun o momen um.
While he ennis ball is lying, we assume ha he ne in lux o mo-
men um h ough he con ol su ace is ze o, o ins ance because he ball
is in a acuum. Bu he e’s a supply o momen um wi hin he olume,
cons an in ime, equal o
[0,0,−0.579]N
. This supply, as we’ll see la e ,
exis because he ennis ball is in he g a i a ional ield o he Ea h.
The ennis ball lies o wo seconds. How much is he momen um o
he ennis ball a he end o his ime lapse? Le ’s call his ime 𝑡1.
F om he desc ip ion abo e we ha e hese da a:
𝑡0=0s , 𝑡1=2s ,
𝑷(𝑡0)=
0
1.70
0.98
Ns ,
𝑭(𝑡)=
0
0
0
N(cons . in ime) ,𝑮(𝑡)=
0
0
−0.579
N(cons . in ime) .
The balance law
(5.3)
allows us o ind he amoun o momen um in he
ennis ball a ime 𝑡1=2s:
𝑷(𝑡1)=𝑷(𝑡0) + ∫𝑡1
𝑡0
𝑭(𝑡)d𝑡+∫𝑡1
𝑡0
𝑮(𝑡)d𝑡
=𝑷(𝑡0) + 𝑭· (𝑡1−𝑡0) + 𝑮· (𝑡1−𝑡0)(because 𝑭,𝑮a e cons an in ime)
=
0
1.70
0.98
Ns +
0
0
0
N·2s +
0
0
−0.579
N·2s
=
0
1.70
−0.18
Ns .
133
5. Physical laws 5.6 Examples
The e o e wo seconds la e he ball’s momen um has he same
𝑥
- and
𝑦
-componen as i had ini ially; bu now i s
𝑧
-componen poin s downwa d
– which means ha he ball is also mo ing downwa d,no only ho izon ally.
S a ic con ol su ace
Take again he p e ious example o a cylind ical block o ice
§5.4 p.127
mo ing downwa d, in a acuum, du ing a lapse o ime o
0.5s
, depic ed
again in he side igu e. In he p e ious example we discussed he ne
amoun o ma e ha c osses a s a ic, closed con ol su ace a he inal
loca ion o he ice block. Now we a e ins ead in e es ed in he ne amoun
o momen um ha c osses he same su ace. We shall he e o e use he
balance o momen um (5.3).
Suppose we ha e his in o ma ion:
𝑡0=0s , 𝑡1=0.5s ,𝑷(𝑡0)=
0
0
0
Ns ,𝑷(𝑡1)=
0
0
−4.72
Ns .
The exp ession o
𝑷(𝑡0)
says he ini ial momen um con en in he con ol
olume is ze o. This makes sense, because he con ol olume is ini ially
emp y o ma e . The exp ession o
𝑷(𝑡1)
says ha he inal momen um
con en in he con ol olume is non-ze o: i di ec ed ully downwa d,
wi h magni ude
4.72Ns
. This also makes sense, because a his ime in
he con ol olume he e’s ma e mo ing downwa d.
Can we ind he ime-in eg a ed in lux o supply o momen um du ing
he ime lapse o
0.5s
? S ic ly speaking, om he da a ma hema ically
exp essed abo e, he answe is no: o ind he ime-in eg a ed in lux we
would need he ime-in eg a ed supply, and ice e sa. This si ua ion
illus a es wha we said p e iously: balance laws gene ally equi e mo e
in o ma ion han conse a ion laws.
Suppose we a e old ha he ime-in eg a ed supply o momen um
wi hin he con ol olume, du ing he ime lapse, is di ec ed downwa d and
has magni ude 1.57Ns. Tha is,
∫𝑡1
𝑡0
𝑮(𝑡)d𝑡=
0
0
−1.57
Ns .
134
5. Physical laws 5.6 Examples
-
Pay a en ion o he ac ha his is no a lux o momen um: his is no
momen um ha “en e s” om he op o he con ol su ace because
he ice block is en e ing he e. This is momen um c ea ed (by g a i y)
in he pa s o he olume whe e he e is ma e .
The in lux h ough he op su ace can now be ound using he balance
law (5.3):
∫𝑡1
𝑡0
𝑭(𝑡)d𝑡=𝑷(𝑡1) − 𝑷(𝑡0) − ∫𝑡1
𝑡0
𝑮(𝑡)d𝑡
=
0
0
−4.72
Ns −
0
0
0
Ns −
0
0
−1.57
Ns
=
0
0
−3.15
Ns
This is he momen um ha “en e s” h ough he closed con ol su ace.
We can in ui i ely deduce ha i is speci ically en e ing h ough he op
pa o he con ol su ace.
Ano he example wi h a mo ing con ol su ace
Le us conside he same physical si ua ion as in he p e ious example, bu
now le ’s choose a mo ing con ol su ace ins ead, as in he i s example
wi h he ennis ball. We choose a closed con ol su ace ha igh ly w aps
he block o ice a all imes be ween 𝑡0and 𝑡1.
An impo an ema k: since we a e now choosing a di e en con ol
su ace and olume om he p e ious example, he alues o momen um
con en s and luxes may be di e en om he p e ious ones as well – hey
e e o di e en egions o space. To a oid ge ing con used, i ’s good
o deno e he amoun s in he p esen example wi h di e en symbols.
Le ’s unde line hem o ins ance; we could also use some o he g aphical
symbol, o simply change he le e s hemsel es.
Suppose ha we wan o know how much is he ime-in eg a ed supply
o momen um in he new, mo ing con ol olume, be ween imes
𝑡0
and
𝑡1
. In o de o ind i om he balance o momen um we need o know:
135
5. Physical laws 5.6 Examples
(a) he ini ial momen um con en , (b) he inal momen um con en , (c) he
ime-in eg a ed in lux.
We a e old ha he ice block ini ially has ze o momen um (because i ’s
eleased, and he e o e has ze o eloci y, exac ly a ha ins an ), and a he
inal ime i has a downwa d momen um o magni ude
4.72Ns
. We a e
also old ha he e is no ne lux o momen um, a any ime, h ough he
con ol su ace ha mo es along wi h he ice block. Ou da a a e he e o e
𝑡0=0s , 𝑡1=0.5s ,
𝑷(𝑡0)=
0
0
0
Ns ,𝑷(𝑡1)=
0
0
−4.72
Ns ,
𝑭(𝑡)=
0
0
0
N(cons . in ime) .
We ha e enough da a o ind he ime-in eg a ed supply o momen um
gene a ed wi hin he mo ing con ol olume, using he balance o mo-
men um:
∫𝑡1
𝑡0
𝑮(𝑡)d𝑡=𝑷(𝑡1) − 𝑷(𝑡0) − ∫𝑡1
𝑡0
𝑭(𝑡)d𝑡
=
0
0
−4.72
Ns −
0
0
0
Ns −∫𝑡1
𝑡0
0
0
0
N d𝑡
=
0
0
−4.72
Ns
As we ema ked abo e, e en i he physical e en is exac ly he same,
he momen um lux and supply in he p esen analysis a e di e en om
he p e ious analysis, because hey e e o di e en imagina y con ol su aces
and olumes. Compa e he ime-in eg a ed in luxes o he wo analyses:
∫𝑡1
𝑡0
𝑭(𝑡)d𝑡=
0
0
−3.15
Ns ≠∫𝑡1
𝑡0
𝑭(𝑡)d𝑡=
0
0
0
Ns .
136
5. Physical laws 5.7 Balance laws: diffe en ial exp ession
and he ime-in eg a ed supplies:
∫𝑡1
𝑡0
𝑮(𝑡)d𝑡=
0
0
−1.57
Ns ≠∫𝑡1
𝑡0
𝑮(𝑡)d𝑡=
0
0
−4.72
Ns
-Fluxes and supplies depend on he con ol su ace and olume
The ime-in eg a ed lux and ime-in eg a ed supply ha appea in
balance laws a e s ic ly dependen on he closed con ol su ace and
olume ha we choose.
The e o e, i we analyse he same physical phenomenon wi h a di e en
se o con ol su aces and olumes, we canno expec he esul s o
calcula ions wi h he old se o be alid o he new one.
«Exe cise 5.3
Conside once mo e he block o ice analysed in he p e ious wo
examples.
This ime choose a s a ic closed con ol su ace ha coincides wi h he
ini ial posi ion o he block. This su ace he e o e includes all ice a ime
𝑡0
, bu is emp y a ime
𝑡1
. The ini ial and inal momen um con en s a e
ze o.
Calcula e he in eg a ed supply o his new con ol su ace be ween
𝑡0
and 𝑡1.
(Hin : conside he esul s om he example o § 5.6, and use he symme y o
lux o ind he lux h ough he bo om pa o he con ol su ace o he p esen
exe cise. Recall also ha he luxes h ough he side and op su aces a e ze o.)
5.7 Balance laws: diffe en ial exp ession
A balance law such as
𝐸(𝑡1)=𝐸(𝑡0) + ∫𝑡1
𝑡0
𝛷(𝑡)d𝑡+∫𝑡1
𝑡0
ℛ(𝑡)d𝑡
o a conse a ion law such as
𝑁(𝑡1)=𝑁(𝑡0) + ∫𝑡1
𝑡0
𝐽(𝑡)d𝑡
137
5. Physical laws 5.8 Se en uni e sal balance laws
cosmology, o who knows wha else. E e y physical phenomenon in ol es
a leas one o hese se en balances in i s physical desc ip ion.
£Concise ma hema ical o m o he uni e sal balances
The se en balances can be exp essed e y concisely i we use he language o di e en ial
o ms
8
. These a e geome ic objec s ha associa e a numbe o any cu e, su ace, o
olumeo ou choice.Thebalances o ma e ,momen um,ene gy,angula momen um,
elec ic cha ge, magne ic lux, en opy hen ake on hese e y concise exp essions:
d𝑁=𝒜d𝒬=ℐdℬ=−ℰ d𝐸=ℛd𝑷=𝑮d𝑳=𝓣d𝑆≥0
whe e he symbols on he le o he equa ions a e aken in a ou -dimensional sense.
I you wan o lea n mo e abou di e en ial o ms, ake a look a he books by Bu ke
1987;1995 and Bossa i 1991.
The se en balances o desc ip ion and p edic ion
The se en uni e sal balances go e n e e y physical phenomenon. Ye his
doesn’ mean ha all o hem a e always used explici ly in he desc ip ion
o p edic ion o physical phenomena.
•
Fo some physical phenomena, all he se en uni e sal balances en e
ou calcula ions.
The desc ip ion o how a com-
mon ligh e wo ks, hanks
o piezoelec ici y
9
, equi es
mo e o less all se en uni e -
sal laws o be explici ly ac-
coun ed o .
•
Fo some o he physical phenomena, some o he se en balances do
no appea explici ly in ou calcula ions. Bu hey s ill en e implici ly in he
way we choose o se up o desc ibe he phenomenon.
Fo example, we may choose con ol olumes o con ol su aces in such
a way ha some conse a ion laws a e au oma ically sa is ied. A ypical
case is he choice o con ol su ace a ound a gi en objec (ma e ), which
gua an ees ha he law o conse a ion o ma e is au oma ically sa is ied.
As ano he example, some imes we simpli y a physical phenomenon o
one spa ial dimension only. Think o when we h ow a ball e ically in
he ai , and only conside i s heigh om he g ound. In such a case we
can make some p edic ions using only he balance o ene gy, appa en ly
a oiding he balance o momen um. Bu in eali y, he ac ha he ball
can be conside ed as mo ing e ically is possible because momen um
is balanced in he ho izon al di ec ions. The balance o momen um is
he e o e s ill necessa y o his p edic ion, bu i has silen ly been aken
ca e o .
144
5. Physical laws 5.8 Se en uni e sal balance laws
in eg al exp ession di e en ial exp ession
ma e
𝑁(𝑡1)=𝑁(𝑡0) + ∫𝑡1
𝑡0
𝐽(𝑡)d𝑡+∫𝑡1
𝑡0
𝒜(𝑡)d𝑡d𝑁(𝑡)
d𝑡=𝐽(𝑡) + 𝒜(𝑡)
elec ic
cha ge
𝒬(𝑡1)=𝒬(𝑡0) + ∫𝑡1
𝑡0
ℐ (𝑡)d𝑡d𝒬(𝑡)
d𝑡=ℐ (𝑡)
magne ic
lux
ℬ(𝑡1)=ℬ(𝑡0) − ∫𝑡1
𝑡0
ℰ(𝑡)d𝑡dℬ(𝑡)
d𝑡=−ℰ(𝑡)
momen um
𝑷(𝑡1)=𝑷(𝑡0) + ∫𝑡1
𝑡0
𝑭(𝑡)d𝑡+∫𝑡1
𝑡0
𝑮(𝑡)d𝑡d𝑷(𝑡)
d𝑡=𝑭(𝑡) + 𝑮(𝑡)
ene gy
𝐸(𝑡1)=𝐸(𝑡0) + ∫𝑡1
𝑡0
𝛷(𝑡)d𝑡+∫𝑡1
𝑡0
ℛ(𝑡)d𝑡d𝐸(𝑡)
d𝑡=𝛷(𝑡) + ℛ(𝑡)
angula
momen um
𝑳(𝑡1)=𝑳(𝑡0) + ∫𝑡1
𝑡0
𝑴(𝑡)d𝑡+∫𝑡1
𝑡0
𝓣(𝑡)d𝑡d𝑳(𝑡)
d𝑡=𝑴(𝑡) + 𝓣(𝑡)
en opy
𝑆(𝑡1) ≥ 𝑆(𝑡0) + ∫𝑡1
𝑡0
𝛱(𝑡)d𝑡d𝑆(𝑡)
d𝑡≥𝛱(𝑡)
Table 5.1 The se en uni e sal balance laws. These o mulae a e alid in New onian
mechanics, Gene al Rela i i y, and e en quan um heo y i hei symbols a e in e p e ed
as ‘s a is ical ope a o s’.
145
5. Physical laws 5.9 Cons i u i e ela ions
•
Fo s ill o he physical phenomena, some o he se en balances may no
be equi ed because we do no need he kind o physical in o ma ion hey
p o ide. We saw an example o his wi h he la - y e p oblem
§5.4 p.126
,
whe e we only used he conse a ion o ma e bu we we en’ in e es ed
in wha happened o o he quan i ies like ene gy o momen um, o in how
he y e was mo ing.
Some physical phenomena may be equally p edic ed using ei he one
pa icula subse o he se en balances, o a di e en subse , as we please.
Fo ins ance, a gi en p oblem migh be sol ed using conse a ion o ma e
and balance o momen um, o al e na i ely by using conse a ion o ma e
and balance o ene gy. We shall see examples o all hese possibili ies in
la e applica ions.
Bu he ac ha he se en uni e sal balances go e n e e y physical
phenomenon doesn’ mean ha hey can be used alone, by hemsel es.
In he as majo i y o cases hey need o be augmen ed by app op ia e
cons i u i e ela ions, which we discuss in he nex sec ion.
In he nex chap e s we shall explo e and apply he se en uni e sal bal-
ance laws in mo e de ail. Fo each o hem we shall ecall i s ma hema ical
exp ession, discuss some cons i u i e ela ions ha a e commonly used
wi h i , and examine some example applica ions.
5.9 Cons i u i e ela ions
In he p e ious sec ions we s udied he ma hema ical o m o balance
and conse a ion laws, and ound ou ha se en balances a e o special
impo ance.
F om hei o mula ion and om he examples, you no iced ha each
balance law connec s he olume con en , lux, supply o one ex ensi e
quan i y a di e en imes. Fo ins ance, he balance law o ene gy
𝐸(𝑡1)=𝐸(𝑡0) + ∫𝑡1
𝑡0
𝛷(𝑡)d𝑡+∫𝑡1
𝑡0
ℛ(𝑡)d𝑡
connec s he olume con en s
𝐸
o ene gy, he lux
𝛷
o ene gy, he supply
ℛ
o ene gy. I does no in ol e, say, he olume con en
𝑁
o ma e , o he
lux 𝑭o momen um.
Bu we know ha he e mus also exis connec ions among he amoun s
o di e en quan i ies. This ac is implici in many exp essions we use
146
5. Physical laws 5.9 Cons i u i e ela ions
e e yday, like “ he p essu e o ai ” (p essu e is momen um lux, ai is
ma e ), o “ he ene gy o he ba e y” ( he ba e y is made o ma e , and
in ol es elec ic cha ge). In p e ious sec ions we o en made s a emen s
such as “ he momen um is ze o, because he e’s no ma e ”.
Physical laws ha connec di e en kinds o quan i y a e called con-
s i u i e ela ions; we b ie ly discussed hem be o e
§5.2 p.120
. As was men-
ioned in ha discussion, he e m ‘cons i u i e’ ac ually e e s no o he
cha ac e is ic o connec ing di e en quan i ies, bu o he ac ha each o
hese laws ypically applies only o speci ic si ua ions:
]Cons i u i e ela ion
Acons i u i e ela ion, also called cons i u i e equa ion, o closu e equa-
ion, o cons i u i e p ope y, is a physical ela ionship o ha is ue
only unde speci ic condi ions; o example: only o speci ic physical
phenomena, o only on speci ic scales o space and ime, o only o
con ol olumes o su aces o pa icula sizes o shapes, o only o
speci ic anges o measu emen p ecision. The e o e hey a e o en used
in speci ic physical heo ies.
Cons i u i e ela ions exp ess he amazing di e si y o physical phenom-
ena ha we obse e a ound and wi hin us. Fo example he ac ha a
body o wa e can easily change shape, as opposed o a block o conc e e;
o ha we can s o e elec ic ene gy in a ba e bu no in a piece o wood.
The di e ences be ween s a es o ma e
11
– solid, liquid, gas, plasma, and
Fou s a es o ma e , a ising
om di e en cons i u i e e-
la ions (image: Spi i 46910)
he e a e o he s – a ise om di e en cons i u i e ela ions.
Cons i u i e ela ions also exp ess app oxima ions ha we use in
desc ibing physical phenomena in pa icula condi ions, and he e o e
ma k he di e ence be ween specialized o app oxima e physical heo ies;
o example be ween New onian mechanics, which applies only o low
speeds and low ene gy-mass concen a ions (hence weak g a i a ional
ields and small space ime cu a u e), and Gene al Rela i i y, which
applies on mos i no all known scales, including cosmological ones.
When we ead ha a new physical phenomenon has been disco e ed,
usually ha means ha a new cons i u i e ela ion has been disco e ed.
Depending on he speci ic scien i ic ield you’ll wo k in, you’ll lea n some
cons i u i e ela ions in mo e de ail han o he s.
Cons i u i e ela ions come in a g ea a ie y o ma hema ical o ms.
Some o hem a e simple algeb aic ela ions be ween he olume con en ,
147
5. Physical laws 5.9 Cons i u i e ela ions
o lux, o supply o one quan i y, and he olume con en , o lux, o
supply o ano he . Some cons i u i e ela ions in ol e de i a i es; some
in ol e in eg als.
Many cons i u i e ela ions connec olume con en s, luxes, supplies
o di e en p imi i e quan i ies, no di ec ly, bu h ough he in e media y
o de i ed o auxilia y quan i ies
§3.11 p.79
such as a eas and olumes,
empe a u e, eloci y, p essu e, pola iza ion, magne iza ion. We shall see
some examples below.
A e y powe ul cha ac e is ic o many cons i u i e ela ions is ha
hey connec he olume con en o a p imi i e quan i y a a gi en ime
wi h he lux o supply o ano he p imi i e quan i y a he same ime. Fo
ins ance, a cons i u i e law may allow us o ind he lux o ma e ac oss
some con ol su ace a ime
𝑡
:
𝐽(𝑡)
, om he con en o momen um in
a neighbou ing con ol olume a he same ime
𝑡
:
𝑷(𝑡)
; his example is
u he discussed below. This cha ac e is ic g ea ly ex ends he p edic i e
powe o balance laws when used oge he wi h cons i u i e ela ions, as
we shall see in chap e 6.
Examples
Le us b ie ly e eal some cons i u i e equa ions ha ha e aci ly been
used in examples o he p e ious sec ions. This is only an o e iew; we
shall s udy hese cons i u i e ela ions a leng h in he nex chap e s. All
hese cons i u i e ela ions a e alid only in “New onian app oxima ion”,
ha is, when speeds a e much lowe han he speed o ligh ; g a i a ion is
weak, as i is on Ea h; and any p esen elec omagne ic ields a e enough
weak. No e how all ela ions a e only alid o pa icula kinds o con ol
olumes o su aces.
Cons i u i e ela ion o mass-ene gy and ma e .
I a small con ol
olume con ains an amoun o ma e
𝑁
, hen i also con ains an amoun
o es mass-ene gy
𝑚=𝜌𝑁
whe e
𝜌
, called mola mass, is app oxima ely a cons an ha depends on
he kind o ma e . This cons i u i e ela ion is he eason why an amoun
o ma e is o en quan i ied in e ms o mass.
Cons i u i e ela ion o momen um and ma e .
One o he mos
used cons i u i e ela ions connec s he amoun o ma e in a small
148
5. Physical laws 5.9 Cons i u i e ela ions
con ol olume, o he lux o ma e h ough i s su ace, wi h he amoun
o momen um in he olume. I can be s a ed in se e al ways. He e a e
wo al e na i es: he i s alid o a mo ing con ol olume, he second o
as a ic one.
Fi s o mula ion. Take a small con ol olume mo ing wi h eloci y
𝒗
,
wi h speed much smalle han ligh ’s. I his con ol olume con ains an
amoun o ma e
𝑁
and he e’s no lux o ma e h ough i s su ace, hen
i also con ains, o a e y good app oxima ion, an amoun o momen um
𝑷=𝑚𝒗.
Old ex books ake his o -
mula as he de ini ion o mo-
men um.
Second o mula ion. Take asmalls a ic cuboidcon ol olumeo cubical
shape and sides pa allel o he coo dina e axes. The olume has size
𝑉
and
each o i s sides a ea
𝐴
. I he luxes o ma e h ough he sides o hogonal
o he h ee coo dina es, in a posi i e di ec ion, a e
[𝐽𝑥, 𝐽𝑦, 𝐽𝑧]
, hen he
con ol olume con ains momen um
𝑷=𝜌[𝐽𝑥, 𝐽𝑦, 𝐽𝑧]𝑉
𝐴
again only i weak elec omagne ic ields a e p esen . This ela ion is
connec ed o he one be ween eloci y and he lux o ma e §4.15 p.116.
Cons i u i e ela ion o supply o momen um.
I a small con ol
olume con ains an amoun o es mass-ene gy
𝑚
, hen i also has a
supply o momen um
𝑮=𝑚𝑔
This cons i u i e ela ion exp esses he g a i a ional olume o ce. The ec o
𝒈
is called he accele a ion o ee all. I exp esses he g a i a ional ield,
ha is, space ime cu a u e. I gene ally depends on ime, posi ion, and
he mo ion o he ma e o elec omagne ic ield ha ing mass-ene gy
𝑚
,
and he coo dina e sys em.
Fo physical phenomena on, o close o, Ea h’s su ace, in a coo dina e
sys em
(𝑡, 𝑥, 𝑦, 𝑧)
ixed he g ound and wi h
𝑧
poin ing upwa d, he ec o
𝒈
can be aken o be app oxima ely cons an ; i poin s owa ds he g ound:
𝒈=−𝑔
0
0
1
wi h 𝑔:=|𝒈|≈9.8N/kg ≡9.8m/s2.
In mo e p ecise applica ions in geodesy
12
his ec o may depend on
The exp ession o
𝒈
used a
NASA o sa elli e and space-
c a mo ion (Moye 2000).
149
5. Physical laws 5.10 Summa y o diffe ences be ween he se en balance laws and cons i u i e ela ions
la i ude, longi ude, al i ude, and e en ime, owing o Ea h’s in e nal
mo ion. Fu he away om Ea h’s, he exp ession o
𝒈
becomes mo e
complex. We shall discuss his u he in chap e 10.
Cons i u i e ela ion o ene gy-mass o ma e .
Take a small con-
ol olume on Ea h’s su ace con aining an amoun o ma e
𝑁
, and such
ha he e is no lux o ma e
𝐽
ac oss i s su ace. In a coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
whe e
𝑧
poin s upwa d, his con ol olume also con ains an
amoun o ene gy-mass
Remembe ha
𝒗2≡𝒗·𝒗≡ |𝒗|2≡𝑣2.
𝐸=𝑚𝑐2+𝑈o+𝑈+1
2𝑚𝒗2+𝑚𝑔𝑧 .
The i s e m in his sum is called es ene gy-mass (a huge amoun
§3.6 p.68
o ene gy in o dina y si ua ions), he hi d e m
𝑈
is called in e nal ene gy,
he ou h is called kine ic ene gy, he i h g a i a ional po en ial ene gy. The
sepa a ion be ween he es - and in e nal-ene gy e ms is ac ually a bi a y;
he a bi a y e m
𝑈o
se es as a ze o o ene gy. The es mass-ene gy
𝑚
and he in e nal ene gy 𝑈depend on he amoun o ma e 𝑁.
5.10
Summa y o diffe ences be ween he se en
balance laws and cons i u i e ela ions
Le ’s summa ize he mos impo an ypical di e ences be ween he se en
uni e sal balance laws o ou se en p imi i e quan i ies on one side, and
cons i u i e ela ions on he o he side:
Uni e sal balance law Cons i u i e ela ion
Se en A i ually in ini e numbe
Valid o e e y phenomenon Valid o a speci ic phenomenon
Valid o any con ol olume and
su ace
O en alid only o con ol olumes o
su aces o speci ic size, shape, loca ion
Connec s con en s, luxes, supplies o
he same quan i y
O en connec s con en s, luxes,
supplies o di e en quan i ies
Connec s con en s, luxes, supplies a
di e en imes
O en connec s con en s, luxes,
supplies a he same imes
Used in all mode n heo ies Used in speci ic heo ies
150
5. Physical laws 5.11 New on’s laws
5.11 New on’s laws
Many books p esen , mainly ou o adi ion, a iew o physical laws based
on New on’s h ee axioms, o laws o mo ion, s a ed in his Philosophiæ
na u alis p incipia ma hema ica. I is he e o e con enien o ge acquain ed
wi h hei s a emen s. In he p esen sec ion we examine New on’s laws
and discuss hei limi a ions as a ounda ions o oday’s physical heo ies.
The s a emen s below a e om he ansla ion by Cajo i (New on 1974);
he o iginal La in, om he hi d edi ion o New on’s P incipia (New-
on 1726b), is epo ed on he side. We mus keep in mind ha hese
s a emen s a e inex icably connec ed wi h he es o de ini ions and eas-
oning gi en in he P incipia, and wi h he knowledge o he ime in which
hey we e w i en. This makes i di icul , o maybe e en meaningless, o
ully in e p e hem. See o ins ance he analysis by Smi h 2024.
Fi s law
LEX I. Co pus omne
pe se e a e in s a u suo
quiescendi el mo endi
uni o mi e in di ec um, nisi
qua enus illud a i ibus
imp essis cogi u s a um suum
mu a e.
LAW I. E e y body con inues in i s s a e o es , o o uni o m mo ion
in a igh line, unless i is compelled o change ha s a e by o ces
imp essed upon i .
This is usually called he ‘law o ine ia’, and i was accep ed al eady
some ime be o e New on. Today his law is o en iewed as a consequence
o he second law below, oge he wi h New on’s de ini ion o momen um:
i he e a e no o ces, hen he e is no change in ‘mo ion’, and i ‘mo ion’
is mass imes eloci y, hen he eloci y is cons an , in magni ude and
di ec ion. Bu he e a e also di e ing iews, which see i as an independen
s a emen abou he p ope ies o space o o absolu e mo ion, o abou
he di e ences be ween “ eal” and “ ic i ious” o ces. Edding on (1958)
eph ased he i s law oughly his way: E e y body con inues in i s s a e o
es o uni o m mo ion in a s aigh line, excep when i doesn’ .
F om a 21s -cen u y poin o iew, cen ed on he ela i i y o mo ion
and coo dina e sys ems and ames, hese deba es abou he i s law lose
some o hei physical impo ance.
Second law
LEX II. Mu a ionem mo us
p opo ionalem esse i mo ici
imp essæ, & ie i secundum
lineam ec am qua is illa
imp imi u .
LAW II. The change o mo ion is p opo ional o he mo i e o ce
imp essed; and is made in he di ec ion o he igh line in which ha
o ce is imp essed.
151
5. Physical laws 5.11 New on’s laws
This law will be u he discussed in chap e 10. I is some imes
ma hema ically epo ed as ‘
𝑭=𝑚𝒂
’, bu New on ne e w o e his
o mula, which mo eo e may no ully ep esen he second law.
Wi h some libe y we may in e p e his law as he balance o mo-
men um, bu wi h wo impo an wa nings.
Fi s , as a as I know, New on did no make any dis inc ion be ween
su ace o ces and olume o ces. The absence o his impo an dis inc ion
caused some di icul ies in applying New on’s laws o bodies capable o
de o ma ion, like luids. These di icul ies we e sol ed when he Be noullis
and especially Eule in he 1700s in oduced mo e clea ly he concep o
con ol su ace and su ace o ce, e en ually de eloped in ull gene ali y by
Cauchy in he 1800s. Mode n mechanical enginee ing and luid mechanics
would ha e been impossible wi hou his concep ual dis inc ion.
Imagina y su ace
𝐸𝐼𝑖𝐹
in
a luid, concei ed by Eule
o desc ibe in e nal p essu e.
F om Eule 1761.
Second, in he P incipia New on de ines ‘quan i y o mo ion’ as “a ising
om he eloci y and quan i y o ma e conjoin ly”, ha is, as ‘
𝑚𝒗
’, so in
his con ex he second law is speci ic o ma e . Bu we know oday ha
he exac exp ession o ma e is di e en , and ha elec omagne ic ields
ha e momen um as well.
Thi d law
LEX III. Ac ioni con a iam
sempe & æqualem esse
eac ionem : si e co po um
duo um ac iones in se mu uo
sempe esse æquales & in pa es
con a ias di igi.
LAW III. To e e y ac ion he e is always opposed an equal eac ion:
o , he mu ual ac ions o wo bodies upon each o he a e always equal,
and di ec ed o con a y pa s.
This is usually called he law o ac ion and eac ion. This law is only alid
o su ace o ces, ha is, o momen um lux; bu no o olume o ces, ha
is, momen um supply. Fo olume o ces his law is gene ally no ue, o
only alid in special app oxima ions and coo dina e choices. We al eady
men ioned ha his law can be iewed as he p inciple o symme y o
lux o momen um
§4.11 p.109
. The e o e an analogous law is ue o he
luxes o all o he quan i ies, no only o momen um.
In he 1900s li e a u e he e was a discussion on whe he he hi d law
could allow pai s o olume o ces on wo bodies as in he op side igu e:
equal and opposi e bu wi h a di ec ion ha doesn’ necessa ily align wi h
a line connec ing he wo bodies. O whe he he hi d law would only
allow o a di ec ion as depic ed in he bo om side igu e.
Today we know ha his discussion is somewha poin less: as al eady
said, he hi d law is in gene ally no ue o supplies o momen um.
152
5. Physical laws 5.11 New on’s laws
Mo eo e in a cu edspace imei becomesunclea wha a ‘line’ connec ing
wo bodies is. Fo su ace o ces – momen um lux – his si ua ion doesn’
a ise, because he lux conce ns he su ace as a whole, so he e is no
“applica ion poin ”.
Fu he limi a ions
Besides he sho comings men ioned in he commen s abo e, he e a e
u he limi a ions ha make New on’s laws insu icien oday as a gene al
basis o unde s and physical phenomena.
One basic limi a ion is ha hey do no co e all balances necessa y o
desc ibe mos mechanical phenomena. I we wan o s udy, o example:
he oceans, he a mosphe e, a ca engine, he s abili y o a b idge (all yea
ound), ae oplane ligh , he beha iou o ma e ials, and simila examples,
hen besides New on’s laws we need he balance o ene gy, o angula
momen um, and o ma e ; all o which a e independen o New on’s laws.
The need o a sepa a e balance o angula momen um, o ins ance, was
I you’ e cu ious abou he
disco e y o he balance o
angula momen um, ake a
look a T uesdell 1968a.
de ini i ely ecognized by he Be noullis and Eule in he 1600–1700s in
ying o s udy he beha iou o luids. The ecogni ion o he balance
o ene gy as a sepa a e law led o mode n he momechanics. Gene al
ela i i y la e showed ha a u he se o h ee balance equa ions mus
be aken in o accoun in conside ing ela i is ic mechanical phenomena.
The e o e New on’s laws accoun o less han hal o he uni e sal equa-
ions necessa y in mechanics. And we’ e no conside ing applica ions
in ol ing elec omagne ics.
Ano he un o una e limi a ion, no o New on’s laws pe se bu o he
way hey a e o en p esen ed and augh , is hei ma hema ical agueness.
You can ind New on’s second law a iously w i en as
𝑭=𝑚𝒂,d𝑚𝒗
d𝑡=𝑭, 𝑚d𝒗
d𝑡=𝑭,d𝑷
d𝑡=𝑭,∂𝜌𝒗
∂𝑡=−∇ · 𝑻+𝜌𝒈,
and o he a ia ions, some o which a e mo e gene al han o he s. The
eason o his a ie y is ha equa ions like he ones abo e come om
he combina ions o one law – he balance o momen um – wi h di e en
cons i u i e ela ions.
Some imes one hea s he s a emen “Jus use New on’s laws!” o “Jus
use New on’s second law!”. Seeing he limi a ions abo e, he ‘jus ’ in ha
s a emen is a li le idiculous. Fi s , one needs mo e laws han New on’s;
and second, i ’s always unclea wha p ecise o mula ha s a emen is
153
11. Balance o ene gy-mass 11.7 The exchange be ween in e nal, kine ic, g a i a ional po en ial ene gies
Bodies in mo ion. In he p eceding chap e s we ha e conside ed small
bodies o ma e such as a ennis ball. In many si ua ions whe e he mo ion
o a body is in ol ed, i s in e nal ene gy
𝑈
is app oxima ely cons an , and
he e o e we only ocus on i s kine ic
1
2𝑚𝒗2
and po en ial
𝑚𝑔𝑧
ene gies.
An exchange o ene gy can happen be ween hem. Fo ins ance, in a ennis
ball alling in a acuum, he g a i a ional ene gy is dec easing while he
kine ic ene gy is inc easing a he same a e: he e ical coo dina e o he
ball is dec easing, and i s downwa d eloci y inc easing.
In o he si ua ions he in e nal ene gy is no cons an , and he e a e
exchanges among all h ee ene gy componen s, as well as changes in he
o al ene gy. A ball bouncing on he loo is an example. We can clea ly
see ha i s e ical posi ion
𝑧
, and he e o e i s g a i a ional ene gy
𝑚𝑔𝑧
,
al e na ely dec eases and inc eases. The same is ue o i s eloci y
𝒗
and
kine ic ene gy
1
2𝑚𝒗2
, which a e ze o a he highes and lowes (bouncing)
poin s. Bu we also obse e ha he highes e ical posi ion o he ball
ge s lowe and lowe , un il he ball es s on he loo , which means ha i s
kine ic and g a i a ional ene gies a e bo h ze o. They ha e been con e ed
pa ly in in e nal ene gy
𝑈
o he ball, and pa ly ans e ed, ia ene gy
lux, o he in e nal ene gy o he loo .
Ano he example whe e his kind o ene gy con e sion and ene gy
lux a e impo an is skydi ing. The g a i a ional ene gy o a skydi e is
ob iously dec easing. I i we e comple ely con e ed o kine ic ene gy,
he skydi e would acqui e a dange ously excessi e alling speed. Ins ead,
his ene gy is luckily mainly ans e ed, ia ic ion, o he in e nal ene gy
o he su ounding ai , and pa ly con e ed in o in e nal ene gy o he
skydi e . Fo his eason he kine ic ene gy, and he e o e he alling
eloci y, o he skydi e emain cons an a e some ime.
«Exe cise 11.2
Askydi e jumps wi h ze o ini ial eloci y a an al i udeo
3000m
.Wha
would be he skydi e ’s eloci y a
1000m
, i he e we e no changes in
in e nal ene gy and no ene gy luxes?
Compa e he eloci y you ound and he ypical eloci y o
200km/h
ha a skydi e may ha e a 1000m, when he pa achu e is deployed.
Sp ings and ubbe bands. In s udying Hookean and non-Hookean
sp ings and ubbe bands in chap e 10, we said ha hey a e usually
256
11. Balance o ene gy-mass 11.8 Hea lux and mechanical powe
modelled as objec s wi hou mass. They he e o e ha e negligible kine ic
and po en ial ene gies. All hei ene gy is he e o e in e nal ene gy.
Gases. Ma e in a gaseous s a e is also o en modelled as ha ing negli-
gible mass. All i s ene gy is he e o e modelled as in e nal ene gy. This
is simila o he modelling o objec s like sp ings and ubbe bands. An
impo an di e ence is ha he in e nal ene gy o gases is ypically highly
dependen on empe a u e, whe eas he he empe a u e dependence o
sp ings can be neglec ed in many applica ions.
Solids and luids. In modelling ex ended ma e in a solid and luid
s a e, usually all h ee o ms o ene gy mus be aken in o accoun . They
can mo eo e be qui e a iable om one small con ol olume o ano he .
This is why hei modelling and simula ion can be ex emely complex.
11.8 Hea lux and mechanical powe
We shall s udy some cons i u i e ela ions o he ene gy lux h ough a
con ol su ace, which a e only alid unde h ee condi ions:
• he e’s ma e on bo h sides o he con ol su ace,
• his ma e has eloci y 𝒗(possibly ze o) on bo h sides,
• he e is no lux o ma e h ough he con ol su ace.
These h ee equi emen s migh appea as con adic o y: i he e’s no lux
o ma e , hen how can ma e ha e a non-ze o eloci y? So le ’s discuss
hem in mo e de ail i s .
One si ua ion is which he condi ions abo e can coexis is when he
con ol su ace is pa allel o he di ec ion o he eloci y
𝒗
, as illus a ed in
he side igu e, whe e he con ol su ace is ep esen ed by he dashed ed
line, and ma e as a g eyish ex u e. In his case he su ace may be s a ic;
ye he e’s no lux o ma e h ough i , and he ma e a ound i is mo ing
wi h eloci y
𝒗
. Imagine o ins ance a ic i ious ho izon al su ace in he
middle o a wa e s eam. No e ha he ma e on he wo sides doesn’
need o be o he same kind, hough. Fo ins ance he e could be wa e on
one side o he con ol su ace, and conc e e on he o he .
Ano he si ua ion in which h ee condi ions can coexis is when he
con ol su ace is also mo ing, wi h he same eloci y
𝒗
as he mo ing
ma e . See side igu e. Ob iously he e’s no lux o ma e h ough he
con ol su ace, ye he e’s ma e on bo h sides o i , mo ing wi h eloci y
257
11. Balance o ene gy-mass 11.8 Hea lux and mechanical powe
𝒗
. No e again ha he ma e on he wo sides doesn’ need o be o he
same kind.
This si ua ion occu s when a pe son pushes a doo , he imagina y
con ol su ace being be ween hand and doo ; o when a hamme hi s and
bends a piece o ho i on, he imagina y su ace being be ween hamme
and i on; o when a bike-pump pis on is p essed down igo ously, he
imagina y su ace being be ween he pis on and he ai wi hin he pump.
O he si ua ions in be ween he wo abo e a e also possible.
Now ha hese possible ela ionships be ween a con ol su ace and
ma e a ound i a e clea e , le ’s s udy an impo an cons i u i e o mula
o he lux o ene gy h ough a con ol su ace
]Hea lux and mechanical powe
Fix a coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
, and conside an imagina y con ol
su ace sa is ying he ollowing condi ions:
•elec ic cha ges and elec omagne ic ields a e negligible;
• he e’s ma e on bo h sides o he con ol su ace;
• he e is no lux o ma e h ough he con ol su ace;
•
ma e in con ac wi h i has coo dina e eloci y
𝒗
, he same a
e e y poin o he su ace;
he las condi ion is ue, o ins ance, i he su ace is small enough.
Now choose a c ossing di ec ion o he su ace, in o de o de ine luxes.
I
𝑭
is he lux o momen um h ough his con ol su ace, hen he
ene gy lux 𝛷 h ough he su ace is gi en by
𝛷=𝑄+𝑭·𝒗.(11.4)
The e m
𝑄
is called he hea lux o hea ing; he e m
𝑭·𝒗
is called he
mechanical powe o wo king ansmi ed by he o ce 𝑭.
Hea lux
𝑄
and o ce
𝑭
a e ypically speci ied by u he cons i u i e
ela ions o bounda y condi ions
§6.5 p.166
. One pa icula bounda y
condi ionis he anishingo hehea lux:
𝑄=0J/s
.When hisbounda y
condi ion applies, he ene gy lux is called adiaba ic.
We see ha he momen um lux, o su ace o ce,
𝑭
appea s in he
cons i u i e ela ion o he ene gy lux. The e o e, he p esence o a
momen um lux h ough a su ace can con ibu e o he lux o ene gy
h ough i : we say ha “con ac o ces do wo k”.
258
11. Balance o ene gy-mass 11.8 Hea lux and mechanical powe
In he in eg al exp ession o he balance o ene gy, he ime-in eg a ed
lux
§4.14 p.114 ∫𝑡1
𝑡0𝛷(𝑡)d𝑡
appea s. Gi en he cons i u i e ela ion
(11.4)
, his
in eg al can be w i en as he sum o wo in eg als:
∫𝑡1
𝑡0
𝛷(𝑡)d𝑡=∫𝑡1
𝑡0
𝑄(𝑡)d𝑡+∫𝑡1
𝑡0
𝑭(𝑡) · 𝒗(𝑡)d𝑡 .
These in eg als ha e special names:
]Hea and mechanical wo k
The ime in eg al o he hea lux be ween imes 𝑡0and 𝑡1,
∫𝑡1
𝑡0
𝑄(𝑡)d𝑡 ,
is called he hea ans e ed h ough he su ace du ing ha ime
in e al.
The ime in eg al o he mechanical powe be ween imes 𝑡0and 𝑡1,
∫𝑡1
𝑡0
𝑭(𝑡) · 𝒗(𝑡)d𝑡 ,
is called he mechanical wo k done by he o ce
𝑭
h ough ha su ace
du ing ha ime in e al.
I ’simpo an okeepinmind se e alpoin sinapplying hecons i u i e
o mula o ene gy lux, and in speaking abou mechanical powe and
wo k. So ge eady now o a s ing o wa nings:
-Momen um supply does no con ibu e o ene gy lux
The momen um supply,o olume o ce,
𝑮
does no appea in he
cons i u i e ela ion o he ene gy lux. In o he wo ds, olume o ces
do no con ibu e o ene gy lux. In ac , since we’ e speaking o a
con ol su ace, such o ces a e no e en clea ly de ined, since hey e e
o a olume.
In pa icula coo dina e sys ems, some kinds o non-g a i a ional
olume o ces con ibu e o he ene gy supply. And i we choose di e en
de ini ions o o al ene gy
§11.2 p.248
, hen g a i a ional olume o ces
may con ibu e o he ene gy supply as well.
259
11. Balance o ene gy-mass 11.8 Hea lux and mechanical powe
-T icky poin s in applying he ene gy- lux o mula
In applying he o mula abo e, keep always in mind he ollowing:
•
The con ol su ace can be pu ely imagina y, wi h no eal physical
sepa a ion be ween he ma e on i s sides.
•
The hea lux could be nega i e and he mechanical powe posi i e,
o ice e sa.
•
When we deal ex ended con ol su aces, he condi ions o apply-
ing he o mula abo e o en do no hold o e he whole su ace.
Fo example, he eloci y
𝒗
o he hea lux
𝑄
could be di e en
on di e en pa s he su ace. To calcula e he o al ene gy lux in
such cases, we mus i s di ide he su ace in o pa s o which
he o mula abo e can be applied, hen add he esul s.
(Image: Leon Zheng 11)
-
Flux symme y does no always apply o hea lux and mechanical
powe
The p inciple o symme y o lux
§4.6 p.98
always applies o he lux o
o al ene gy
𝛷
. Bu symme y o lux does no always apply o he hea
lux 𝑄and he mechanical powe 𝑭·𝒗sepa a ely.
Le ’s pu his in symbols. Conside a con ol su ace wi h sides
𝑎
and
𝑏
. Fo he hea lux
𝑄𝑎⇝𝑏
in he c ossing di ec ion
𝑎⇝𝑏
, and he hea
lux 𝑄𝑏⇝𝑎in he c ossing di ec ion 𝑏⇝𝑎i may happen ha
𝑄𝑎⇝𝑏≠−𝑄𝑏⇝𝑎.
In ac in some si ua ions he hea lux may be posi i e in bo h c ossing
di ec ions! Tha is, posi i e hea lows o bo h sides. Jus ub he palms
o you hands agains each o he o e i y his: bo h ge s wa me , bo h
ecei ed hea .
The same is ue o he mechanical powe
𝑭·𝒗
. The eason why he
symme y o lux may b eak down is clea e in his case: i happens when
ma e has wo di e en eloci ies
𝒗𝑎
,
𝒗𝑏
on he wo sides o he con ol su ace,
as i may happen when kine ic ic ion §10.14 p.231 is p esen :
𝑭𝑎⇝𝑏·𝒗𝑏≠−𝑭𝑏⇝𝑎·𝒗𝑎.
No e ha he momen um lux does always sa is y he symme y p in-
ciple: 𝑭𝑎⇝𝑏=−𝑭𝑏⇝𝑎.
260
11. Balance o ene gy-mass 11.8 Hea lux and mechanical powe
The e o e be ca e ul in d awing sepa a e conclusions abou hea lux
and mechanical powe on wo sides o a con ol su ace. We shall
see examples o his cu ious possibili y when discussing su aces o
discon inui y §11.15 p.294.
«Exe cise 11.3
1.
Wa e is lowing downwa d in a pipe, as illus a ed in he side igu e
by he blue squiggly a ows. Take a con ol su ace mo ing wi h
eloci y
𝒗
as depic ed in ed. Can we apply o mula
(11.4)
o he
ene gy lux h ough his con ol su ace?
2.
Suppose ha he e is no powe
𝑭·𝒗
ansmi ed h ough a mo ing
con ol su ace. Does his mean ha he momen um lux
𝑭
is ze o?
O does his mean ha he ma e ’s eloci y 𝒗is ze o?
3.
A con ol su ace, wi h a gi en c ossing di ec ion, is mo ing wi h
eloci y
[0,2,−3]m/s
. Th ough his su ace we ha e a hea lux o
−3J/s
and a momen um lux o
[1,−2,1]N
. How much is he ene gy
lux h ough he su ace?
4.
Conside he con ol su ace ha sepa a es ai om he pis on wi hin
a bike pump, schema ized by he ed line in he side illus a ion. The
pis on and he gas in con ac wi h i , on he wo sides o he con ol
su ace, a e mo ing downwa d wi h a eloci y
[0,0,−0.5]m/s
. The
edownwa d lux o momen um is
[0,0,−20]N
. The ene gy lux
h ough his su ace is adiaba ic. How much is he downwa d lux
o ene gy h ough his su ace?
5.
Conside again he con ol su ace o ques ion 3. abo e. We a e now
old ha he e is no ma e lux h ough his su ace, and he ma e
in con ac wi h he su ace on bo h sides is mo ing wi h he same
eloci y as he su ace. How much is he ene gy lux h ough he
su ace?
6.
Imagine a cylind ical closed con ol su ace en eloping he ai wi hin
he bike pump o ques ion 4. abo e. The su ace p e iously con-
side ed is pa o his closed con ol su ace. We a e old ha ac oss
he es o he closed su ace he e is a o al hea e lux o
2J/s
. How
much is he o al ene gy in lux h ough he closed con ol su ace?
261
11. Balance o ene gy-mass 11.9 Hea and powe : obse a ion scales
Volume hea ing and olume mechanical powe
{To be w i en in a la e e sion
Snippe om a ex on
he momechanics (Mülle &
Mülle 2009) showing h ee
special e sions o he bal-
ance o ene gy. The symbols
¤
𝑊
,
¤
𝑊s ess
,
¤
𝑊in
s and o h ee
di e en de ini ions o ‘ o al
mechanical powe ’. The hea
lux is
¤
𝑄
, kine ic ene gy
𝐾
,
po en ial ene gy 𝐸po .
-Con as ing de ini ions o ‘wo k’
The de ini ion abo e is abou he mechanical wo k done by a su ace
o ce
𝑭
– no e he explici men ion o ‘ o ce’. This de ini ion seems o be
s anda d ac oss all physics and enginee ing ex s.
Many ex s also gi e addi ional de ini ions o di e en kinds o powe
and wo k, which do no explici ly men ion a speci ic o ce; o example
‘ne ’, o ‘ o al’, o ‘in e nal’, o ‘s ess’ wo k and powe . Un o una ely
hesede ini ionscanbed as icallydi e en om ex o ex .Fo ins ance
some ex s de ine he ‘ o al’ o ‘ne ’ wo k on a con ol olume as ollows:
ne / o al wo k :=wo k done by su ace o ces on whole con ol su ace .
Bu o he ex s ins ead call ‘s ess’ wo k he de ini ion abo e, and de ine
‘ne ’ o ‘ o al’ wo k as ollows:
ne / o al wo k :=(wo k done by su ace o ces on whole con ol su ace) −
(change in o al kine ic ene gy) −
(change in o al g a i a ional po en ial ene gy).
S ill o he ex s call he abo e de ini ion ‘in e nal’ wo k. The e a e o he
al e na i e de ini ions.
The e o e, whene e you pe use a new ex o esea ch o job pu poses,
and you ead he simple wo d ‘wo k’, make su e o check he de ini ion
o o mula o ha e m. You don’ wan o use inco ec alues in you
o mulae!
11.9 Hea and powe : obse a ion scales
The o al-ene gy con en in a con ol olume depends on ou coo dina e
sys em, bu i does no depend on he spa ial scale o ou desc ip ion; he
sepa a ion in o in e nal and kine ic ene gy, howe e , does depend on he
spa ial scale. A comple ely analogous si ua ion, and o analogous easons,
occu s o he lux o o al ene gy h ough a con ol su ace, and o i s
sepa a ion in o hea lux and mechanical powe .
262
11. Balance o ene gy-mass 11.9 Hea and powe : obse a ion scales
Conside wo bodies o ma e in con ac wi h each o he . Fo one
esea che A, who obse es hese bodies mac oscopically, he e may be a
lux o o al ene gy
𝛷=1J/s
h ough he con ac su ace, and no isible
mo ion o ma e . This esea che he e o e conside s his ene gy lux
o consis comple ely in a hea lux
𝑄A=1J/s
, and in no mechanical
powe
𝑭A·𝒗A
because he eloci y
𝒗A
measu ed by A is ze o. Fo ano he
esea che B, who can keep ack o molecula mo ions and eloci ies, he
o al-ene gy lux is s ill
𝛷=1J/s
; bu his lux consis s comple ely in
1J/s
o mechanical powe , ob ained by summing up
𝑭B·𝒗B
o all molecules a
he con ac su ace; he hea lux
𝑄B
is ze o. In he no a ion jus used, ‘A’
and ‘B’ e e , no o di e en con ol su aces, bu o di e en esea che s,
who measu e he same con ol su ace wi h ins umen s o di e en spa ial
esolu ions.
Examples o hea and mechanical-powe luxes
Le us see some examples in which o al-ene gy lux appea s as hea lux,
lux o mechanical-powe , o bo h. In all hese examples we choose a
coo dina e sys em a es wi h he g ound.
Holding a cup o ho ea. When we hold a cup o ho ea o co ee, we
can eel a lux o ene gy om he cup o ou hands. How much o i is hea
lux, and how much is mechanical powe ?
Le ’s conside a con ol su ace be ween ou hands and he cup, and
le ’s choose a hands
⇝
cup c ossing di ec ion. The e is ob iously a
momen um lux om ou hands o he cup. The cup has a cons an supply
o downwa d momen um om g a i y
§10.2 p.192
, bu i isn’ alling. This
means ha he e mus be an in lux o upwa d momen um: i comes om
ou hands. We can also eel he p essu e ha he cup exe s on ou hands;
by he symme y o lux his means ha he e’s a lux o momen um om
ou hands o he cup. So he o al in lux o momen um
𝑭
is no ze o. The
cup, howe e , is no mo ing: i s eloci y
𝒗
is ze o. The mechanical powe
𝑭·𝒗is he e o e ze o as well.
The ene gy lux h ough ou con ol su ace mus he e o e be com-
ple ely in he o m o hea lux
𝑄
. The hea in lux h ough he con ol
su ace is nega i e, because ene gy is lowing om he cup o ou hands.
I we mo e he cup a ound, hen i s eloci y is no longe ze o, and
some mechanical powe
𝑭·𝒗
can be ansmi ed. I shows as an inc ease
in he kine ic ene gy o he ea, which can e en splash ou o he cup.
263
11. Balance o ene gy-mass 11.9 Hea and powe : obse a ion scales
Cooking. When we cook some hing we c ea e a hea in lux in o he i em
being cooked. No mechanical powe is ans e ed, since he eloci y o
ma e is ze o.
Sp ing and body. In a Hookean o non-Hookean sp ing o ubbe band
he e is a momen um lux om one end o he sp ing o he body a ached
a ha end. I he momen um in lux o he body is
𝑭
, and he body is
mo ing wi h eloci y
𝒗
, hen he e is an ene gy lux in o he body, possibly
nega i e, in he o m o mechanical powe
𝑭·𝒗
. This ene gy lux changes
he kine ic ene gy and g a i a ional po en ial ene gy o he a ached body.
By he symme y o lux, his sameamoun o ene gy (possibly nega i e)
lows ou o he sp ing, whose in e nal ene gy changes. The change
in in e nal ene gy can be e i ied by a simple expe imen
12
: p ess an
uns e ched ubbe band agains you lips (which a e especially sensi i e
o empe a u e changes), and quickly s e ch i . Do you eel any change in
he empe a u e o he ubbe band? an inc ease o a dec ease? Now allows
some seconds o he s e ched ubbe band o each ambien empe a u e
again. Keeping i p essed agains you lips, quickly un-s e ch i . Wha
empe a u e change do you eel?
Gases. In desc ibing and using ma e in a gaseous s a e, usually hea
lux and mechanical powe mus bo h be aken in o accoun . Many physical
heo ies and echnologies a e indeed ocused on he p oblem o gene a ing
e luxes o powe as la ge as possible om a body o ma e , by p o iding
in luxes o hea o i .
Foo , wheel, ack on g ound. When we walk, he e is a lux o mo-
men um
𝑭
om he g ound o ou bodies. This momen um poin s in
he di ec ion o ou mo ion and upwa d. The e ical componen nulli-
ies he g a i a ional supply o momen um; he ho izon al componen is
wha keeps ou o wa d mo ion. (Ob iously his is equi alen o a lux o
momen um om ou body o he g ound, ha ing opposi e o ien a ion.)
Wha abou he lux o ene gy ac oss he imagina y con ol su ace ha
sepa a es ou oo om he g ound? To calcula e he mechanical powe
𝑭·𝒗
we mus conside he eloci y
𝒗
o ou oo – mo e p ecisely, o he sole o
ou oo o shoe – as i ’s in con ac wi h he g ound. The oo ac ually isn’
mo ing wi h espec o he g ound while he lux o momen um occu s;
i s eloci y
𝒗
is he ze o ec o . This means ha he mechanical powe
𝑭·𝒗
is also ze o. The e may be a sligh hea lux be ween oo and g ound, bu
264
11. Balance o ene gy-mass 11.10 Summa y: cons i u i e ela ions o ene gy in he p esence o ma e
usually i is neglec ed,
𝑄≈0J/s
, when conside ing he ene gies in ol ed
in he mo ion.
(Image: Rajamani 2012)
The lux o o al ene gy be ween he g ound and ou body is he e o e app ox-
ima ely ze o (in he p esen coo dina e sys em). The same conclusion is ue
also o a ca . In pa icula , he e is no lux o mechanical powe o wo k done
om he g ound o he ca , because he pa o he y e momen a ily ouching
he g ound, called ‘con ac a ea’ o ‘con ac pa ch’, has ze o eloci y in he
p esen coo dina e sys em. Simila ly o a ehicle mo ing on acks like a
ca e pilla o a bulldoze .
11.10 Summa y: cons i u i e ela ions o
ene gy in he p esence o ma e
So a we ha e discussed wo gene al cons i u i e ela ions: one o ene gy
con en
𝐸
, o mula
(11.3)
; and one o ene gy lux
𝛷
, o mula
(11.4)
. We
also said ha ou de ini ion o o al ene gy app oxima ely sa is ies a con-
se a ion law
§11.2 p.250
, so he ene gy supply
ℛ
is ze o. Le us ew i e he e
hese cons i u i e ela ions, explici ly indica ing hei ime dependence:
𝐸(𝑡)=𝑈(𝑡) + 1
2𝑚𝒗(𝑡)2+𝑚𝑔𝑧(𝑡)
𝛷(𝑡)=𝑄(𝑡) + 𝑭(𝑡) · 𝒗(𝑡)
ℛ(𝑡)=0
Le us ecall he condi ions unde which hey a e alid:
•
The coo dina e sys em is a es wi h he g ound, and wi h a e ical
𝑧-coo dina e poin ing upwa d.
•Ma e is p esen wi hin and di ec ly ou side he con ol olume.
•The con ol olume and su ace a e such ha he e ical posi ion 𝑧,
he ma e eloci y
𝒗
and mola mass
𝜌
, and he e o e also he es
mass 𝑚, a e he same h oughou .
•No ma e lux occu s h ough he con ol su ace.
•Elec ic cha ges o elec omagne ic ields a e negligible.
•
The speeds in ol ed a e much lowe han ligh ’s and he con ol
olume is close o he Ea h’s su ace.
In some si ua ions we mus di ide a con ol olume o closed con ol
su ace in o se e al pa s whe e hese condi ions a e alid.
Unde hese condi ions, he o mulae abo e can be used oge he wi h
he balance o ene gy
(11.1)
in o de o desc ibe many common physical
265
11. Balance o ene gy-mass 11.12 Rigid mo ion and igid bodies
explici ly depends on a coo dina e eloci y, which in u n is comple ely
dependen on he choice o coo dina es.
Analyses simila o he one abo e can be made o all mechanical
sys ems, wi h he same conclusion abou coo dina e-dependence. This
shows ha we mus ake “explana ions” abou ene gy lowing he e o
he e always wi h a g ain o sal : hey may be ue in one coo dina e
sys em, bu alse in ano he .
(Image: Bike umo 13)
«Exe cise 11.4
The mo emen o a bicycle wi h espec o he g ound is connec ed
wi h he bu ning o in e nal chemical ene gy in he bike ’s muscles.
In ui i ely we would say ha ene gy lows om he bike ’s ee o he
back wheel; bu along which pa h does i low he e?
Analyse, in he same way as o he ca example, he low o mechanical
ene gy, a a gi en ime ins an
𝑡
, h ough he h ee con ol su aces
shown by he blue lines in he side igu e:
(1)
ac oss he uppe , au pa o he chain; no e ha he e is a ensile
o ce §4.12 p.111 he e;
(2) ac oss he lowe , slack pa o he chain; no e ha , app oxima ely,
he e is no su ace o ce he e;
(3) be ween he g ound and he back wheel;
(4)
conside con ol su aces ac oss o he igid pa s on which he
back wheel in moun ed.
Do he analysis in h ee di e en coo dina e sys ems:
(a) coo dina e sys em whe e he g ound is a es ;
(b) coo dina e sys em whe e main body o he bike is a es ;
(c)
coo dina e sys em whe e, momen a ily, he uppe , au pa o he
chain is a es .
11.12 Rigid mo ion and igid bodies
Choose a ec angula coo dina e sys em, and conside a body o ma e ;
o ins ance a ennis ball, o some a bi a ily de ined pa wi hin a block
o conc e e o wi hin a pa cel o wa e . In o dina y ci cums ances we can
associa e a coo dina e eloci y
§4.15 p.116
o he ma e a each poin in he
272
11. Balance o ene gy-mass 11.12 Rigid mo ion and igid bodies
body, o mo e p ecisely o he ma e in any small con ol olume wi hin
he body.
Now imagine o compu e he physical dis ances
§2.4 p.48
be ween all
pai s o poin s in he body a a gi en ins an o ime
𝑡
. Then check i hese
dis ances change as you ollow, o a e y sho ime lapse
Δ𝑡
, each poin
as i mo es wi h i s eloci y.
]Rigid mo ion
I he ne wo k o dis ances a ime
𝑡
be ween all pai s o poin s in a body
o ma e does no change du ing a e y small ime lapse
Δ𝑡
, hen we
say ha he body is unde going an ins an aneously igid mo ion a
ime 𝑡.
I he ne wo k o dis ances does no change be ween wo imes
𝑡0
and
𝑡1
, hen we say ha he body unde goes a igid mo ion be ween hose
wo imes.
Clea ly in a igid mo ion he body has a cons an olume and shape; ha
is, i isn’ de o ming.
We don’ ac ually need o conside he dis ances be ween each poin
and e e y o he poin in he body: i ’s enough i we conside hose be ween
a poin and, loosely speaking, i s immedia e neighbou s. In ac o some
pai s o poin s he physical dis ance may no e en be de ined §2.4 p.48.
When we s udy a body o ma e in ci cums ances in which i only
unde goes igid mo ions, hen we call i a igid body. Fo ins ance, a block
o conc e e, a block o ice, a ba o ha d i on, a pen unde go only igid
mo ions in o dina y si ua ions, so we may call hem igid bodies in hese
si ua ions. No so wi h a pa cel o wa e o ai in a cylinde . No body is
uly igid. Whe he we call i ‘ igid’ o no depends on he ci cums ances
and he p ecision equi ed. Mo eo e , unde ex eme condi ions a body
may no longe be igid: we may mel he block o conc e e and b eak he
pen. A ennis ball may be conside ed as a igid body as long a i ’s eely
lying; bu no as i bounces on he loo o is hi by a ennis acke . In
gene al i ’s be e o speak o a ‘ igid mo ion o a body’ a he han o a
‘ igid body’.
Rigid mo ions ha e special p ope ies as ega ds he balance o ene gy.
273
11. Balance o ene gy-mass 11.12 Rigid mo ion and igid bodies
]Balance o ene gy o ma e in igid mo ion
Fo ma e unde going a igid mo ion, he balance o ene gy ake on a
special o m:
•The hea lux con ibu es only o changes in in e nal ene gy.
•The in e nal ene gy can only change h ough hea luxes.
•The mechanical powe con ibu es only o changes in kine ic &
g a i a ional po en ial ene gy.
•
Kine ic & g a i a ional po en ial ene gy can only change h ough
mechanical powe .
Le ’s exp ess hese condi ions ma hema ically. Conside a con ol olume
con aining all o pa o a body unde going igid mo ion, and o which
he eloci y
𝒗
is he same h oughou , e en i i can change wi h ime.
This will be app oxima ely ue a leas i he con ol olume is small
enough. The mass and in e nal ene gy con ained in he con ol olume
a e
𝑚
and
𝑈
, he o al momen um in lux and hea lux ac oss he closed
con ol su ace a e
𝑭
and
𝑄
. Use a e ical coo dina e
𝑧
. Then, in in eg al
exp ession:
𝑈(𝑡1)=𝑈(𝑡0) + ∫𝑡1
𝑡0
𝑄(𝑡)d𝑡(11.6a)
1
2𝑚𝒗(𝑡1)2+𝑚𝑔𝑧(𝑡1)=1
2𝑚𝒗(𝑡0)2+𝑚𝑔𝑧(𝑡0) + ∫𝑡1
𝑡0
𝑭(𝑡) · 𝒗(𝑡)d𝑡(11.6b)
o equi alen ly in di e en ial exp ession:
d𝑈(𝑡)
d𝑡=𝑄(𝑡)(11.7a)
d
d𝑡1
2𝑚𝒗(𝑡)2+𝑚𝑔𝑧(𝑡)=𝑭(𝑡) · 𝒗(𝑡)(11.7b)
In o he wo ds, mechanical and he mal phenomena ge s comple ely
sepa a ed unde a igid mo ion. This sepa a ion does no occu when he
mo ion is no igid: in gene al, mechanical powe and changes in in e nal
ene gy a e ela ed o each o he .
Mo eo e , i u ns ou ha unde a igid mo ion he equa ions ela ing
kine ic & g a i a ional po en ial ene gy and mechanical powe ,
(11.6b)
and (11.7b), can be de i ed om he balance o momen um.
274
11. Balance o ene gy-mass 11.13 Cons i u i e ela ions o ideal gases
The special sepa a ion be ween “ he mal” and “mechanical” ene gy
changes unde igid mo ions has an impo an consequence. I we a e no
in e es ed in he changes o he in e nal ene gy o a body in igid mo ion, hen we
do no need o conside he balance o ene gy. The balance o momen um is all
we need.
This ac explain why we we e able o sol e many p oblems o mo ion
in he p e ious chap e s: many mo ions we conside ed, such as he ligh
o a ennis balls o he all o a block o ma e ial, we e p ac ically igid,
and we we e no in es iga ing he in e nal ene gies o hose bodies. The
balance o momen um was he e o e su icien o desc ibe hose mo ions.
Sp ings and ubbe bands do no unde go igid mo ions, bu o hem we
implici ly assumed ha hei in e nal ene gy was cons an , and no hea
luxes occu ed.
«Exe cise 11.5
1.
Suppose someone ells you he posi ions ha all ma e poin s o a
body ha e a a gi en ime
𝑡
(bu you a en’ old abou o he imes).
F om hese posi ions you can calcula e all dis ances a ime
𝑡
. F om
his in o ma ion can you ell whe he ha body is unde going an
ins an aneously igid mo ion?
2. Conside a body unde going a igid mo ion. Does his mean ha
•all o i s poin s ha e he same eloci y?
•all o i s poin s ha e he same mass?
Think o a boome ang o ins ance.
Long exposu e o he ajec -
o y o a boome ang ou i ed
wi h LEDs14
11.13 Cons i u i e ela ions o ideal gases
The e is a se o simple cons i u i e ela ions which can be used as
app oxima ions o many gases, especially when hey a e a e ied, ha
is, when hei amoun o ma e pe olume
𝑁/𝑉
is low. We say ha
hese cons i u i e ela ions apply o ideal gases. When we say ha some
ma e ial can be modelled as an ideal gas – o simply say “i ’s an ideal gas”,
we mean ha we can model i wi h good enough app oxima ion using
hese ela ions.
275
11. Balance o ene gy-mass 11.13 Cons i u i e ela ions o ideal gases
The cons i u i e ela ions o ideal gases in ol e cons an s and a couple
o unc ions ha can be di e en depending on he kind o gas. So he e
isn’ jus one ideal gas, bu a amily o hem. Some ex s speak o he ideal
gas, as i he e was only one; bu hey do so because hey discuss p ope ies
ha a e common o all ideal gases.
In discussing he cons i u i e ela ions below, we conside a closed
con ol su ace ha encloses an amoun o some ideal gas. We assume
ha he e is no lux o he gas h ough he enclosing su ace, and mo e
gene ally ha he condi ions lis ed in he p e ious 11.10summa y a e
sa is ied. These condi ions a e app oxima ely ue o many p ac ical cases,
like ai unde comp ession o expansion in a bike pump.
I we wan o model a body o gas accu a ely, howe e , we gene ally
need o use a la ge numbe o such con ol olumes, each one enough
small ha he condi ions abo e a e app oxima ely sa is ied wi hin i . This
way we can y o simula e many in e es ing physical beha iou s ha ideal
gases can ha e, such as u bulence.
P essu e (momen um lux) o an ideal gas
We p e iously discussed he peculia luxes o momen um ha ake place
in a gas, which we gene ally call he in e nal p essu e
§10.20 p.243
o he gas.
Now we discuss a cons i u i e ela ion ha connec s:
•
he lux o momen um
𝑭
h ough he closedcon ol su acecon aining
an ideal gas
• he amoun o ideal gas wi hin
• he empe a u e wi hin
• he olume enclosed by he con ol su ace
This cons i u i e ela ion is mo e easily exp essed in e ms o he
momen um lux h ough a su ace, di ided by he a ea o he su ace:
]P essu e ec o , s ess ec o , p essu e
Fo a con ol su ace o a ea
𝐴
whe e he momen um lux
𝑭
is uni o m
h oughou , and h ough which he e is no ma e lux, we call p essu e
ec o o s ess ec o
𝒑
he momen um lux di ided by he a ea:
𝒑:=𝑭/𝐴.
I he momen um lux
𝑭
, and he e o e he p essu e ec o , a e o ho-
gonal o he su ace and poin away om he su ace, hen we call p essu e
276
11. Balance o ene gy-mass 11.13 Cons i u i e ela ions o ideal gases
he magni ude o he p essu e ec o , and usually deno e i 𝑝:
𝑝:=|𝒑| ≡ |𝑭|
𝐴(11.8)
The name ension is used ins ead o p essu e i he momen um lux poin s
owa ds he su ace. Compa e ou discussion abou comp essi e and
ensile momen um luxes §4.12 p.110.
P essu e has physical dimensions o o ce pe a ea, wi h uni s
N/m2
;
his uni also called pascal (Pa).
-‘P essu e’ has many di e en meanings
Be awa e ha he e m ‘p essu e’ is used in many di e en and some-
imes e en opposi e ways in he physics li e a u e. Some ex s use
‘p essu e’ o indica e no jus he magni ude o
𝑭/𝐴
, bu he ull ec o ;
so o hem ‘p essu e’ is no jus a numbe , bu a se o h ee ec o
componen s. Some ex s use ‘p essu e’ o deno e he p essu e ec o ; so
o hem a p essu e need no be o hogonal o a con ol su ace. Some
ex s use ‘p essu e’ also o indica e ‘ ension’, o ice e sa.
So when you ead a ex ha uses o discusses ‘p essu e’, make su e o
ge co ec ly in which sense he ex is using his wo d.
In hese no es we shall some imes use ‘p essu e’ in a sligh ly di e en
o mo e gene al meaning, which should be clea om he con ex .
Now ha he no ion o p essu e is in oduced, we can o mula e a
pa icula cons i u i e ela ion o he momen um lux:
]Ideal-gas law wi h iscosi y
I a closed con ol su ace encloses a olume
𝑉
wi h an amoun
𝑁
o
an ideal gas, a uni o m empe a u e
𝑇
, and he momen um lux is
o hogonal o he su ace and uni o m h oughou , and he e is no
ma e lux h ough he su ace, hen he p essu e 𝑝is gi en by
𝑝=𝑅𝑁𝑇
𝑉−𝜇1
𝑉
d𝑉
d𝑡(11.9)
277
11. Balance o ene gy-mass 11.13 Cons i u i e ela ions o ideal gases
whe e
d𝑉
d𝑡
is he a e o change o olume,
𝜇
isa iscosi y coe icien ,
and 𝑅is he mola gas cons an 15, ha ing uni e sal alue
𝑅=8.31446261815324J/(mol ·K)(exac ly)
The equa ion abo e is called he ideal-gas law wi h iscosi y.
I he su ace has a ea
𝐴
, he magni ude o he momen um lux is
he e o e |𝑭|=𝑝𝐴.
No e ha he o mula abo e o p essu e and momen um lux is only
alid o su aces h ough which no ma e lux occu s.
In he exp ession abo e o he p essu e you may ecognize he amous
“
𝑝𝑉 =𝑁𝑅𝑇
” o mula, called he ideal-gas law. This amous o mula is
s ic ly speaking only alid when he ideal gas is a es , which means ha
i s olume is no changing, so ha d𝑉
d𝑡=0.
I he ideal gas is in mo ion, o ins ance expanding o con ac ing and
i s olume
𝑉
changes wi h ime, hen addi ional e ms mus be added o
he amous “
𝑝𝑉 =𝑁𝑅𝑇
” o mula. This is wha we see in he ela ion
(11.9)
abo e. In many cases hese addi ional e ms a e ex emely small and
he e o e neglec ed.
The ac ha he iscosi y coe icien s mus be posi i e is a consequence
o he balance o en opy – he second law o he modynamics – which we
shall discuss la e .
In e nal ene gy o an ideal gas
Nex we discuss a cons i u i e ela ion ha connec s he amoun o
in e nal ene gy in he con ol olume wi h he amoun o ideal gas and i s
empe a u e:
]In e nal ene gy o an ideal gas
I a small con ol olume con ains an amoun
𝑁
o ideal gas a uni o m
empe a u e
𝑇
, hen i also con ains an amoun o in e nal ene gy
𝑈
gi en by
𝑈=𝐶 𝑁 𝑇 (11.10)
whe e he cons an
𝐶
is called mola hea capaci y and depends on he
kind o ideal gas.
278
11. Balance o ene gy-mass 11.13 Cons i u i e ela ions o ideal gases
This o mula is e y impo an : o an ideal gas, he absolu e empe a u e
is a di ec measu e o he in e nal ene gy, so i can o en be used as a p oxy
o he la e . Fo ai , when i can be modelled as an ideal gas, he alue o
he mola hea capaci y is app oxima ely 𝐶ai ≈20J/(molK).
-Validi y o he in e nal-ene gy o mula
•
The pa icula cons i u i e ela ion
(11.10)
is alid only o an ideal
gas. The in e nal ene gy o a gene ic ma e ial is ela ed o o he
quan i ies besides amoun o ma e and empe a u e.
•
The cons an
𝐶
is also called ‘mola hea capaci y a cons an
olume’, bu despi e his name he cons i u i e ela ion abo e is
also alid when he olume o he gas is changing!
•
Some books exp ess he o mula abo e saying “ he in e nal ene gy
o an ideal gas depends only on empe a u e”. This s a emen is
somewha ague and easy o misunde s and. Fi s , he in e nal
ene gy clea ly “depends” also on he amoun o ma e
𝑁
. Second,
i in some applica ion we exp ess he amoun o ma e o he
empe a u e as unc ions o o he quan i ies, such as olume, hen
he in e nal ene gy also becomes a unc ion o hose quan i ies.
The o mula abo e can be simply exp essed as ollow: i we know
he amoun (and kind) o ma e in a olume, and he empe a u e
in ha olume, hen we also know he amoun o in e nal ene gy
he ein.
Hea lux be ween sides a diffe en empe a u es
Las ly we discuss a cons i u i e ela ion ha connec s hea lux and
empe a u e. This ela ion applies o many physical phenomena and
ma e ials: no only o ideal gases, bu o many o he luids and solids as
well.
]New on’s law o cooling
Conside a con ol su ace o a ea
𝐴
and wi h wo sides
𝑎
and
𝑏
. I he
ma e on side
𝑎
has empe a u e
𝑇𝑎
e e ywhe e close o he su ace,
and he ma e on side
𝑏
has empe a u e
𝑇𝑏
e e ywhe e close o he
su ace, hen he hea lux
𝑄𝑎⇝𝑏
in he
𝑎⇝𝑏
c ossing di ec ion is gi en
279
11. Balance o ene gy-mass 11.13 Cons i u i e ela ions o ideal gases
by New on’s ‘law o cooling’:
𝑄𝑎⇝𝑏=𝐴ℎ (𝑇𝑎−𝑇𝑏)(11.11)
whe e
ℎ
is called he su ace coe icien o hea ans e and depends on
he pa icula physical condi ions o he ma e on he wo sides o he
su ace. This coe icien is usually posi i e.
The cons i u i e o mula abo e wi h
ℎ>0
says ha i he empe a u e
on side
𝑎
is highe han he one on side
𝑏
, hen a posi i e hea lux occu s
om
𝑎
o
𝑏
. In o he wo ds, posi i e hea lows om he ho e o he
colde side o he su ace.
New on’s law o cooling implies ha we can app oxima ely conside
empe a u e as ha ing a jump o discon inui y be ween he wo sides o
he su ace. In si ua ions whe e his app oxima ion is oo g oss, ano he
cons i u i e equa ion is o en used: Fou ie ’s law o hea conduc ion,
which we can w i e as
𝑄=−𝐴𝑘 ∂𝑇
∂𝑥
In his exp ession we imagine he su ace o be o hogonal o he
𝑦𝑧
di ec ions, and he c ossing di ec ion o be he posi i e-
𝑥
di ec ion. The
de i a i e
∂𝑇
∂𝑥
is he g adien o he empe a u e, exp essing how much he
empe a u e changes om one poin o ano he e y close one.
Le ’s make clea ha New on’s law o cooling and Fou ie ’s law a e
no uni e sal. The e a e physical si ua ions and ma e ials o which he
hea lux is connec ed o empe a u e in mo e complex ways; and no only
o empe a u e, bu also o momen um lux, ma e lux, elec omagne ic
quan i ies. The moelec ic coole s
16
a e an example applica ion o hese
mo e complex cons i u i e ela ions o he hea lux.
-Hea can low om cold o ho
Some ex s say ha “hea canno low om cold o ho ”, and p esen
his ague s a emen as a consequence o , o equi alen o, he second
law o he modynamics. This s a emen is ac ually no ue, om se e al
poin s o iew.
The ac ha hea can low om cold o ho is e y clea in de ices such
as e ige a o s and eeze s, which would no be possible o he wise:
I is ob ious ha , on a mac oscopic scale, hea lows om a cold sou ce
280
11. Balance o ene gy-mass 11.13 Cons i u i e ela ions o ideal gases
o a ho e sink in a e ige a o : hus he idea ha he second law
equi es hea o in a iably low om ho o cold is belied by he ac ha
e ige a o s do wo k. (As a i a 1990)
E en mo e in e es ing examples, in which Fou ie ’s law does no always
apply, occu o ma e ials such as polyme s {add ex and e e ence
O he common assump ions abou ideal gases
Besides he pa icula cons i u i e ela ions o momen um lux, in e nal
ene gy, and hea lux jus discussed, i is ypically assumed ha a no - oo-
la ge olume o ideal gas has a negligible mass. This is usually a easonable
assump ion (see exe cise below).
I he mass
𝑚
o a olume o ideal gas is assumed o be ze o, hen ou
o he quan i ies a e ze o as well in ha olume, a all imes:
• he o al momen um con en 𝑷=𝑚𝒗
• he momen um supply om g a i y 𝑮=𝑚𝒈
• he kine ic ene gy 1
2𝑚𝒗2
• he g a i a ional po en ial ene gy 𝑚𝑔𝑧
This assump ion has an impo an consequence o he balance o
momen um o a con ol olume con aining an ideal gas: we ha e
d𝑷(𝑡)
d𝑡
=0
=𝑭(𝑡) + 𝑮(𝑡)
=0
=⇒𝑭(𝑡)=0
ha is, he o al momen um in lux is always ze o. Bu no e ha his does no
mean ha he momen um in lux is ze o e e ywhe e ac oss he closed con ol
su ace; i only means ha he momen um in luxes o opposi e pa s o
he su ace cancel ou pe ec ly, as schema ized in he side igu e.
Remembe ha when we use he ideal-gas law
(11.9)
, we a e assuming
ha any small su ace o a ea
𝐴
o he con ol olume has a momen um
in lux 𝑭𝐴o magni ude
𝐹𝐴=𝐴 𝑝 =𝐴𝑅𝑁𝑇
𝑉−𝐴𝜇1
𝑉
d𝑉
d𝑡
and di ec ed inwa ds, acco ding o he ideal-gas law §11.13 p.277 (11.9).
281
11. Balance o ene gy-mass 11.14 Example applica ions: ideal gas and pis on
-Uni o m condi ions in he gas
Keep in mind ha his cons i u i e ela ion can only be used when he
empe a u e and he amoun o ma e pe olume a e app oxima ely
he same e e ywhe e in he con ol olume. I we ha e e y comp ession
o decomp ession, his assump ion won’ be ue. Wha coun s as ‘ as ’
in his physical si ua ion? The ele an speed u ns ou o be he speed
o sound in he gas. Fo ai i is app oxima ely
340m/s
. The e o e,
as long as he pis on is mo ing wi h a eloci y well below ha , ou
app oxima ion o uni o m empe a u e and uni o m gas dis ibu ion
shouldn’ be oo bad.
Fo he o al ene gy in lux
𝛷
we can use he gene al cons i u i e ela ion
ha exp esses i as a sum o hea lux and mechanical powe
§11.8 p.258
.
Bu his cons i u i e ela ion canno be used o he gas’s su ace as a
whole, because di e en hea luxes, eloci ies, and su ace o ces occu on
di e en pa s o i . We he e o e need o apply he p inciple o ex ensi i y.
We mus conside h ee pa ial su aces:
Side:
The e could be a hea lux h ough he side walls, o which we
could use New on’s law o cooling
§11.13 p.279
. No e ha as he pis on
mo es up o down, he a ea o he side walls changes, and his should
be aken in o accoun in New on’s law o cooling. Fo simplici y le ’s
assume ha he hea lux h ough he side walls is ze o, ha is, he
ene gy lux is adiaba ic §11.8 p.258.
Le ’s ocus on he mechanical powe now. The e is a momen um in lux
𝑭side
, o su ace o ce, on side walls. Bu acco ding o he ideal-gas
law his su ace o ce is o hogonal o he con ol su ace. As he pis on
mo es up o down, we assume ha he gas is app oxima ely mo ing
only e ically. This means ha , on he side walls, he mechanical
powe
𝑭side ·𝒗side
is app oxima ely ze o, because
𝑭side
and
𝒗side
a e
o hogonal he e.
The ene gy in lux h ough he bo om su ace is he e o e
𝛷side(𝑡)≈0J/s.
Bo om:
Le ’s conside he possibili y o a hea lux
𝑄bo (𝑡)
h ough he
bo om su ace, o which we can use New on’s law o cooling holds
288
11. Balance o ene gy-mass 11.14 Example applica ions: ideal gas and pis on
as a cons i u i e ela ion. Since he a ea
𝐴
o his su ace is cons an ,
his lux is
𝑄bo (𝑡)=𝐴 ℎ [𝑇ex −𝑇(𝑡)] ,
whe e
𝑇ex
is he empe a u e o he cylinde walls. In p inciple his
empe a u e could be unde ou con ol, and we could choose i s
ime dependence. Fo simplici y le ’s assume ha i ’s kep cons an .
Rega ding he mechanical powe , he momen um in lux
𝑭bo
h ough
he bo om wall is di ec ed upwa d. The eloci y o he gas
𝒗bo
a he
bo om su ace mus be ze o, o he wise he gas would de ach om i
and a acuum would o m, and his is no wha ’s ypically obse ed.
Since he gas eloci y
𝒗bo
a his su ace is ze o, hen he mechanical
powe is also ze o: 𝑭bo ·𝒗bo =0J/s.
The ene gy in lux h ough he bo om su ace is he e o e
𝛷bo (𝑡)=𝑄bo (𝑡) + 𝑭bo ·𝒗bo =𝐴 ℎ [𝑇ex −𝑇(𝑡)] .
Top:
We could conside a hea lux h ough he op su ace, be ween gas
and pis on. The analysis would be simila o he one o he bo om
su ace, bu we would need o conside he empe a u e o he pis on,
and in u n possibly i s ene gy balance, as p e iously discussed. Fo
simplici y le ’s assume ha his hea lux is ze o: 𝑄 op =0J/s.
Le ’s analyse he mechanical powe . The momen um in lux
𝐹gas(𝑡)
h ough he op su ace is has a nega i e e ical componen , and
has magni ude gi en by he ideal-gas law as in o mula
(11.17)
. The
eloci y o he ideal gas a his su ace mus be equal o ha o he
su ace i sel ,
𝑣(𝑡)
, o he wise he gas would de ach om he pis on
and a acuum would o m – and hen we would need e y di e en
cons i u i e ela ions. Fo ce and eloci y a e he e o e pa allel, and
a he op su ace he e is a non-ze o in lux o mechanical powe
𝐹gas(𝑡) · 𝑣(𝑡). The ene gy in lux ac oss his su ace is he e o e
𝛷 op(𝑡)=𝐹gas(𝑡) · 𝑣(𝑡)
whe e 𝐹gas(𝑡)is gi en by o mula (11.17).
289
11. Balance o ene gy-mass 11.14 Example applica ions: ideal gas and pis on
Summing up he in luxes om all pa ial su aces, he o al ene gy
in lux in o he gas’s con ol olume is he e o e
𝛷(𝑡)=𝛷bo +𝛷 op wi h
𝛷bo =𝐴 ℎ [𝑇ex −𝑇(𝑡)]
𝛷 op =𝐹gas(𝑡) · 𝑣(𝑡), 𝐹gas(𝑡)=−𝑅𝑁𝑇(𝑡)
𝑧(𝑡)+𝜇𝐴
𝑧(𝑡)𝑣(𝑡).
(11.19)
The ene gy equa ions o he gas’s con ol olume a e he e o e
d𝐸(𝑡)
d𝑡=𝛷(𝑡)
𝐸(𝑡)=𝐶𝑁𝑇(𝑡)
𝛷(𝑡)=𝛷bo +𝛷 op wi h
𝛷bo =𝐴 ℎ [𝑇ex −𝑇(𝑡)]
𝛷 op =𝐹gas(𝑡) · 𝑣(𝑡), 𝐹gas(𝑡)=−𝑅𝑁𝑇(𝑡)
𝑧(𝑡)+𝜇𝐴
𝑧(𝑡)𝑣(𝑡).
(11.20)
We inally ha e all he equa ions and bounda y condi ions o desc ibe
and nume ically ime-in eg a e ou physical sys em. We only need o
speci y he ini ial condi ions and some addi ional cons an s. Le ’s w i e
290
11. Balance o ene gy-mass 11.14 Example applica ions: ideal gas and pis on
hem all oge he he e:
balances:
d𝑧(𝑡)
d𝑡=𝑣(𝑡)
d𝑃pis(𝑡)
d𝑡=𝐹pis(𝑡) + 𝐺pis(𝑡)
d𝐸(𝑡)
d𝑡=𝛷(𝑡)
cons i . ela ions:
𝑃pis(𝑡)=𝑚𝑣(𝑡)
𝐹pis(𝑡)=𝐹a m −𝐹gas(𝑡)
𝐺pis =−𝑚 𝑔
𝐸(𝑡)=𝐶𝑁𝑇(𝑡)
𝛷(𝑡)=𝛷bo +𝛷 op wi h
𝛷bo =𝐴 ℎ [𝑇ex −𝑇(𝑡)] 𝛷 op =𝐹gas(𝑡) · 𝑣(𝑡)
𝐹gas(𝑡)=−𝑅𝑁𝑇(𝑡)
𝑧(𝑡)+𝜇𝐴
𝑧(𝑡)𝑣(𝑡).
bounda y condi ions: 𝐺pis =−𝑚 𝑔 , 𝐹a m , 𝑇ex
physical cons an s: 𝐴 , 𝑚 , 𝑔 , 𝑁 , 𝐶 , ℎ , 𝑅 , 𝜇.
No e ha he e isn’ a clea dis inc ion be ween some cons i u i e ela ions
and bounda y condi ions. Fo ins ance
𝐺pis =−𝑚 𝑔
is a cons i u i e
ela ion, bu i can also be conside ed a bounda y condi ion because i ’s
cons an an known be o ehand in his se up.
«Exe cise 11.7
1.
Use he basic sc ip -w i ing s a egy
§10.10 p.213
o w i e a sc ip ha
simula es he sys em o ideal gas & pis on.
2. How can we choose he s a e a iables §10.10 p.217 o he sys em?
291
11. Balance o ene gy-mass 11.14 Example applica ions: ideal gas and pis on
3.
Simula e he sys em wi h he ollowing nume ical cons an s, bound-
a y condi ions, and ini ial condi ions:
𝐴=0.01m2, 𝑚 =10kg , 𝑔 =9.8N/kg ,
𝑁=0.04mol , 𝐶 =20J/(molK)
ℎ=8×103J/(Ksm2),𝜇=4×10−5Ns/m2
𝑅=8.31446261815324J/(molK),
𝑇ex =296.15K , 𝐹a m =−𝐴·105N/m2
𝑡0=0s , 𝑡1=1s ,Δ𝑡=0.0001s
𝑧(𝑡0)=0.1m , 𝑣(𝑡0)=0m/s, 𝑇(𝑡0)=296.15K
Plo he posi ion
𝑧(𝑡)
o he pis on and he empe a u e
𝑇(𝑡)
o he
ideal gas as unc ions o ime. Wha do you obse e?
4.
Simula eagainwi h hesame aluesasbe o ebu asu acecoe icien
o hea ans e
ℎ=0J/(Ksm2)
, ha is, assuming ha all ene gy
luxes a e adiaba ic. Plo again posi ion and empe a u e agains
ime. Wha do you obse e?
5.
Keeping all ene gy luxes adiaba ic, simula e now by se ing he
iscosi y coe icien 𝜇=400Ns/m2. Wha do you obse e?
6. Play and simula e wi h o he alues!
He e is an example sc ip o simula ing he sys em o ideal gas &
pis on and plo ing 𝑧(𝑡)and 𝑇(𝑡):
Download Oc a e e sion
idealgas_pis on.m17
Download Py hon e sion
idealgas_pis on.py18
1%%% Simula ion o ideal gas & mass pis on in 1D
2%% Coo dina es ( , z)
3
4%% Cons an s
5R = 8.31446261815; % J/(K*mol): mola gas cons an
6g = 9.8; % N/kg: g a i a ional accele a ion
7C = 20; % J/(K*mol): mola hea capaci y
8mu = 4e-5; % N*s/m^2: gas iscosi y
9h = 8e3; % J/(K*m^2): hea - ans e coe icien
10 N = 0.04; % mol: amoun o ideal gas
11 A = 0.1^2; % m^2: a ea o pis on
12 m = 10; % kg: mass o pis on
13
14 %% Ini ial condi ions. S a e: (z, , T)
15 = 0; % s: ini ial ime
16 z = 0.15; % m: ini ial posi ion o pis on
292
11. Balance o ene gy-mass 11.14 Example applica ions: ideal gas and pis on
17 = 0; % m/s: ini ial eloci y o pis on
18 T = 273.15 + 23; % K: ini ial empe a u e o gas
19
20 %% Bounda y condi ions
21 Gpis = -m*g; % N: g a i y supply o momen um o pis on
22 Fa m = -1e5 * A; % N: o ce on pis on by a mosphe e
23 Tex = 273.15 + 23; % K: empe a u e o en i onmen
24 % o he luxes a e ze o
25
26 %% Pa ame e s o ime loop
27 1 = 1; % s: inal ime
28 d = 0.0001; % s: ime s ep
29
30 ## Plo ing
31 d plo = 1/360; # ime in e al be ween plo s
32 plo = d plo ; # ime o nex plo
33 igu e
34 subplo (2, 1, 1); plo ( , z, '.b')
35 xlim([0, 1])
36 xlabel('{ i }/s'); ylabel('{ i z}/m')
37 axis(' igh '); g id on; hold on
38 subplo (2, 1, 2); plo ( , T, '. ')
39 xlim([0, 1])
40 xlabel('{ i }/s'); ylabel('{ i T}/K')
41 axis(' igh '); g id on; hold on
42
43 %% Nume ical ime in eg a ion
44 while < 1
45 %% cons i u i e ela ions
46 Fgas = -(R * N * T / z - mu * A * / z);
47 Fpis = -Fgas + Fa m;
48 Ppis = m * ;
49 Phibo = A * h * (Tex - T);
50 Phi op = Fgas * ;
51 Phi = Phibo + Phi op;
52 E=C*N*T;
53
54 %% s ep o wa d in ime wi h balance laws
55 = + d ;
56 Ppis = Ppis + (Fpis + Gpis) * d ;
57 z=z+ *d ;
58 E = E + Phi * d ;
59
60 %% cons i u i e ela ions: calcula e s a e
61 T = E / (C * N);
62 = Ppis / m;
63
293
11. Balance o ene gy-mass 11.15 Su aces o discon inui y
64 %% plo
65 i > plo
66 subplo (2, 1, 1); plo ( , z, '.b')
67 subplo (2, 1, 2); plo ( , T, '. ')
68 pause(0)
69 plo = plo + d plo ;
70 end
71 end
11.15 Su aces o discon inui y
F om ou discussion abou ic ion
§10.14 p.228
, we know wha happens
om he poin o iew o momen um when a pe son is pushing a hea y
objec , say a c a e, on he loo . The pe son is p o iding a cons an in lux
o ho izon al momen um
𝑭p
o he c a e, bu he loo is also p o iding
ho izon al-momen um in lux
𝑭
, kine ic ic ion, ha ing opposi e o ien a-
ion. Suppose ha he in lux by he pe son is equal and opposi e o he
ic ion:
𝑭p=−𝑭
. Then, acco ding o New on’s cons i u i e equa ion o
momen um
𝑷=𝑚𝒗
, he c a e mo es wi h cons an ho izon al eloci y
𝒗c a e, because i s ime- a e change o ho izon al momen um is ze o:
d𝑷
d𝑡=𝑭p+𝑭 =−𝑭 +𝑭p=0N .
Le usconside wha happens om hepoin o iewo ene gy.Th ough
he pa o con ol su ace whe e he pe son pushes he c a e he e’s an
in lux o mechanical powe
𝑭p·𝒗c a e
. Le ’s say ha he e’s no hea lux
he e. The o al-ene gy in lux h ough ha pa o su ace is hen
𝛷p=𝑭p·𝒗c a e >0.
This ene gy in lux is posi i e because he o ce
𝑭p
exe ed by he pe son
and he eloci y 𝒗c a e ha e he same o ien a ion.
Wha abou he con ac su ace o c a e and loo ? The e should be
an ene gy lux h ough he e oo. In ac , he mo ion o he c a e is igid
§11.12 p.272
; he e o e he posi i e mechanical powe coming om he pe son
should inc ease he kine ic o he g a i a ional po en ial ene gy o he
c a e. Bu hese wo ene gies a e cons an ; his means ha he e mus be a
nega i e in lux o mechanical powe somewhe e else. Ob iously i mus
occu h ough he su ace be ween c a e and loo . Wha kind o ene gy
lux occu s he e? is i only mechanical powe ? is i also hea ing?
294
11. Balance o ene gy-mass 11.15 Su aces o discon inui y
The cons i u i e ela ion
𝛷=𝑄+𝑭·𝒗
does no apply o he con ol
su ace be ween c a e and loo , because he eloci y o ma e
𝒗
is no
he same on bo h sides o he su ace. On he uppe side, he ma e
cons i u ing he c a e is mo ing wi h ho izon al eloci y
𝒗c a e
; on he
lowe side, he ma e cons i u ing he loo is a es , wi h ze o eloci y.
This con ac su ace is an example o su ace o discon inui y:
]Su ace o discon inui y
I a physical quan i y
𝑋
which is a unc ion o posi ion
𝒓
has wo di e en
limi alues as we conside posi ions close and close o he wo sides o
asmallcon olsu ace, hen hela e iscalleda su ace o discon inui y
o he quan i y 𝑋.
No e ha i doesn’ make sense o speak o su ace o discon inui y o
a quan i y which is no a unc ion o posi ion, such as he olume con en s
o ma e , momen um, ene gy, and so on. Volume con en s a e ela ed
o pa icula con ol olumes, no o posi ions. Quan i ies o which i
makes sense o speak o su aces o discon inui ies a e o example eloci y,
empe a u e, p essu e, ma e densi y, ene gy densi y, and simila .
A con ol su ace may be a su ace o discon inui y o some quan i y
bu no o ano he . In ou p esen example, o ins ance, he ma e eloci y
is di e en on he wo sides, bu he empe a u e o he loo and o he
bo om o he c a e migh be he same. This con ac su ace is hen a su ace
o discon inui y o he eloci y bu no o he empe a u e.
We shall now lea n a common and impo an echnique o s udy
su aceso discon inui y.This echnique allowsus oex end heapplica ion
o some cons i u i e ela ions also o cases whe e hey canno be applied
a i s . Ou discussion o he echnique is la gely in ui i e, bu i can be
made ma hema ically igo ous.
Le ’s e u n o he example wi h he c a e and he loo . Zoom in on
he imagina y con ol su ace ha sepa a es hem. Replace his su ace
wi h a e y hin imagina y con ol olume; see he side igu e. Two
ho izon al sides (ligh - ed dashed lines) o his con ol olume ha e he
same ex ension as he o iginal su ace and a e pa allel o i . One o hem
lies comple ely wi hin he c a e, and he o he comple ely wi hin he loo .
The la e al sides (da k- ed dashed lines) o he con ol olume ha e a e y
small heigh ℎ.
295
11. Balance o ene gy-mass 11.15 Su aces o discon inui y
Conside he luxes o momen um o his con ol olume. Thanks o
ex ensi i y, hey can be sepa a ed in o h ee con ibu ions:
Su ace wi hin c a e:
Ho izon al momen um (ha ing le wa d o ien a-
ion in he igu e), as ic ion, is coming om he loo and he lowe
pa s o he c a e, and is passed on o he uppe pa s o he c a e.
Th ough his su ace he e is he e o e an e lux o momen um equal
o 𝑭 .
All ma e close o his su ace has eloci y
𝒗c a e
, on bo h sides,
because his su ace is comple ely wi hin he c a e. Fo he ene gy
e lux
𝛷c a e
h ough his su ace we can he e o e use he cons i u i e
ela ion
𝛷c a e =𝑄c a e +𝑭 ·𝒗c a e
whe e 𝑄c a e is he hea e lux, possibly ze o.
Su ace wi hin loo :
Ho izon al momen um (ha ing igh wa d o ien a-
ion in he igu e), as ic ion, is coming om he c a e and he uppe
pa s o he loo , and is passed on o he lowe pa s o he loo .
Th ough his su ace he e is he e o e an e lux o momen um equal
o −𝑭 .
All ma e close o his su ace has ze o eloci y, on bo h sides, because
his su ace is comple ely wi hin he loo . Fo he ene gy e lux
𝛷 loo
h ough his su ace we can he e o e use he cons i u i e ela ion o
ma e , bu wi h ze o mechanical powe , owing o he ze o eloci y:
𝛷 loo =𝑄 loo
whe e 𝑄 loo is he hea e lux, possibly ze o.
Side su aces:
We imagine o ake he heigh
ℎ
o be smalle and smalle :
so small ha he a ea o he side con ol su aces is negligible. All
luxes h ough hese su aces a e, in ui i ely, negligible in he limi .
Now le ’s examine he balance o ene gy o his hin con ol olume.
Owing o he e y small heigh
ℎ
, he olume is ex emely small. The e o e
he ene gy con en
𝐸
and i s ime- a e o change
d𝐸/d𝑡
a e, in ui i ely,
negligible. The balance o ene gy hen yields
d𝐸
d𝑡=−𝛷c a e −𝛷 loo
0J/s=−(𝑄c a e +𝑭 ·𝒗c a e) − 𝑄 loo
296
11. Balance o ene gy-mass 11.15 Su aces o discon inui y
om which we ind
𝛷c a e =−𝛷 loo
𝑄c a e +𝑭 ·𝒗c a e =−𝑄 loo
This is a e y in e es ing esul . Zoom ou om he hin con ol olume
jus analysed, so ha i looks like he ini ial con ol su ace be ween c a e
and loo . Wha happens a his con ol su ace is he ollowing:
•
The symme y o lux
§4.6 p.98
is s ill alid o he lux o o al-ene gy:
he lux om loo o c a e,
𝛷c a e
, and he lux om c a e o loo ,
𝛷 loo , a e equal in magni ude bu opposi e: 𝛷c a e =−𝛷 loo .
•
Howe e , he ene gy lux appea s as hea lux plus mechanical powe
𝑄c a e +𝑭 ·𝒗c a e
on one side (c a e) o he su ace, bu only as hea lux
𝑄 loo on he o he side ( loo ).
In o he wo ds, he con ac su ace ac s as a so o con e e be ween
mechanical powe and hea ing. When we discuss he balance o en opy
chap e 15 p.325
we shall see ha his su ace can be conside ed as a peculia
so o o he modynamic engine. We shall also p o e ha posi i e in lux o
mechanical powe on one side can be con e ed in o posi i e ou lux o
hea on he o he side – bu he opposi e con e sion canno happen.
This con e sion, a a con ac su ace, be ween mechanical powe and
hea ing makes sense om a molecula poin o iew. Roughly speaking, he
isible kine ic ene gy o he mac oscopic, coo dina ed pa o he mo ion
o he molecules ha make up he c a e – wha we call ‘ he eloci y o
he c a e’ – is ans o med in o in isible kine ic ene gy o a mic oscopic,
uncoo dina ed mo ion o he molecules o c a e and loo . This is why
objec s ypically become wa me unde ic ion.
Bu no e an amazing ac : we ound ou abou his con e sion be ween
mechanical powe and hea ing by applying he balance laws, wi hou
in oking any molecula pic u e!
The esul abo e also shows, in acco dance wi h a p e ious wa ning
§11.8 p.260
, ha he p inciple o ex ensi i y only holds o he lux o o al-
ene gy and o he o al quan i ies, bu no necessa ily o he indi idual
pa s in o which we may decompose hose quan i ies; like hea lux and
mechanical powe in he case o ene gy lux.
297
12. Balance o angula momen um 12.4 Cons i u i e ela ions o angula momen um
Owing o in ellec ual ine ia, many books a e a aid o using wis ed
ec o s, and ely on o dina y ec o s ins ead, in oducing he ‘ igh -hand
ule’ o de e mine he sense o o a ion om he o ien a ion o he o dina y
ec o . I you’ e e e asked you sel “why he igh hand, and no he
le hand?”, he answe is ha i ’s pu ely a con en ion indeed; one could
ha e in oduced a le -hand ule ins ead. Using wis ed ec o s we don’
need hese a bi a y con en ions and mnemonics: he sense o o a ion is
unequi ocally indica ed.
Use whiche e ec o ep esen a ion you p e e !
12.4 Cons i u i e ela ions o angula
momen um
The peculia capaci y o he balance o angula momen um o be exp essed
in wo di e en ways has igh connec ions wi h he cons i u i e ela ions
o angula momen um. In ac i s cons i u i e ela ions could almos be
aken as de ini ions o he con en , lux, and supply o angula momen um.
They a e used in essen ially all si ua ions and heo ies: wi h ma e
and wi h elec omagne ic ields, in Gene al Rela i i y and in New onian
app oxima ion.Theonlyexcep iona ephysicalphenomenawi hpa icula
kinds o ma e ials a iously called pola ,mic opola ,o wi h in insic spin.
These gene al cons i u i e ela ions o con en , lux, and supply ha e
common o m, which in ol es he con en , lux, supply o momen um,
as well as he posi ion ec o in a gi en coo dina e sys em. We he e o e
p esen all o hem he e.
Choose a coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
; i does no need o be Ca esian
§2.6 p.54
. In he ollowing we shall speak o momen um; keep i can be he
momen um associa ed o ma e , o o elec omagne ic ield, o bo h –
he e a e no es ic ion wha soe e .
]Angula -momen um con en
Take a small con ol olume ha ing posi ion ec o
𝒓=[𝑥, 𝑦, 𝑧]
– i ’s
necessa y o he con ol olume o be small, o he wise we could no
associa e jus one posi ion o i . Deno e by
𝑷=[𝑃𝑥, 𝑃𝑦, 𝑃𝑧]
he con en
o momen um in his small con ol olume.
304
12. Balance o angula momen um 12.4 Cons i u i e ela ions o angula momen um
The con en o angula momen um
𝑳=[𝐿𝑥, 𝐿𝑦, 𝐿𝑧]
, de ined wi h espec
o he o igin o coo dina es is gi en by he ec o p oduc
𝑳=𝒓×𝑷o equi alen ly
𝐿𝑥=𝑦 𝑃𝑧−𝑧 𝑃𝑦
𝐿𝑦=𝑧 𝑃𝑥−𝑥 𝑃𝑧
𝐿𝑧=𝑥 𝑃𝑦−𝑦 𝑃𝑥
(12.3a)
The componen s
[𝐿𝑥, 𝐿𝑦, 𝐿𝑧]
can al e na i ely be deno ed
[𝐿𝑦𝑧, 𝐿𝑧𝑥 , 𝐿𝑥𝑦]
, as some ex s do. The la e no a ion makes he o -
mulae abo e easie o emembe :
𝐿𝑦𝑧 =𝑦 𝑃𝑧−𝑧 𝑃𝑦
𝐿𝑧𝑥 =𝑧 𝑃𝑥−𝑥 𝑃𝑧
𝐿𝑥𝑦 =𝑥 𝑃𝑦−𝑦 𝑃𝑥
(12.3b)
Choose whiche e you p e e .
]Angula -momen um lux
Take a small con ol su ace ha ing posi ion ec o
𝒓=[𝑥, 𝑦, 𝑧]
– i ’s
necessa y o he con ol su ace o be small, o he wise we could no
associa e jus one posi ion o i . Deno e by
𝑭=[𝐹𝑥, 𝐹𝑦, 𝐹𝑧]
he lux o
momen um, o su ace o ce, h ough his small con ol su ace.
The lux o angula momen um
𝑴=[𝑀𝑥, 𝑀𝑦, 𝑀𝑧]
, de ined wi h espec
o he o igin o coo dina es, is also called su ace o que and is gi en by
he ec o p oduc
𝑴=𝒓×𝑭o equi alen ly
𝑀𝑥=𝑦 𝐹𝑧−𝑧 𝐹𝑦
𝑀𝑦=𝑧 𝐹𝑥−𝑥 𝐹𝑧
𝑀𝑧=𝑥 𝐹𝑦−𝑦 𝐹𝑥
(12.4)
Analogously o angula -momen um con en , you can deno e he com-
ponen s by [𝑀𝑦𝑧, 𝑀𝑧𝑥 , 𝑀𝑥𝑦]ins ead.
305
12. Balance o angula momen um 12.4 Cons i u i e ela ions o angula momen um
]Angula -momen um supply
Take a small con ol olume ha ing posi ion ec o
𝒓=[𝑥, 𝑦, 𝑧]
. Deno e
by
𝑮=[𝐺𝑥, 𝐺𝑦, 𝐺𝑧]
he supply o momen um, o olume o ce, in his
small con ol olume.
The supply o angula momen um
𝓣=[𝒯𝑥,𝒯𝑦,𝒯𝑧]
, de ined wi h espec
o he o igin o coo dina es, is also called olume o que and is gi en by
he ec o p oduc
𝓣=𝒓×𝑮o equi alen ly
𝒯𝑥=𝑦 𝐺𝑧−𝑧 𝐺𝑦
𝒯𝑦=𝑧 𝐺𝑥−𝑥 𝐺𝑧
𝒯𝑧=𝑥 𝐺𝑦−𝑦 𝐺𝑥
(12.5)
As usual you can deno e he componen s by [𝒯𝑦𝑧,𝒯𝑧𝑥 ,𝒯𝑥𝑦]ins ead.
So he con en , lux, and supply o angula momen um o small con ol
olumes o su aces a e ob ained by he c oss-p oduc o posi ion
𝒓
and
he co esponding momen um quan i ies. Wha ’s “small” is de e mined
by he dis ances and eloci ies in ol ed in he phenomenon unde s udy.
Fo example, billia d balls canno be conside ed “small” when we s udy
hei collisions on a billia d able: hei o a ions lead o special mo ions;
on he o he hand, a plane can in some ci cums ances be conside ed as
“small” when we s udy i s mo ion in he sola sys em.
The cons i u i e ela ions abo e he e o e canno be used di ec ly o
ex ended con ol olumes o su aces. An ex ended con ol olume o su ace
doesn’ ha e a de ini e, single posi ion ec o
𝒓
. The only way o apply
he cons i u i e ela ions abo e o an ex ended con ol olume o su ace
is he e o e by ex ensi i y
§3.2 p.63
: di ide i in o pa s so small ha each
o hem can be cha ac e ized by a single posi ion ec o . This means
an in ini e numbe o small pa s: we a e pe o ming a h ee-dimensional
in eg a ion, possibly o e a complica ed shape.
Bu he di icul ies do no end he e. I he con ol olume o su ace
is mo ing, o a ing, de o ming, hen he posi ion ec o s
𝒓(𝑡)
o i s small
pa s will be changing wi h ime. This means ha we may need o pe o m
he h ee-dimensional in eg a ion anew a e e y ins an 𝑡.
I is because o hese complica ions wi h h ee-dimensional in eg a ion
ha he al e na i e enso ial exp ession
(12.2)
o he balance o momen um
is used when s udying ma e ials. Bu he e is a special case whe e he con-
s i u i e ela ions abo e may be sligh ly simpli ied: o s udy he beha iou
306
12. Balance o angula momen um 12.4 Cons i u i e ela ions o angula momen um
o ma e ials ha always unde go igid mo ion
§11.12 p.272
, ha is, igid bodies.
We shall s udy hese simpli ica ions in he nex sec ions.
«Exe cise 12.1
1.
Conside a GPS sa elli e as a poin ha has, a a gi en ins an , he
ollowingposi ionandmomen umcon en inageocen iccoo dina e
sys em (𝑡, 𝑥, 𝑦, 𝑧):
𝒓=1.4×107,1.6×107,0m
𝑷=−3.1×106,3.4×106,0Ns .
Calcula e he sa elli e’s angula momen um
𝑳
wi h espec o he
o igin o he coo dina e sys em ( emembe ha you should ob ain a
ec o wi h h ee componen s).
2.
Assume ha he GPS sa elli e has he ollowing ime-dependen
posi ion ec o and momen um:
𝒓(𝑡)=cos(𝜔𝑡),sin(𝜔𝑡),0]·2.1×107m
𝑷(𝑡)=−sin(𝜔𝑡),cos(𝜔𝑡),0] · 1.2×103Ns wi h 𝜔=3.64s−1.
Calcula e he ime dependence o he sa elli e’s angula momen um
𝑳(𝑡)
wi h espec o he o igin o he coo dina e sys em. T y o
explain he peculia esul you ob ain.
3.
A lying ennis ball, conside ed as a poin , has he ollowing ime-
dependen posi ion ec o in a coo dina e sys em (𝑡, 𝑥, 𝑦, 𝑧):
𝒓(𝑡)=𝑣𝑡, 0,−1
2𝑔𝑡2+𝑧0
wi h 𝑣=1m/s, 𝑔 =9.8m/s2, 𝑧0=3m .
(a)
Find i s ime-dependen momen um
𝑷(𝑡)
by using New on’s
cons i u i e ela ion ela ion o momen um, assuming a mass-
ene gy 𝑚=0.058kg.
(b)
Then ind i s ime-dependen angula momen um
𝑳(𝑡)
wi h
espec o he o igin o he coo dina es.
(c)
Rep esen he angula momen um you ound as a s aigh o
wis ed ec o .
307
12. Balance o angula momen um 12.5 P elimina y ema ks on cons i u i e ela ions o igid mo ion
12.5 P elimina y ema ks on cons i u i e
ela ions o igid mo ion
As al eady s a ed, he cons i u i e ela ions
(12.3a)
,
(12.4)
,
(12.5)
e ec i ely
co e almos all physical phenomena, e en in ol ing elec omagne ic
ields. Bu he use o hese cons i u i e ela ions ypically equi es di iding
any ex ended con ol olume in e y small pa s – a h ee-dimensional
spa ial in eg a ion. This di ision can be become compu a ionally expensi e
in simula ions.
In he case o ma e unde going a igid mo ion, he cons i u i e
ela ions o angula momen um and i s balance in in eg al o di e en ial
o m can be used a oiding some o he spa ial in eg a ion. We shall now
discuss hese simpli ied o m o he cons i u i e ela ions. In o de o do so
we need o in oduce h ee concep s: cen e o mass-ene gy,angula eloci y,
and enso o ine ia. These concep s apply o any mo ion, bu become
especially use ul wi h igid mo ion.
Ou pu pose will be o ge jus a gene al acquain ance wi h hese
concep s and ge a glimpse o how hey a e used in applica ions.
12.6 Cen e o mass-ene gy
The cen e o mass-ene gy can be de ined o any con ol olume ha ing a
non-ze o ene gy-mass con en : he e o e o con ol olumes ha con ain
ma e , o ha con ain only elec omagne ic ields, o bo h.
]Cen e o mass-ene gy
Conside a coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
and a con ol olume ha ing a
non-ze o mass-ene gy con en 𝑚.
Di ide he con ol olume in o smalle and smalle sub- olumes, which
wecannumbe
1,2,3, . . .
.When heya esmall enough,wecanassocia e
wi h hem he posi ions
𝒓1(𝑡),𝒓2(𝑡),𝒓3(𝑡), . . .
. Call
𝑚1, 𝑚2, 𝑚3, . . .
hei
mass-ene gy con en s; some o hese may be ze o, i some sub- olumes
a e emp y. By ex ensi i y we ha e ha 𝑚=𝑚1+𝑚2+𝑚3+ · · · .
308
12. Balance o angula momen um 12.6 Cen e o mass-ene gy
The cen e o mass-ene gy
𝒓c(𝑡)
o he con ol olume a coo dina e
ime 𝑡is de ined as
𝒓c(𝑡):=1
𝑚lim
smalle
sub- olumes𝑚1𝒓1(𝑡) + 𝑚2𝒓2(𝑡) + 𝑚3𝒓3(𝑡) + · · · .(12.6)
In he de ini ion o he cen e o mass-ene gy, all mass-ene gy is o be
coun ed: no only he es mass-ene gy, bu also in e nal ene gy, kine ic
ene gy, and so on. Fo a con ol olume con aining ma e he es mass-
ene gy only will be an ex emely good app oxima ion; o a con ol olume
con aining elec omagne ic ields, all he elec omagne ic ene gy mus be
conside ed.
The pen’s cen e o mass-
ene gy is igh abo e he bal-
ancing inge .
We ha e an in ui i e unde s anding o he cen e o mass-ene gy o
objec s in e e yday li e. When we y o balance an objec on ou hands o
inge s, we’ e ying o keep he objec ’s cen e o mass-ene gy di ec ly on
he e ical abo e he balancing poin . Fo con ol olumes o pa icula
shapes, he cen e o mass-ene gy may lie ou side he con ol olume i sel ;
hink o example abou he cen e o mass-ene gy o a ho seshoe.
The cen e o mass-ene gy in gene al does no s ay in a ixed posi ion
wi h espec o he con ol olume, bu mo es a ound as he olume
changes i s shape and he mass-ene gy wi hin lows in di e en di ec ions.
The cen e o mass-ene gy o a body in igid mo ion
§11.12 p.272
, howe e ,
does s ay in a ixed posi ion wi h espec o ha body.
«Exe cise 12.2
In he ICRS coo dina e sys em
§2.5 p.50
cen ed on he Sun, he Ea h and
Moon a coo dina e ime 𝑡ha e posi ions
𝒓E(𝑡)=
−2.759
13.20
5.727
·1010 m,𝒓M(𝑡)=
−2.744
13.17
5.710
·1010 m.
Ea h’s mass-ene gy is 5.972 ×1024 kg and Moon’s is 7.346 ×1022 kg.
1.
Calcula e he cen e o mass-ene gy o he Ea h-Moon sys em a
he same coo dina e ime, in his sys em o coo dina es.
2.
Conside ing ha Ea h’s mean adius is
6371km
, de e mine how
many kilome es abo e o below Ea h’s su ace is ha cen e o
mass-ene gy loca ed.
309
12. Balance o angula momen um 12.7 Angula eloci y
12.7 Angula eloci y
Gi en a coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
, we can de ine he angula eloci y o
a posi ion ec o
𝒓(𝑡)
wi h espec o ano he posi ion ec o
𝒓0(𝑡)
, ela i e
o he coo dina e sys em. I ep esen s how as , in e ms o
angle
ime
, he i s
posi ion ec o is ins an aneously o a ing wi h espec o he second poin .
]Angula eloci y
In a Ca esian coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
, ake a poin ha ing posi ion
ec o
𝒓(𝑡)
a coo dina e ime
𝑡
, and ano he posi ion ec o
𝒓0(𝑡)
dis inc
om he i s . Conside hei ec o di e ence
𝒓(𝑡) − 𝒓0(𝑡)
, which can be
seen as a ec o going om he second poin o he i s .
A e a sho ime lapse
Δ𝑡
he ec o
𝒓(𝑡+Δ𝑡) − 𝒓0(𝑡+Δ𝑡)
may ge
sligh ly sho e o longe , and i may ge sligh ly o a ed wi h espec
o he coo dina es
(𝑥, 𝑦, 𝑧)
. The axis o o a ion is pe pendicula o he
plane o med by 𝒓(𝑡) − 𝒓0(𝑡)and 𝒓(𝑡+Δ𝑡) − 𝒓0(𝑡+Δ𝑡).
Theangula eloci y
𝝎(𝑡)
o he posi ion
𝒓(𝑡)
wi h espec o he posi ion
𝒓0(𝑡)a ime 𝑡is a ( wis ed) ec o de ined as ollows:
magni ude:
he angle be ween
𝒓(𝑡) − 𝒓0(𝑡)
and
𝒓(𝑡+Δ𝑡) − 𝒓0(𝑡+Δ𝑡)
,
di ided by Δ𝑡;
di ec ion: he ins an aneous axis o o a ion;
o ien a ion: he sense o o a ion;
all o hem obse ed a ime 𝑡.
Angula eloci y has physical dimension ime−1, and uni ad/s.
The de ini ion abo e is only alid in si ua ions whe e we can use he
New onian app oxima ion. Fo he mos gene al de ini ion, which is
ma hema ically mo e complica ed and holds in any coo dina e sys em,
we mus conside how he me ic
§3.12 p.81
ge s de o med a ound he
posi ion 𝒓(𝑡)in he ime lapse Δ𝑡.
The angula eloci y is he eloci y ha he ec o poin ing om
𝒓0
o
𝒓
has in a di ec ion pe pendicula o i sel , also called angen ial eloci y,
scaled by he leng h o ha ec o .
I we conside he posi ion ec o s
𝒓1(𝑡),𝒓2(𝑡),𝒓3(𝑡), . . .
o di e en
poin s o a body o ma e , and one gi en e e ence posi ion
𝒓0(𝑡)
, hen
in gene al he angula eloci ies o he poin s wi h espec o
𝒓0(𝑡)
a e all
di e en and change wi h ime in di e en ways. Bu o a se o poin s
310
12. Balance o angula momen um 12.8 Tenso o ine ia
unde going a igid mo ion some hing special happens: all poin s ha e he
same angula eloci ies wi h espec o he cen e o mass-ene gy (o any o he
poin in he se ac ually).
The angula - eloci y ec o
𝝎≡ [𝜔𝑥,𝜔𝑦,𝜔𝑧]
can also ep esen ed as an angula - eloci y ma ix 𝜴:
𝜴=
0−𝜔𝑧𝜔𝑦
𝜔𝑧0−𝜔𝑥
−𝜔𝑦𝜔𝑥0
ha ing a ze o diagonal and elemen s o opposi e signs on opposi e sides
o he diagonal.
«Exe cise 12.3
A poin wi h ime-dependen posi ion
𝒓(𝑡)
is o a ing a ound he ixed
poin
𝒓0=[2,−3,0]
. The ajec o y o
𝒓(𝑡)
is pa allel o he
𝑧𝑥
plane.
The o a ion is clockwise i we look a
𝒓(𝑡)
om some poin a away on
he posi i e-
𝑦
axis. The poin
𝒓(𝑡)
does wo comple e o a ions e e y
second.
1. W i e he angula - eloci y ec o co esponding o his o a ion.
2. W i e he co esponding angula - eloci y ma ix.
12.8 Tenso o ine ia
The enso o ine ia is a ma hema ical objec ha encodes some in o ma ion
abou he dis ibu ion o mass-ene gy wi hin a con ol olume. This
ma hema ical objec is ep esen ed by a ma ix. I is de ined wi h espec
o a coo dina e sys em and a e e ence poin , which o ou pu pose we’ll
choose as he cen e o mass-ene gy.
]Tenso o ine ia
In a coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
, he enso o ine ia o momen o
ine ia
𝑰c(𝑡)
a ime
𝑡
o a se o poin s wi h espec o hei cen e o
311
12. Balance o angula momen um 12.8 Tenso o ine ia
mass-ene gy is gi en by a ma ix:
𝑰c(𝑡) ≡
𝐼𝑥𝑥 𝐼𝑥𝑦 𝐼𝑧𝑥
𝐼𝑥𝑦 𝐼𝑦𝑦 𝐼𝑦𝑧
𝐼𝑧𝑥 𝐼𝑦𝑧 𝐼𝑧𝑧
(12.7)
whe e all componen s depend on ime. This ma ix has equal elemen s
on opposi e sides o he diagonal, so i can ha e a mos six di e -
en componen s ( h ee on he diagonal and h ee on one side o he
diagonal).
I all coo dina es ha e physical dimension leng h, hen he componen s
o he enso o ine ia ha e dimension mass ·leng h2and uni kgm2.
The enso o ine ia is calcula ed
1
om he dis ibu ion o he mass-
ene gy in he igid body in he chosen coo dina e sys em. I he e o e also
depends on he shape o he body. Fo ins ance, he enso o ine ia o a
solid ball o adius 𝑅and o al mass-ene gy 𝑚, uni o mly dis ibu ed, is
𝑰c,ball =2
5𝑚𝑅2
1 0 0
0 1 0
0 0 1
,
he cen e o he ball being also i s cen e o mass-ene gy. In his pa icula
case he enso o ine ia does no depend on ime.
Gi en i s ela ion wi h mass-ene gy and possibly wi h ma e , we
suspec ha he enso o ine ia mus be somehow ela ed o hei balances
as well. Indeed i s ime de i a i e is gi en by a special equa ion:
]Time de i a i e o enso o ine ia
The ime de i a i e o he enso o ine ia, de ined wi h espec o he
coo dina e sys em
(𝑡, 𝑥, 𝑦, 𝑧)
and he cen e o mass-ene gy, is gi en by:
d
d𝑡𝑰c(𝑡)=𝜴(𝑡)𝑰c(𝑡) − 𝑰c(𝑡)𝜴(𝑡)(12.8)
whe e
𝜴
is he angula - eloci y ma ix. The igh side o his o mula
has wo mul iplica ions o wo ma ices. This equa ion is essen ially a
consequence o he balances o ma e and mass-ene gy.
312
12. Balance o angula momen um 12.9 Rigid mo ion II
12.9 Rigid mo ion II
Recall ha i a body o ma e unde goes a igid mo ion, hen all dis ances
be ween i s poin s emain cons an in ime. In o he wo ds, he body can
mo e a ound and o a e, bu i doesn’ change shape o expand o con ac .
We call a body ha only unde goes a igid mo ion a igid body.
In o de o desc ibe he mo ion o a gene ic body o ma e , like a pa cel
o wa e , we need o speci y he posi ion o eloci y o each o i s poin s,
because any wo poin s could suddenly mo e a apa o close oge he .
In p inciple his means an in ini y o posi ions o be speci ied.
To desc ibe he mo ion o a igid body, ins ead, we only need six
pa ame e s, hanks o i s igidi y. Th ee o hese six pa ame e s co espond
o he mo ion o one ixed poin in he igid body. The emaining h ee
co espond o he o a ion o he igid body wi h espec o he coo dina e
sys em.
These six pa ame e s can be de ined in di e en bu equi alen ways.
We choose he ollowing: he cen e o mass-ene gy:
𝒓c(𝑡) ≡ 𝑥c(𝑡), 𝑦c(𝑡), 𝑧c(𝑡),
om which we can calcula e he eloci y
𝒗c(𝑡):=d𝒓c(𝑡)
d𝑡≡d𝑥c(𝑡)
d𝑡,d𝑦c(𝑡)
d𝑡,d𝑧c(𝑡)
d𝑡;
and he angula eloci y o any poin o he igid body wi h espec o i s
cen e o mass-ene gy:
𝝎(𝑡) ≡ 𝜔𝑥(𝑡),𝜔𝑦(𝑡),𝜔𝑧(𝑡).
Toge he , hey allow us o ind he eloci y o any poin o he igid
body:
]Veloci y o a poin in a igid body
I
𝒓(𝑡)
is he posi ion ec o o a poin o a igid body a ime
𝑡
in a
coo dina e sys em (𝑡, 𝑥, 𝑦, 𝑧), hen i s eloci y 𝒗(𝑡)is gi en by
𝒗(𝑡)=𝒗c(𝑡) + 𝝎(𝑡) × [𝒓(𝑡) − 𝒓c(𝑡)] (12.9)
whe e
𝒓c(𝑡)
and
𝒗c(𝑡)
a e he posi ion and eloci y o he cen e o
mass-ene gy, and 𝝎(𝑡)is he angula eloci y.
313
12. Balance o angula momen um 12.12 Cons i u i e ela ions o o ces and o ques in a igid body
URLs o chap e 12
1. h ps://sciencewo ld.wol am.com/physics/Momen o Ine ia.h ml
2. h ps://ma hwo ld.wol am.com/Qua e nion.h ml
320
13
Balance o boos momen um
> and i Noe he ’s heo em s ill wo ks in hese cases, why
he heck do books no alk abou he conse ed quan i ies
co esponding o such symme ies?
Ei he 1) he ex book w i e s a e oo s upid o ha e
hough abou his issue, o 2) hey ha e decided i ’s be e
o le e e ybody igu e his ou o hemsel es, o 3) hey
eel he answe is no su icien ly impo an o was e a
p ecious pa ag aph on i . I don’ know. When I become
king o he uni e se, I will make all books on mechanics
men ion his issue.
J. Baez 2006
13.1 Fo mula ion and gene ali ies
]Balance o boos momen um o ene gy-mass momen
Volume con en : ?Flux: ?Supply: ?
(𝑡1)=(𝑡0) + ∫𝑡1
𝑡0
(𝑡)d𝑡+∫𝑡1
𝑡0
(𝑡)d𝑡
in eg al exp ession
d(𝑡)
d𝑡=(𝑡)+(𝑡)
di e en ial exp ession
(13.1)
{To be w i en in a la e e sion
321
13. Balance o boos momen um 13.1 Fo mula ion and gene ali ies
322
14
Rema ks on momen um and
ene gy
I hold in ac
(1) Tha small po ions o space a e in ac o a na u e
analogous o li le hills on a su ace which is on he a e age
la ; namely, ha he o dina y laws o geome y a e no
alid in hem.
(2) Tha his p ope y o being cu ed o dis o ed is
con inually being passed on om one po ion o space o
ano he a e he manne o a wa e.
(3)
Tha his a ia ion o he cu a u e o space is wha
eally happens in ha phenomenon which we call he
mo ion o ma e , whe he ponde able o e he ial.
(4) Tha in he physical wo ld no hing else akes place
bu his a ia ion, subjec (possibly) o he law o con inui y.
W. K. Cli o d 1876
14.1 Common misunde s andings on
momen um, ene gy, angula momen um
Momen um o wha ? Ene gy o wha ?
{To be w i en in a la e e sion
323
14. Rema ks on momen um and ene gy 14.1 Common misunde s andings on momen um, ene gy, angula momen um
324
15
Balance o en opy
Thei a ious “second laws” sound mo e like wa nings o
h ea s han p inciples o a a ional science.
C.A. T uesdell, III 1984
15.1 Fo mula ion and gene ali ies
]Balance o en opy
Volume con en : 𝑆Flux: 𝛱
𝑆(𝑡1) ≥ 𝑆(𝑡0) + ∫𝑡1
𝑡0
𝛱(𝑡)d𝑡
in eg al exp ession
d𝑆(𝑡)
d𝑡≥𝛱(𝑡)
di e en ial exp ession
(15.1)
The balance o en opy exp esses wha ’s commonly called “second
law o he modynamics”. En opy and i s balance a e success ully used in
many applica ions, bu ou unde s anding o hem and o hei physical
ounda ion is s ill incomple e. This s a e o a ai s is e lec ed in he many
and wildly di e en p esen a ions o en opy and i s balance: di e en
in wo ding, ma hema ical o mula ion, scope, and some imes e en in
physical consequences.
Many ex books only p esen limi ed and specialcaseso p ope iesand
uses o en opy and i s balance, and un o una ely hey o en make hese
limi ed, special cases appea as mo e gene al, o o b oade applica ion,
han hey ac ually a e. Such ex books also ypically es ic hemsel es
o si ua ions whe e he e he con en s o quan i ies in a con ol olume
do no change wi h ime, and he luxes a e ze o. We call his a si ua ion
325
15. Balance o en opy 15.1 Fo mula ion and gene ali ies
o equilib ium. The discipline ha s udies equilib ium si ua ions is called
he mos a ics.
In hese lec u e no es, en opy and i s balance a e p esen ed om a
poin o iew, ac i ely de eloped and used since he 1960s, ha ing he
ollowing ea u es:
•
I has been used o many yea s in conc e e echnological applica ions
(some examples a NASA: Chang & Haddad 1971; Hughes, F anca,
e al. 1986; Tu on, Camanho, e al. 2004; Diosady, Mu man, e
al. 2018; Ka o & Rose 2020), and in he s udy complex ma e ials such
as polyme s and mix u es.
•
I has led o new physical cons i u i e ela ions, o o he physical
and ma hema ical ounda ion o exis ing ones, om i s p inciples.
•
I is o mula ed wi h he same ma hema ics, and a he same ma hem-
a ical le el, as he physics o ma e , momen um, angula momen um,
ene gy, and elec omagne ism.
•
I includes ime-dependen phenomena and is ully connec ed wi h
phenomena in ol ing he o he six basic quan i ies.
In he echnical li e a u e he en opy balance abo e, used in i s ull
gene ali y, goes unde he name o Clausius-Duhem inequali y1.
The modynamic en opy and s a is ical en opy
One added di icul y is ha en opy and i s balance can also be app oached
om a comple ely di e en di ec ion, especially when we s udy physical
sys ems on small scales. I ’s he app oach o s a is ical mechanics
2
, which
conside s physical si ua ions in whe e we lack in o ma ion abou ini ial
condi ions, o bounda y condi ions, o cons i u i e ela ions. In s a is ical
mechanics, a concep ually di e en en opy appea s, no as a physical
quan i y, bu as a measu e o ou lack o in o ma ion abou he physical
sys em, in he s ic echnical sense o In o ma ion Theo y
3
. Also his
en opy sa is ies a balance law e y simila o (15.1) abo e.
One o he easons o he bewilde men which is some imes el a an
unhe alded appea ance o he e m en opy is he supe abundance o
objec s which bea his name. On he one hand, he e is a la ge choice o
mac oscopic quan i ies ( unc ions o s a e a iables) called en opy, on he
o he hand, a a ie y o mic oscopic quan i ies, simila ly named, associa ed
wi h he loga i hm o a p obabili y o he mean alue o he loga i hm o a
densi y. Each one o hese concep s is sui ed o a speci ic pu pose. Mo e
con using, howe e , han he lack o imagina ion in e minology is he ac
326
15. Balance o en opy 15.1 Fo mula ion and gene ali ies
ha se e al o hese dis inc concep s, di e en in meaning and in nume ical
alue, may be signi ican in a single p oblem. (G ad 1961 §1 p. 323)
We hus ha e a he modynamic en opy and a s a is ical en opy.
Thei ascina ing ela ion is only pa ly unde s ood, and s ill he objec o
some deba e. In he p esen no es we shall ocus on he modynamic en opy.
En opy depends on he obse a ion scale
The en opy con en in a con ol olume and he en opy lux ac oss
a con ol su ace a e no uniquely de ined; ha is, se e al choices a e
possible, each one co ec in a speci ic si ua ion. This non-uniqueness is
ac ually wo- old. In he p esen sec ion we discuss a i s sense in which
en opy is non-unique; in a la e sec ion we’ll discuss a second sense.
Fi s le ’s make clea ha he en opy con en in a gi en con ol olume,
and he en opy lux h ough a gi en con ol su ace, a a gi en coo dina e
ime, do no depend on he coo dina e sys em chosen. In his ega d hey a e
like he con en & lux o ma e , elec ic cha ge, and magne ic lux; and
unlike momen um, angula momen um, and ene gy.
On he o he hand, en opy con en & lux do depend on he de ail and
scale o obse a ion o a physical phenomenon. In his ega d hey a e unlike
all o he six undamen al quan i ies, whose o al con en and lux do no
depend on he obse a ion scale.
As an example, ake a con ol olume con aining ai . This con ol
olume could be s udied, desc ibed, and measu ed in h ee di e en ways:
(a) as con aining a con inuous, luid amoun o ma e o one kind: ‘ai ’;
(b) as con aining a con inuous, luid mix u e o amoun s o ma e o
di e en kinds: ni ogen, oxygen, and se e al o he s; (c) as con aining a
bunch o molecules in mo ion.
The ene gy con en measu ed wi hin his con ol olume, a a pa icula
ime ins an , will be exac ly he same in all h ee cases. Wha changes
among hem is he di ision o his o al ene gy in o in e nal and kine ic
§11.6 p.255, bu he o al is he same.
The en opy con en assigned o his con ol olume, on he o he hand,
will be di e en in each case.
A gi en objec o s udy canno always be assigned a unique alue, i s
“en opy”. I may ha e many di e en en opies, each one wo hwhile. The
p ope choice will depend on he in e es s o he indi idual, he pa icula
phenomena unde s udy, he deg ee o p ecision a ailable o a bi a ily
327
15. Balance o en opy 15.2 The physical ole o he balance o en opy
decided upon, o he me hod o desc ip ion which is employed; and each
o hese c i e ia is la gely subjec o he disc e ion o he indi idual. [. . .]
Fo ano he example we u n o ae odynamics. The exis ence o di usion
be ween oxygen and ni ogen somewhe e in a wind unnel will usually be
o no in e es . The e o e he ae odynamicis uses an en opy which does
no ecognize he sepa a e exis ence o he wo elemen s bu only ha o
“ai ”. In o he ci cums ances, he possibili y o di usion be ween elemen s
wi h a much smalle mass a io (e.g., 238/235) may be conside ed qui e
ele an . (G ad 1961 §1 pp. 323, 325)
We shall now see ha his peculia dependence o en opy on he
de ails and scale o obse a ion ac ually makes a lo o sense when we
unde s and how he en opy balance is used.
15.2
The physical ole o he balance o en opy
The en opy balance
(15.1)
has an appa en peculia i y, compa ed o he
gene al o m o a balance law §5.5 p.129: i is no an equali y
. . . =. . .
bu an inequali y
. . . ≥. . .
Why? and wha a e he consequences o his peculia i y?
Many ex s and media y o summa ize he meaning o he en opy
balance and i s inequali y sign in simple e ms. Bu he eali y is ha his
inequali y leads o an amazing a ie y o e y di e en phenomena, and
canno be summa ized in wo ds.
This should no be su p ising. Take he balance o momen um o
example. I leads o all so s o mo ions and de o ma ions o objec s, bu
also o he s abili y o objec s: om he ex emely complica ed mo ion o
he a mosphe e o he s illness o a pen es ing on a able. This balance
law could no be summa ized in some simple sen ence, like “objec s
spon aneously all downwa d”. Fi s , such a sen ence would be alse in
many si ua ions: jus look a he mo ion o ocks expelled by a olcano,
o a cosmological expansion. Second, i would be useless o p ecise
p edic ions and nume ical simula ions.
Themixingo woliquids and
he g owing o li e a e bo h
consequences o he second
law o he modynamics.
The same ema k holds ue o he balance o en opy. This balance also
leads o a wild a ie y o physical consequences. The mixing o wo liquids
can be said o be a consequence o he balance o en opy ( oge he wi h he
328
15. Balance o en opy 15.3 I e e sibili y
o he balances). Bu also he appea ance o li e on Ea h is a consequence o
he balance o en opy; and all complex physical mechanisms unde lying
li e a e consequences o he balance o en opy, oo ( oge he wi h he
o he balances). Any simplis ic summa ies, like common ones men ioning
“ endency o diso de ” o “spon aneously doing his o ha ” o simila ,
a e (a) simply alse in many physical si ua ions; (b) ague: wha does ‘ o
end’ mean? how is ‘diso de ’ de ined and quan i ied? o wha ini ial
alues o posi ion, eloci y, and so on, does ‘spon aneous’ e e o? wha ’s
‘spon aneous’ and wha ’s no ? (c) useless o quan i a i e p edic ions.
15.3 I e e sibili y
F om a quali a i e poin o iew, he inequali y sign in he balance o
en opy exp esses i e e sibili y; bu we mus unde s and his wo d in he
igh way.
Conside i s a gene ic balance o conse a ion law; le ’s ake conse -
a ion o ma e o ins ance:
𝑁(𝑡1)=𝑁(𝑡0) + ∫𝑡1
𝑡0
𝐽(𝑡)d𝑡
Suppose a con ol olume con ains an amoun
20mol
o ma e . Du ing an
in e al o ime we p o ide a ne low o ma e
∫𝑡1
𝑡0𝐽(𝑡)d𝑡=3mol
in o he
con ol olume. By he conse a ion law, a he end we ha e an amoun
23mol
o ma e in he con ol olume. This also means ha in p inciple
we could e e o an amoun o
20mol
by p o iding a nega i e amoun o
he same low as be o e,
−3mol
. We can, in p inciple, change he ma e
con en in he olume be ween
20mol
and
23mol
by p o iding a ne low
o
+3mol
o i s opposi e
−3mol
. I we conside a balance wi h a supply,
he conclusion emains he same, i we in e also he sign o he supply.
Now conside he balance o en opy wi h a s ic inequali y:
𝑆(𝑡1)>𝑆(𝑡0) + ∫𝑡1
𝑡0
𝛱(𝑡)d𝑡
Suppose a con ol olume con ains an amoun
20J/K
o en opy. Du ing
an in e al o ime we p o ide a ne low o en opy
∫𝑡1
𝑡0𝛱(𝑡)d𝑡=3J/K
in o
he con ol olume. By he inequali y abo e, a he end we canno ha e an
amoun
23J/K
o en opy in he con ol olume: we mus ha e a la ge
amoun , o ins ance
23.1J/K
. I we wan o e e o an en opy con en o
329
15. Balance o en opy 15.7 The mal engines
o ma e , momen um, ene gy be ween he engine and i s ex e io ; in
pa icula , exchanges o hea and o mechanical powe . And wha e e
happens wi hin he engine, ha is, wi hin ou imagina y con ol olume,
mus obey he se en uni e sal balance laws.
This powe ul eedom in choosing a con ol olume, when combined
wi h he balances o en opy and ene gy, can lead o amazingly gene al
physical esul s. Thanks o hese esul s we can o example p e en
was ing ou e o s in echnological ideas ha would e en ually u n ou
o be un easible. We shall now see a couple o examples.
Fi s o all le ’s de ine wha we mean by ‘engine’.
]Cyclic he mal engine
Acyclic he mal engine is de ice ha can abso b o emi bo h hea and
mechanical wo k, and ha can ope a e epea edly, in p inciple o e e .
The abili y o ope a e o e e means ha a ecu ing poin s in ime he
de ice mus be ind i sel in he same physical s a e
§10.10 p.217
, so as o
s a o e .
A he mal engine can also ecei e o elease ma e , momen um, angula
momen um, and elec omagne ic quan i ies.
No e he equi emen o ope a ing cyclically. A de ice consis ing o
an elec ic ba e y ac ua ing a mechanical a m, o ins ance, is no a
cyclic he mal engine, because i will cease ope a ing once he ba e y
is exhaus ed. Some o he heo ems ha we discuss below a e no alid
wi hou he condi ion o cyclic ope a ion.
«Exe cise 15.2
1.
Take a con ol olume w apping a ca . Can his con ol olume be
conside ed as a cyclic he mal engine? Why o why no ?
2.
Take a con ol olume w apping he engine o adi ional in e nal-
combus ion ca : he sys em o pis ons, al es, sha s, bu excluding
he uel. Can his con ol olume be conside ed as a cyclic he mal
engine? A e he e any luxes o ma e du ing an ope a ion cycle?
336
15. Balance o en opy 15.8 One- empe a u e he mal engines: hea e s
15.8
One- empe a u e he mal engines: hea e s
Would i be possible o build a cyclic he mal engine ha akes some
ene gy in he o m o hea om an inle a cons an empe a u e, and eleases
ene gy in he o m o wo k, say by li ing an objec ?
The ope a ion o such an engine is schema ized in he side igu e.
Imagine o w ap he engine, no ma e how complex i is, in a closed
con ol su ace, de ining a con ol olume. A pa o he con ol su ace, in
ed in he igu e, delimi s he inle h ough which he engine ecei es a
hea lux
𝑄(𝑡)
, possibly a iable in ime. The empe a u e
𝑇
a he inle is
cons an in ime. Ano he pa o he con ol su ace, in blue in he igu e,
delimi s he mo able componen s h ough which he engine is eleasing
mechanical powe
−𝑭(𝑡) · 𝒗(𝑡)
, whe e
𝑭(𝑡)
is he in lux o momen um
h ough ha pa , and
𝒗(𝑡)
is he eloci y o he ma e se in o mo ion;
bo h can a y wi h ime. The exp ession o he mechanical powe has a
minus sign because i ’s he powe we ecei e om he engine, so i ’s an
e lux o he engine. Th ough he es o he con ol su ace he e, in g ey,
he e a e no exchanges o hea o mechanical powe , bu he e may be luxes o
ma e , momen um, angula momen um, and elec omagne ic quan i ies.
A he end o a cycle he olume con en s o all hese quan i ies a e, by
de ini ion o ‘cycle’, he same as a he beginning. F om he balance laws,
he ime-in eg a ed luxes o all hese quan i ies mus be he e o e be ze o
o e a cycle.
Bu no e ha he ime-in eg a ed lux o hea
𝑄
and he mechanical
powe
−𝑭·𝒗
can bo h be non-ze o o e a cycle: he balance o ene gy
equi es only ha hei sum be ze o.
The engine s a s a cycle a ime
𝑡0
and ope a es un il ime
𝑡1
, a which
ime i s physical s a e is exac ly he same as a
𝑡0
. The cycle is comple e,
and he engine is eady o s a a new ope a ion cycle. Le ’s in oduce
symbols o he ime-in eg a ed amoun o hea we p o ide o he engine
and he o al amoun o wo k we ecei e om i :
Δ𝐻:=∫𝑡1
𝑡0
𝑄(𝑡)d𝑡Δ𝑊:=−∫𝑡1
𝑡0
𝑭(𝑡) · 𝒗(𝑡)d𝑡
The ne amoun o o al ene gy lowing in o he con ol olume be ween
𝑡0and 𝑡1is he e o e
∫𝑡1
𝑡0
𝛷(𝑡)d𝑡=∫𝑡1
𝑡0
𝑄(𝑡)d𝑡+∫𝑡1
𝑡0
𝑭(𝑡) · 𝒗(𝑡)d𝑡≡Δ𝐻−Δ𝑊
337
15. Balance o en opy 15.8 One- empe a u e he mal engines: hea e s
We al eady ema ked ha his ime-in eg a ed lux mus be ze o: om
he balance o ene gy and he condi ion o cyclic ope a ion:
𝐸(𝑡1)=𝐸(𝑡0) + ∫𝑡1
𝑡0
𝛷(𝑡)d𝑡≡𝐸(𝑡0)+Δ𝐻−Δ𝑊
we ind
Δ𝑊= Δ𝐻(15.4)
ha is, he mechanical wo k p oduced in a cycle mus be equal o he o al
amoun o hea p o ided in a cycle, as expec ed.
The balance o ene gy doesn’ equi e mo e han his. Acco ding o he
balance o ene gy his engine is he e o e admissible, as long as “hea in
=
wo k ou ”.
Le ’s see wha he balance o en opy says. We ha e
𝑆(𝑡1) ≥ 𝑆(𝑡0) + ∫𝑡1
𝑡0
𝛱(𝑡)d𝑡=𝑆(𝑡0) + ∫𝑡1
𝑡0
𝑄(𝑡)
𝑇d𝑡cons i . ela ion o en opy lux
=𝑆(𝑡0) + 1
𝑇∫𝑡1
𝑡0
𝑄(𝑡)d𝑡 empe a u e was assumed cons an
=𝑆(𝑡0) + Δ𝐻
𝑇no a ion o o al hea in low
=⇒𝑆(𝑡1) ≥ 𝑆(𝑡0) + Δ𝐻
𝑇
Also in his case he ini ial and inal en opy con en in a cycle mus be he
same: 𝑆(𝑡1)=𝑆(𝑡0). We ind
Δ𝐻
𝑇≤0=⇒
because 𝑇>0
Δ𝐻≤0(15.5)
This is a ema kable esul : i ’s impossible o gi e a posi i e ne amoun o hea ,
in a cycle, o such an engine. O he wise he en opy-balance law would be
b oken.
Toge he wi h he equali y
(15.4)
ob ained om he balance o ene gy,
we also ind ha
Δ𝑊≤0(15.6)
ha is, i ’s impossible o ecei e a posi i e ne amoun o wo k, in a cycle, om
such an engine.
We conclude ha an engine designed in his way canno gi e us any posi i e
ne wo k. No ma e which kind o ingenious echnology o ma e ials we
338
15. Balance o en opy 15.9 Two- empe a u e he mal engines: bounds on mechanical wo k
ied o use, we would ne e be able o cyclically gain posi i e mechanical
wo k om i .
No e ha he opposi e use, hough, is physically possible: we can,
cyclically, p o ide posi i e wo k o he engine and ecei e posi i e hea
ou a a cons an empe a u e. Indeed his is how many hea ing sys ems
ope a e.
I ’s impo an o unde s and co ec ly wha ’s possible and wha ’s
no . We can o cou se p o ide posi i e hea o he de ice, a a cons an
empe a u e, as much as we please. The esul abo e says ha we won’
be able o e u n he de ice o i s ini ial s a e as long as we do so. The
incompa ibili y is be ween:
•posi i e ne amoun o hea ,
•cons an empe a u e,
•cyclic ope a ion.
Bu no e ha he con e se is possible: we can ex ac a posi i e ne
amoun o hea om he de ice, a cons an empe a u e, as much as we
please, and e u n he de ice o i s ini ial s a e.
«Exe cise 15.3
An impo an esul o his sec ion is he inequali y
(15.5)
, which says
ha his engine canno abso b a posi i e amoun o hea in a cycle. Now
hink o when, in a cold day, you a e s anding s ill in on o a i e,
o unde he sun. The e’s no doub ha you a e abso bing a posi i e
amoun o hea , and he empe a u e could be cons an and uni o m.
Why isn’ his si ua ion in con adic ion wi h inequali y (15.5)?
15.9
Two- empe a u e he mal engines: bounds
on mechanical wo k
Le ’s y a di e en design, whe e one o he condi ions in he p e ious
sec ion is d opped. The abili y o ope a e cyclically is e y con enien ,
so we keep i . Wha happens i we le hea be exchanged a di e en
empe a u es?
Le ’s modi y he design so ha he exchange o hea happens a wo
di e en empe a u es. This could be done by changing he empe a u e
o one pa o he su ace o e ime (wi hin a cycle), o by allowing hea
339
15. Balance o en opy 15.9 Two- empe a u e he mal engines: bounds on mechanical wo k
o be exchanged a wo di e en inle s, ha ing cons an bu di e en
empe a u es. We choose he second op ion because i ’s easie o analyse
and leads o he same esul s as he i s .
The new engine design is schema ized in he side igu e. An in lux o
hea
𝑄+(𝑡)
occu s h ough a pa o he closed con ol su ace, in da k ed,
a cons an empe a u e
𝑇+
; ano he in lux o hea
𝑄−(𝑡)
occu s h ough
ano he pa o he su ace, in ligh ed, a cons an empe a u e
𝑇−
. Ei he
lux can be posi i e o nega i e. Fo de ini eness le ’s say ha
𝑇+>𝑇−
bu we a e no making assump ions abou ela i e magni udes o
𝑄+(𝑡)
and
𝑄−(𝑡)
. Th ough ano he , mo able pa o he con ol su ace, in blue,
he engine is eleasing mechanical powe
−𝑭(𝑡) · 𝒗(𝑡)
, which can also be
posi i e o nega i e. Th ough he es o he su ace, in g ey, he e may be
luxes o ma e and o he quan i ies, excep hea and mechanical powe ;
he ime-in eg a ed luxes o such quan i ies a e ze o o e an ope a ion
cycle.
Conside a cycle o he engine be ween imes
𝑡0
,
𝑡1
. Employ simila
symbols as be o e:
Δ𝐻+:=∫𝑡1
𝑡0
𝑄+(𝑡)d𝑡Δ𝐻−:=∫𝑡1
𝑡0
𝑄−(𝑡)d𝑡
Δ𝑊:=−∫𝑡1
𝑡0
𝑭(𝑡) · 𝒗(𝑡)d𝑡
so ha he ne amoun o ene gy lowing in o he con ol olume in his
cycle is ∫𝑡1
𝑡0
𝛷(𝑡)d𝑡= Δ𝐻++Δ𝐻−−Δ𝑊
The balance o ene gy applied o he engine’s con ol olume leads o:
𝐸(𝑡1)=𝐸(𝑡0)+Δ𝐻++Δ𝐻−−Δ𝑊 , 𝐸(𝑡1)=𝐸(𝑡0)
=⇒Δ𝑊= Δ𝐻++Δ𝐻−.(15.7)
Tha is, he ne wo k eleased in a cycle mus equal o he ne hea p o ided,
as expec ed. Ob iously we would like
Δ𝑊
o be posi i e; le ’s see i his is
possible wi h ou new engine design.
340
15. Balance o en opy 15.9 Two- empe a u e he mal engines: bounds on mechanical wo k
The ime-in eg a ed lux o en opy o he engine’s con ol olume is
∫𝑡1
𝑡0
𝛱(𝑡)d𝑡=∫𝑡1
𝑡0𝑄+(𝑡)
𝑇++𝑄−(𝑡)
𝑇−d𝑡≡Δ𝐻+
𝑇++Δ𝐻−
𝑇−
whe e we ha e again used ou sho symbols o he ime-in eg a ed hea ,
and aken in o accoun ha he empe a u es a e cons an .
The balance o en opy, du ing a cycle, he e o e says
𝑆(𝑡1) ≥ 𝑆(𝑡0) + Δ𝐻+
𝑇++Δ𝐻−
𝑇−, 𝑆(𝑡1)=𝑆(𝑡0)
=⇒Δ𝐻+
𝑇++Δ𝐻−
𝑇−≤0.
Fo his new engine, he en opy balance is no saying ha he ne amoun
o hea p o ided in a cycle mus be nega i e o ze o. I looks like he new
engine design migh wo k.
Wi h a li le algeb a, and ecalling ha a he modynamic empe a u e
is always posi i e, so ha
𝑇+>𝑇−>0
, we can ew i e he inequali y
abo e as ollows:
Δ𝐻−≤ −𝑇−
𝑇+Δ𝐻+.(15.8)
This inequali y has se e al in e es ing consequences. Suppose o ins ance
ha he ne amoun o hea
Δ𝐻+
p o ided a he highe empe a u e is
s ic ly posi i e. The ac ion
𝑇−/𝑇+
is also posi i e. Then
Δ𝐻−
mus be
s ic ly nega i e, because i mus be smalle han a s ic ly nega i e quan i y.
The en opy balance he e o e says ha in a cycle, i he ne amoun o hea
p o ided a he highe empe a u e is posi i e, hen he ne amoun p o ided a
he lowe empe a u e mus be nega i e.
Bu he mo e in e es ing consequence o he inequali y abo e conce ns
ou main ques ion: does his new wo- empe a u e design allow us o
ecei e posi i e ne wo k
Δ𝑊
om he engine, in a cycle? To answe his
ques ion, add Δ𝐻+ o bo h sides o he inequali y:
Δ𝐻++Δ𝐻−≤Δ𝐻+−𝑇−
𝑇+Δ𝐻+.
The le side is equal o he ne wo k gained in a cycle, om o mula
(15.7)
,
which comes om he balance o ene gy. The igh side can be ew i en
wi h a li le algeb a. We ob ain he ollowing impo an inequali y:
Δ𝑊≤1−𝑇−
𝑇+Δ𝐻+(15.9)
This is he combined consequence o he balances o ene gy and en opy
o his cyclic he mal engine. The ac o 1−𝑇−
𝑇+on he igh side is called
341
15. Balance o en opy 15.9 Two- empe a u e he mal engines: bounds on mechanical wo k
he e iciency o he he mal engine. Since he modynamic empe a u e
canno be nega i e, he e iciency canno be g ea e han
1
; and since
𝑇−<𝑇+by design, he e iciency canno be nega i e:
0<1−𝑇−
𝑇+<1.
Does he inequali y
(15.9)
allow us o ge posi i e ne wo k in a cycle?
Yes! The he mal-e iciency ac o is posi i e. I he hea
Δ𝐻+
p o ided o
he engine a he highe empe a u e is also posi i e, hen he igh side o
ha inequali y is posi i e. The ne wo k can he e o e be posi i e oo, as
long as i ’s less han o equal o he posi i e quan i y on he igh side. Fo
ins ance,
i 𝑇−=300K , 𝑇+=400K ,Δ𝐻+=1000J hen Δ𝑊≤250J.
and we could he e o e ob ain, say,
249J
o mechanical wo k a e e y cycle.
Ou new engine design is success ul!
No e he uppe bound ‘
Δ𝑊≤. . .
’ in o mula
(15.9)
, on how much
wo k we can gain in a cycle. We can only ge s ic ly less han he hea
we p o ide a he highes empe a u e. A i s , one migh wonde why
his mus be he case, because we could also p o ide hea a he lowe
empe a u e, hea which could be con e ed in o wo k as well. Bu we
mus emembe he inequali y
(15.8)
o
Δ𝐻−
: i
Δ𝐻+
is posi i e – and we
need i o be posi i e i we wan o gain posi i e wo k – hen
Δ𝐻−
mus be
nega i e. So we’ e losing some ene gy as hea he e. I we y o elimina e
ha hea loss by making
Δ𝐻−
ze o, hen he inequali y
(15.8)
says ha
Δ𝐻+
mus be ze o oo – and hen he wo k gained is ze o o nega i e. In
ac i
Δ𝐻−
we e ze o hen we would e ec i ely be back o ou p e ious
ine ec i e one- empe a u e engine design.
Le ’s see how o maximize he amoun o wo k
Δ𝑊
ha we can gain,
o a gi en amoun o hea
Δ𝐻+
p o ided o he engine. Inequali y
(15.9)
shows ha he e a e wo ways, bo h o which inc ease he e iciency; he
wo a e no mu ually exclusi e:
•
dec ease as much as possible he empe a u e
𝑇−
a which hea is
eleased om he engine;
•
inc ease as much as possible he empe a u e
𝑇+
a which hea is
p o ided o he engine.
I is amazing ha we can p edic how much wo k can be ob ained
om such an engine, wi hou knowing o needing o speci y wha kind
342
15. Balance o en opy 15.10 Example applica ion: ic ion coefficien
o echnology, ma e ials, and way o ope a ion he engine could be based
upon. You see he s eng h o he consequences o he “
≥
” sign in he
en opy balance.
The he mal-engine example abo e also hin s a he ole o he balance
o en opy as a me a-law abou cons i u i e ela ions. In a eal applica ion
and cons uc ion o an engine, he hea lux
𝑄
and momen um lux
𝑭
will
be speci ied by cons i u i e ela ions; hink o ins ance abou New on’s
law o cooling
§11.13 p.279
o
𝑄
, o abou he ideal-gas law
§11.13 p.277
o
𝑭
.
Bu i a limi a ion such as he maximal wo k e iciency
(15.9)
, which we
can ew i e in ull as
−∫𝑡1
𝑡0
𝑭(𝑡) · 𝒗(𝑡)d𝑡≤1−𝑇−
𝑇+∫𝑡1
𝑡0
𝑄+(𝑡)d𝑡 ,
is o be uni e sally alid, hen he speci ic ma hema ical o mulae o
𝑄
and
𝑭
canno be a bi a y. Thei ma hema ical exp essions mus ha e
se e e es ic ions.
«Exe cise 15.4
1.
Find an uppe bound o he amoun o wo k gained simila o
o mula
(15.9)
, bu in e ms o
𝑇+
,
𝑇−
, and he hea
Δ𝐻−
p o ided
a he lowe empe a u e. (Hin : use o mula
(15.7)
o exp ess
Δ𝐻+
in
e ms o Δ𝑊and Δ𝐻−; eplace in o mula (15.9); sol e o Δ𝑊.)
2.
A wo- empe a u e engine ecei es, in a cycle, an amoun o hea o
500J
a a empe a u e o
400K
. You wan o gain
300J
o wo k om
his engine o e a cycle. Wha ’s he maximum empe a u e
𝑇−
a
which hea can be discha ged om he engine, in o de o achie e
you goal?
15.10 Example applica ion: ic ion coefficien
In a p e ious discussion abou su aces o discon inui y
§11.15 p.294
we
analysed he lux o o al-ene gy a he su ace o con ac be ween loo
and a c a e pushed o d agged on i .
343
15. Balance o en opy 15.10 Example applica ion: ic ion coefficien
URLs o chap e 15
1. h ps://encyclopediao ma h.o g/wiki/Clausius-Duhem_inequali y
2. h ps://pla o.s an o d.edu/en ies/s a phys-s a mech/
3. h ps://www.b i annica.com/science/in o ma ion- heo y
344
Pos ace o he eache
“Wha you do in his wo ld is a ma e o no consequence,”
e u ned my companion, bi e ly. “The ques ion is, wha
can you make people belie e ha you ha e done?. . . ”
She lock Holmes (A. C. Doyle) 1887
Themajo i yo e e ydayand o e on echnologiesisbasedonphysical
phenomena a he in e sec ion o adi ional ca ego ies such as “mech-
anics”, “elec omagne ics”, “ he modynamics”. In many cases i is no
clea whe he hese phenomena should be labelled as belonging o one
ca ego y a he han ano he . Some imes such labelling is a i icial, mo e
misleading han help ul.
How o p epa e s uden s in physics o enginee ing o he my iad
o possible physics specializa ions and applica ions lying ahead? The
p oblem is no only ha he s uden s may no know ye which physics ield
hey’ll wan o pu sue, bu also ha hey’ll likely need some knowledge o
all o he ields anyway. I belie e ha he bes app oach is o each hem
physical no ions and physical laws ha a e common o as many physical
phenomena as possible, and can be used in as many physical applica ions
as possible. No ions and laws ha he s uden s will always be able o
use a e wa ds, and upon which he s uden s can g adually build mo e
specialized knowledge.
As poin ed ou in he P e ace,we do ac ually ha e such no ions and laws,
al hough some ex books almos seem o o ge hei exis ence:
•wha we can call ‘ma e ’ o ‘subs ance’ o ‘pa icle numbe ’
•ene gy
•momen um
•angula momen um (including boos momen um)
•elec ic cha ge
•magne ic lux
345
Pos ace o he eache
352
Bibliog aphy
Belie e no hing, O monks, me ely because you ha e been
old i , o because i is adi ional, o because you
you sel es ha e imagined i . Do no belie e wha you
eache ells you me ely ou o espec o he eache .
(A ibu ed o Gau ama Buddha)
(“de
𝑋
” is lis ed unde D, “ an
𝑋
” unde V, and so on, ega dless o na ional
con en ions.)
Sci-Hub, LibGen, Anna’s
A chi e a e ou iends. Help
keeping scien i ic knowledge
ee!
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