MEXANIKA MASALALARINI GRAFIK USULDA WOLFRAM MATHEMATICA DASTURI ASOSIDA YECHISH

GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 4 DOI: https://doi.org/10.5281/zenodo.18048020 MEXANIKA MASALALARINI GRAFIK USULDA WOLFRAM MATHEMATICA DASTURI ASOSIDA YECHISH Elmurodov O‘tkir Tolibovich Oriental universiteti, Samarqand kampusi fizika fani o‘qituvchisi [email protected] Ro‘ziqulov Anvar Ravshanovich Oriental universiteti, Samarqand kampusi talabasi ANNOTATSIYA Ushbu ilmiy maqolada klassik mexanikaning asosiy bo‘limlariga oid masalalarni Wolfram Mathematica dasturi yordamida grafik va sonli usullarda yechish hamda tahlil qilish masalalari ko‘rib chiqilgan. Mexanik jarayonlarni matematik modellashtirish ko‘pincha murakkab differensial tenglamalar va analitik yechimlar bilan bog‘liq bo‘lib, bunday holatlarda grafik va sonli yondashuvlar muhim ahamiyat kasb etadi. Maqolada otilgan jism harakati, havo qarshiligi bilan harakat, qiya sirt bo‘yicha sirpanish, oddiy mayatnik, garmonik tebranuvchi, ikki massa-prujina tizimi, markaziy kuch maydonida harakat, shuningdek impuls va energiya saqlanish qonunlari misollar orqali tahlil qilingan. Har bir mexanik model uchun tegishli matematik tenglamalar tuzilib, Wolfram Mathematica muhitida ularning grafik tasvirlari, faza portretlari va sonli yechimlari olingan. Olingan natijalar mexanik jarayonlarning fizik mohiyatini chuqurroq tushunishga, parametrlarning o‘zgarishi harakat xarakteriga qanday ta’sir ko‘rsatishini aniqlashga imkon beradi. Tadqiqot natijalari Wolfram Mathematica dasturidan mexanika fanini o‘qitishda, laboratoriya GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 5 mashg‘ulotlarida hamda ilmiy tadqiqotlarda samarali foydalanish mumkinligini ko‘rsatadi. Kalit so‘zlar: Klassik mexanika, Wolfram Mathematica, otilgan jism harakati, grafik va raqamli usullar, havo qarshiligi, garmonik tebranuvchi, markaziy kuch maydoni ABSTRACT This scientific article considers the solution and analysis of problems from the main sections of classical mechanics using graphical and numerical methods implemented in the Wolfram Mathematica software. The mathematical modeling of mechanical processes is often associated with complex differential equations and analytical solutions; therefore, graphical and numerical approaches play a crucial role in understanding such systems. The paper analyzes projectile motion, motion with air resistance, sliding on an inclined plane, the simple pendulum, the harmonic oscillator, a two mass-spring system, motion in central force fields, as well as the laws of conservation of momentum and energy through illustrative examples. For each mechanical model, the corresponding mathematical equations are formulated and their graphical representations, phase portraits, and numerical solutions are obtained in the Wolfram Mathematica environment. The results clearly demonstrate the physical nature of mechanical processes and show how variations in system parameters influence the character of motion. The study confirms that Wolfram Mathematica is an effective and versatile tool for visualizing, modeling, and analyzing both linear and nonlinear mechanical systems. The obtained results indicate that this software can be efficiently used in teaching classical mechanics, conducting laboratory work, and performing scientific research. KIRISH Mexanika jismlarning harakati, kuchlarning ta’siri ostida ularning o‘zgarishi va dinamik xususiyatlarini o‘rganuvchi fundamental fandir. Ushbu jarayonlar matematik GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 6 modellar orqali ifodalanadi. Bunday modellar ko‘pincha differensial tenglamalar, integral hisoblash, vektor funksiyalar, faza fazosida tahlil kabi murakkab matematik apparatni talab qiladi. ADABIYOTLAR TAHLILI VA METODOLOGIYA Wolfram Mathematica dasturi murakkab mexanik jarayonlarni modellashtirish, vizualizatsiya qilish va sonli yechimlar olish uchun eng qulay ilmiy dasturiy vositalardan biridir. Mathematica yordamida mexanik tizimlarning grafik modeli, faza trayektoriyalari, kuch vektor maydonlari, energiyaning vaqt bo‘yicha o‘zgarishi, tebranishlar va murakkab nochiziqli tizimlarning yechimlari oson aniqlanadi [1-4]. Grafik metod mexanika masalalarini o‘rganishda juda muhim. Chunki u jarayonni ko‘z oldiga keltirish, parametrlarning o‘zgarishi natijasida harakat shakllarining qanday o‘zgarishini kuzatish, murakkab tenglamalarning fizik mazmunini anglashga yordam beradi. Aynan shuning uchun, ushbu maqolada grafik yechimlar asosiy o‘rinda turadi [5-6]. NATIJALAR Mathematica mexanikada qo‘llanishi mumkin bo‘lgan kuchli vositalarga ega. Ulardan ayrimlari: -Symbolik hisoblash. - Sonli yechim: NDSolve. - Grafik chizish: Plot, ParametricPlot, VectorPlot, StreamPlot. - Interaktiv modellar: Manipulate. - Dinamik animatsiya: Animate. - Jadvallar: Table, Grid, Dataset. Mathematica mexanikada qo‘llanishi mumkin bo‘lgan kuchli vositalarga ega. Ulardan ayrimlari: - Symbolik hisoblash. - Sonli yechim: NDSolve. - Grafik chizish: Plot, ParametricPlot, VectorPlot, StreamPlot. - Interaktiv modellar: Manipulate. GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 7 - Dinamik animatsiya: Animate. - Jadvallar: Table, Grid, Dataset. Bu imkoniyatlar mexanik jarayonlarni chuqur va aniq o‘rganish uchun sharoit yaratadi. Masalan, differensial tenglama analitik yechimi mavjud bo‘lmagan holatlarda NDSolve yordamida raqamli yechim olinadi. Otilgan jism harakati Otilgan jism harakatida jism boshlang‘ich tezlik v0 va burchak θ ostida otib yuboriladi. Analitik model quyidagicha: x(t)=v0 cos(θ) t, y(t)=v0 sin(θ) t − (1/2) g t². Traektoriya parabola hosil qiladi. ClearAll["Global`*"]; v0=20; (*m/s*) thetaDeg=45; (*degree*) g=9.81; theta=thetaDeg Degree; x[t_]:=v0 Cos[theta] t; y[t_]:=v0 Sin[theta] t-1/2 g t^2; Tflight=2 v0 Sin[theta]/g; Range=v0 Cos[theta] Tflight; Hmax=v0^2 Sin[theta]^2/(2 g); Print["Flight time T = ",N[Tflight]," s"]; Print["Range R = ",N[Range]," m"]; Print["Max height Hmax = ",N[Hmax]," m"]; GraphicsRow[{ParametricPlot[{x[t],y[t]},{t,0,Tflight},AxesLabel->{"x (m)","y (m)"},PlotRange->All,AspectRatio->1/3,PlotLabel- >"Trayektoriya"],Plot[y[t],{t,0,Tflight},AxesLabel->{"t (s)","y (m)"},PlotLabel- >"Balandlik"]}] GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 8 Havo qarshiligi bilan otilgan jism harakati Havo qarshiligi mavjud bo‘lgan holda harakat tenglamalari nochiziqli bo‘lib ketadi. Qarshilik kuchi −c v² ko‘rinishida olinadi. Bu tizimni faqat sonli usulda yechish mumkin. NDSolve yordamida x(t) va y(t) funksiyalari olinadi. Grafik chizmalar havo qarshiligi trayektoriyani qanday qisqartirishini aniq ko‘rsatadi. ClearAll["Global`*"]; m=0.15; (*kg*) cd=0.05; (*drag coefficient (effective)*) v0=20;thetaDeg=45; g=9.81; theta=thetaDeg Degree; (*System:x’’=-(cd/m) v x’,y’’=-g-(cd/m) v y’ where v=Sqrt[x’^2+y’^2]*) tmax=5; sol=NDSolve[{x’’[t]==-(cd/m) Sqrt[x’[t]^2+y’[t]^2] x’[t],y’’[t]==-g-(cd/m) Sqrt[x’[t]^2+y’[t]^2] y’[t],x[0]==0,y[0]==0,x’[0]==v0 Cos[theta],y’[0]==v0 Sin[theta]},{x,y},{t,0,tmax},MaxSteps->20000,AccuracyGoal->8]; traj=ParametricPlot[Evaluate[{x[t],y[t]}/. sol],{t,0,tmax},AxesLabel->{"x (m)","y (m)"},PlotRange->All,PlotLabel->"Havo qarshilik kuchining trayektoriyaga ta’siri"]; (*Compare with no-drag parabola*) par=ParametricPlot[{v0 Cos[theta] t,v0 Sin[theta] t-1/2 g t^2},{t,0,2 v0 Sin[theta]/g},PlotStyle->{Dashed},PlotRange->All]; Show[traj,par,ImageSize->Large] GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 9 Qiya sirt bo‘yicha harakat Qiya sirt mexanikada asosiy modeldir. Jism sirpanayotganda unga og‘irlik kuchi, normal reaksiya va ishqalanish kuchi ta’sir qiladi. Harakat energiya tenglamasi orqali ham, kuchlarning proyeksiyasi orqali ham o‘rganiladi. Mathematica yordamida jismning yo‘li, tezligi, tezlanishi, energiyasi vaqt bo‘yicha grafik tarzda chiqariladi. ClearAll["Global`*"]; m=1;g=9.81; alphaDeg=30;mu=0.1;v0=0; alpha=alphaDeg Degree; a=g Sin[alpha]-mu g Cos[alpha]; (*acceleration along plane*) s[t_]:=1/2 a t^2+v0 t; (*displacement along incline*) v[t_]:=D[s[t],t]; Kinetic[t_]:=1/2 m v[t]^2; Potential[t_]:=m g s[t] Sin[alpha]; (*height=s Sin alpha*) TotalE[t_]:=Kinetic[t]+Potential[t]; Print["Acceleration a = ",N[a]," m/s^2"]; GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 10 GraphicsRow[{Plot[s[t],{t,0,10},AxesLabel->{"t (s)","s (m)"},PlotLabel- >"Displacement vs t"],Plot[v[t],{t,0,10},AxesLabel->{"t (s)","v (m/s)"},PlotLabel- >"Velocity vs t"]},ImageSize->Large] Plot[{Kinetic[t],Potential[t],TotalE[t]},{t,0,10},PlotLegends- >{"Kinetic","Potential","Total"},AxesLabel->{"t (s)","Energy (J)"},PlotLabel- >"Energies vs t",ImageSize->Large] Oddiy mayatnik Oddiy mayatnik nochiziqli tizim bo‘lib, tenglama θ’’ + (g/l) sin θ = 0 ko‘rinishida bo‘ladi. Kichik burchaklarda chiziqli model qo‘llanadi, katta burchaklarda esa to‘liq nochiziqli yechim talab qilinadi. Mathematica mayatnikning faza portretlari, energiya funksiyalari, tebranish davrini aniqlash imkonini beradi. ClearAll["Global`*"]; g=9.81;l=1; theta0=0.9;omega0=0;tmax=20; solPend=NDSolve[{theta’’[t]+(g/l) Sin[theta[t]]==0,theta[0]==theta0,theta’[0]==omega0},theta,{t,0,tmax},MaxSteps- >Infinity]; (*theta(t)*) Plot[Evaluate[theta[t]/. solPend],{t,0,tmax},AxesLabel- >{"t","theta(t)"},PlotLabel->"Mayatnik burchagi va vaqt"] GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 11 (*Phase portrait:theta vs omega*) phase=ParametricPlot[Evaluate[{theta[t],theta’[t]}/. solPend],{t,0,tmax},AxesLabel->{"theta","omega"},PlotRange->All,PlotLabel- >"Phase portrait"]; (*Energy*) Energy[t_]:=1/2 l^2 (theta’[t])^2+g l (1-Cos[theta[t]]); Plot[Evaluate[Energy[t]/. solPend],{t,0,tmax},AxesLabel- >{"t","E(t)"},PlotLabel->"Energiya va t"] Ikki massa — prujina tizimi Ikki massa bir-biriga prujina bilan ulanib, murakkab garmonik tizim hosil qiladi. Bu tizimda energiya almashinuvi, rezonans, normal modalar mavjud. Mathematica yordamida x1(t) va x2(t) yechimlari grafik chiziladi. ClearAll["Global`*"]; m1=1.2;m2=0.8; v1i=2;v2i=0; (*initial velocities*) (*Elastic collision formulas*) v1f=((m1-m2) v1i+2 m2 v2i)/(m1+m2); v2f=(2 m1 v1i+(m2-m1) v2i)/(m1+m2); Print["After elastic collision: v1f = ",N[v1f]," , v2f = ",N[v2f]]; (*Check conservation*) Print["Initial momentum: ",N[m1 v1i+m2 v2i]]; GOLDEN BRAIN ISSN: 2181-4120 VOLUME 3 | ISSUE 19 | 2025 Multidisciplinary Scientific Journal December, 2025 12 Print["Final momentum: ",N[m1 v1f+m2 v2f]]; Print["Initial KE: ",N[1/2 m1 v1i^2+1/2 m2 v2i^2]]; Print["Final KE: ",N[1/2 m1 v1f^2+1/2 m2 v2f^2]]; Markaziy kuch maydonida harakat Gravitatsion yoki elektrostatik markaziy kuch ostida jismning harakati r(θ)=p/(1+e cos θ) formulasi bilan ifodalanadi. Mathematica yordamida planetar orbitaning barcha turlari - elliptik, parabolik va giperbolik trayektoriyalar chiziladi. p=1; (*semi-latus rectum*) eList={0.6,1.0,1.5}; (*eccentricities:ellipse,parabola,hyperbola*) orbits=Table[ParametricPlot[Evaluate[{(p/(1+e Cos[theta])) Cos[theta],(p/(1+e Cos[theta])) Sin[theta]}],{theta,-Pi+0.01,Pi-0.01},PlotRange->All,AxesLabel- >{"x","y"},PlotLabel->"e = "<>ToString[e]],{e,eList}]; GraphicsRow[orbits,ImageSize->Large] Garmonik tebranuvchi