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RollingVsInertia.gif (506 × 311 pixels, file size: 695 KB, MIME type: image/gif, looped, 74 frames)

Summary

Description
English: Cylinders with the same mass but different moments of inertia will roll down a flat surface at different accelerations, as they have to "convert" different amounts of potential energy into kinetic energy to achieve a given angular velocity.
Date
Source https://twitter.com/j_bertolotti/status/1225050271384002561
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code

s1[t_] := R1 \[Phi]1[t];
x1[t_] := x10 + s1[t] Cos[\[Alpha]] + R1 Sin[\[Alpha]];
y10 = Abs[x10] Tan[\[Alpha]];
y1[t_] := y10 - s1[t] Sin[\[Alpha]] + R1 Cos[\[Alpha]];
T1[t_] := 1/2 m1 (D[x1[t], t]^2 + D[y1[t], t]^2) + 1/2 I1 (D[\[Phi]1[t], t])^2;
V1[t_] := m1 g y1[t];
L = FullSimplify[T1[t] - V1[t]]
eq1 = FullSimplify[D[D[L, \[Phi]1'[t]], t] - D[L, \[Phi]1[t]]]

m1 = 1; R1 = 1; g = 1; \[Alpha] = \[Pi]/6; x10 = -20;
sol = Table[
   NDSolve[{(eq1 == 0) /. {I1 -> j}, \[Phi]1[0] == 0, \[Phi]1'[0] == 1}, {\[Phi]1[t]}, {t, 0, 20}], {j, 0, 5}];
dim = Dimensions[sol][[1]];
\[CapitalDelta] = 2;
p1 = Table[ Legended[Graphics3D[{
Gray, Polygon[{{0, \[CapitalDelta], 0}, {1.2*x10, \[CapitalDelta], 1.2 y10}, {1.2*x10, (dim + 1) \[CapitalDelta], 1.2 y10}, {0, (dim + 1) \[CapitalDelta], 0}}], Table[{Hue[j/dim], Cylinder[({{x1[t], \[CapitalDelta] j, y1[t]}, {x1[t], \[CapitalDelta] (j + 1), y1[t]}} /. sol[[j]]) /. {t -> \[Tau]}, R1]}, {j, 1, dim}]}, Boxed -> False],
    PointLegend[Table[Hue[j/dim], {j, 1, dim}] , Table[StringJoin["m=\!\(\*SubscriptBox[\(m\), \(0\)]\)   I=", ToString[j], "\!\(\*SubscriptBox[\(I\), \(0\)]\)"], {j, 1, dim}], LegendMarkerSize -> 20,      LabelStyle -> {FontFamily -> "Times"}]  ]
   , {\[Tau], 0, 7.3, 0.1}];
ListAnimate[p1]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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5 February 2020

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Date/TimeThumbnailDimensionsUserComment
current09:24, 6 February 2020Thumbnail for version as of 09:24, 6 February 2020506 × 311 (695 KB)BertoUser created page with UploadWizard

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