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Physics LibreTexts

25.4: Cylinder Rolling Inside another Cylinder

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Small cylinder of radius a rolling inside a larger one of radius R. The angle between the centers is theta and the velocity of the center of the smaller cylinder tangent to the circumference of the larger one is V
Figure 25.4.1

Now consider a solid cylinder radius a rolling inside a hollow cylinder radius R, angular distance from the lowest point θ, the solid cylinder axis moving at V=(Ra)˙θ and therefore having angular velocity (compute about the point of contact) Ω=V/a.

The kinetic energy is

12MV2+12I(V/a)2=12(M+Ia2)(Ra)2˙θ2

The potential energy is Mg(Ra)cosθ.

The Lagrangian L=TV, the equation of motion is

(M+Ia2)(Ra)2¨θ=Mg(Ra)sinθMg(Ra)θ

so small oscillations are at frequency ω=g(1+IMa2)(Ra).


This page titled 25.4: Cylinder Rolling Inside another Cylinder is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.

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