Question

A thin-walled hollow cylinder with mass m₁ and radius R₁ rolls without slipping on the inner wall of a larger thin-walled hollow cylinder with mass m₂ and radius R₂ >> R₁. Both cylinders' axes are aligned and horizontal, and some magical mechanism ensures this perfect friction contact between them. Unfortunately for you, the larger cylinder rolls without slipping on a horizontal surface. Describe the motion including small oscillation frequencies near any equilibria if the systems is released from rest with the inner cylinder at some angle α₀.

Answer 1

In the case of the inner cylinder, it seems to roll down on a plane at any instant (since radius of outer cylinder is much greater). Then:

The outer cylinder rolls over a plane. Then:

In this case, since we can consider that effect of moment of inertia of cylinder 1 is negligible (this is why it does not appear in moment of inertia Ib). Later:

Now let's suppose that angles theta are small, then: , which lead us to two differential equations that describes simple harmonic motion. Then:

(for simple harmonic motion)

Notice also that:

Since:

We conclude that bigger cylinder has a very low frequency when compared with smaller cylinder (smaller cylinder oscillates faster).