- #1

CAF123

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## Homework Statement

A hoop of mass ##m## and radius ##a## rolls without slipping inside a pipe of radius ##R## (the motion is 2D). Write down the kinetic and potential energy. Hence find the frequency of small oscillations about equilibrium.

## Homework Equations

Moment of inertia of a hoop: I=Mr

^{2}

Rotational kinetic energy

## The Attempt at a Solution

I envisaged the problem as a hoop rolling backwards and forwards in a pipe shaped like an arc of radius R.

Kinetic energy is sum of rotational and kinetic, so $$T = \frac{1}{2}mv^2 + \frac{1}{2} (ma^2) \left(\frac{v^2}{a^2}\right) = mv^2.$$

The path of the C.O.M of the hoop is arc shaped. Let ##\theta## be the angle between a vertical passing through the centre of the hoop and the C.O.M. Draw a horizontal at the base of the pipe. After a little time, the C.O.M is at angle ##\theta##. For small oscillations the increase in height of the C.O.M from its initial position (at a height a above the base of the pipe) is a + a(1-cosθ). So V = mg(a+a(1-cosθ)).

Is this correct? I am wondering if this can be solved more easily using the centre of momentum frame.