Question

A disk of radius a and mass m is suspended at its centre by a vertical torsion wire which exerts a couple -cθ on the disk when it is twisted through an angle θ from its equilibrium position. Show that oscillations of the disk are simple harmonic, and obtain an expression for the period. A wire ring of mass m and radius a/2 is dropped concentrically onto the disk and sticks to it. Calculate the changes to (a) the period (b) the amplitude (c) the energy of the oscillations for the two cases where the ring is dropped on (i) at the end of the swing when the disk is instantaneously at rest (ii) at the midpoint of the swing when the disk is moving with its maximum angular velocity.

Answer 1

Given

Moment of inertia of the disk is of radius a is

for small theta the solution for this equation is

Where

This the oscillation of the disk is simple harmonic motion

As a concentrated ring sticks to the disk, the moment of inertia of the ring adds up to the disk

The Moment of inertia of the ring is

Total moment of Inertia is

Thus

i)

Change to the period is

The total energy in the oscillating disk is

b)

The Amplitude of the oscillation does not change

C)

I)the energy of the oscillations when the disk is instantaneously at rest is

The Force

Thus

Energy will be

ii) the energy of the oscillations when the disk moving with its maximum angular velocity.

Initial Kinetic energy is equal to final kinetic energy taking this situation as a collision

But the energy is

The Energy of the system remain same