Physics kinetic energy rotational motion?

Question

A circular disk with mass m and radius R is mounted at its center, about which it can rotate freely. The disk has a moment of inertia I = 1/2 m (R squared) . A light cord wrapped around it supports weight mg. Find the total kinetic energy of the system when the weight is moving at a speed v.

Answer 1

The total K.E. of the system is 3/4 m v^2, obtained as follows:

The angular speed w of the disk is v/R, so the total K.E. is

1/2 I w^2 + 1/2 m v^2 = 1/2 I (v/R)^2 + 1/2 m v^2

= 1/4 m v^2 + 1/2 m v^2 = 3/4 m v^2.

Live long and prosper.

Answer 2

The kinetic energy of the moving mass is 1/2 m v^2
The kinetic energy of the disk is 1/2 I w^2
where I is the moment of inertia and w is the angular velocity.

You have the information required to calculate I for the disk. What about the angular velocity w? Well, since the edge of the disk must move at the same velocity as the mass, the angular and linear velocities are related by v = r w.

Now you can compute both kinetic energies and add them up to get the total.