Question

A disk is spinning at a rate of 10 rad/s. A second disk of the same mass and shape with no spin is placed on the top of the frist disk friction acts between the two of the disk until both are eventually travelling at the same speed. What is the final angular velocity of the two disks?

Answer 1

Assuming conservation of angular momentum:

Angular momentum of first disk before interaction:
L1 = I1 * w1
where I1 is moment of inertia and w1 is initial angular velocity

Angular momentum of second disk before interaction:
L2 = I2 * w2
where I2 is moment of inertia and w2 is initial angular velocity (which is 0)

Total angular momentum before interaction:
L = L1 + L2 = I1 * w1

After interaction, the two disks will have the same final angular velocity w:

Angular momentum of first disk after interaction:
L1' = I1 * (w1 - w)

Angular momentum of second disk after interaction:
L2' = I2 * w

Total angular momentum after interaction:
L' = L1' + L2' = I1 * (w1 - w) + I2 * w

Since angular momentum is conserved, L = L':
I1 * w1 = I1 * (w1 - w) + I2 * w

Simplifying:
I1 * w1 = (I1 + I2) * w - I1 * w
2 * I1 * w1 = (I1 + I2) * w
w = 2 * I1 * w1 / (I1 + I2)

Since the two disks have the same mass and shape, they have the same moment of inertia:
I1 = I2

Substituting:
w = 2 * w1 / 2 = w1

Therefore, the final angular velocity of the two disks is 10 rad/s.