Question

A ring (2 kg, r = 1 m) rotates in a CW direction with initial angular velocity 10 s-1. A disk (8 kg, r = 1 m) rotates in a CCW direction with initial angular velocity 50 s-1. The ring and disk "collide" and eventually rotate together. Assume that positive angular momentum and angular velocity values correspond to rotation in the CCW direction.

What is the initial angular momentum Li of the ring+disk system?

What is the final angular velocity ωf of the ring+disk system?

Answer 1

Angular momentum, like linear momentum, is always conserved. The angular momentum of the 2 objects rotating together is equal the total angular momentum of the 2 objects before the collision. The total angular momentum of the 2 disks before the collision is the difference of the 2 angular momentums, because they are rotating in opposite directions

Angular momentum = moment of inertia * angular velocity
For ring moment of inertia = mass * radius^2 = 2 * 1^2 = 2
Angular momentum = 2 * 10 = 20 = L1

For disk moment of inertia = ½ * mass * radius^2 = ½ * 8 * 1^2 = 4
Angular momentum = 2 * 50 = 100 = L2

Total angular momentum before the collision = 100+20 = 120

The angular momentum of the 2 disks rotating together is equal the total angular momentum of the 2 disks before the collision.

Moment of inertia after the collision = 4 + 2 = 6
Angular momentum after the collision 6 * ωf

6 * ωf = 120
ωf = 20 rad/s