Question

Disk A, with a mass of 2.0 kg and a radius of 90 cm , rotates clockwise about a frictionless vertical axle at 40 rev/s . Disk B, also 2.0 kg but with a radius of 50 cm , rotates counterclockwise about that same axle, but at a greater height than disk A, at 40 rev/s . Disk B slides down the axle until it lands on top of disk A, after which they rotate together. After the collision, what is magnitude of their common angular velocity (in rev/s)? What direction do disks rotate after collison?

Answer 1

Moment of inertia of disk: I = ½mr²
Angular momentum: L = I?

Note we can work with units of rev/s (instead of rad/s) as the unit-conversions would cancel.

Moment of inertia of disk A = ½mr² = ½*2.0*0.9² = 0.81 kgm²
Moment of inertia of disk B = ½mr² = ½*2.0*0.5² = 0.25 kgm²
Total moment of inertia of combination = 0.81 + 0.25= 1.06kgm²

Using positive for clockwise and negative for anticlockwise:
Initial angular momentum of disk A = 0.81 x 40 = 32.4 kgm²/s
Initial angular momentum of disk B = 0.25 x -40 = -10 kgm²/s
Final angular momnetum = 1.06?

From the law of conservation of angular momentum:
Initial angular momentum = final angular momentum

32.4 + (-10) = 1.06?
(if we had worked in rad/s, the conversion factor would have cancelled here)
? = 21.132rev/s

The value is positive, so the rotation is clockwise.