Solving the Angular Momentum & Kinetic Energy Equations: Find ωa & ωb

[mcheung4]

Homework Statement

A ring of mass M_b, radius b, is mounted to a smaller ring of mass M_a, radius a and with the same centre, and they are free to rotate about an axis which points through this centre and is perpendicular to the rings. Dust of mass M_s is distributed uniformly on the inner surface of M_a. At t=0, M_a rotates clock-wise at angular speed Ω while M_b is stationary. At t=0, small perforations in the inner ring are opend, and the dust start to fly out at a constant rate λ and sticks to the outer ring. Find the subsequent angular velocities of the 2 rings ω_a and ω_b. Ignore the transit time of the dust.

Homework Equations

Conservation of angular momentum.
Conservation of kinetic energy.

The Attempt at a Solution

(M_a + M_s)a²Ω = (M_a+M_s-λt)a²ω_a + (M_b+λt)b²ω_b

(M_a + M_s)a²Ω² = (M_a+M_s-λt)a²ω_a² + (M_b+λt)b²ω_b²

I tried to solve the system and both velocities turned out to be zero. Am I doing anyhting wrong?

 

[TSny]

At t=0, small perforations in the inner ring are opend, and the dust start to fly out at a constant rate λ and sticks to the outer ring. Find the subsequent angular velocities of the 2 rings ω_a and ω_b.

Hello, mcheung4.
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When a piece of dust flies out and sticks to the outer ring, what type of collision is that?

Would you expect kinetic energy to be conserved in this type of collision?

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Think about how the angular velocity of the inner ring will change with time. It might help to consider a related situation. Suppose you're standing on a platform that is free to rotate. You are initially rotating at an angular speed Ω with your arms outstretched and a mass m held in each hand. You then release the masses in your hands (without "throwing" the masses). What happens to your angular speed?