Rotation: mass transfer and angular momentum conservation

[Avi Nandi]

Homework Statement

a drum of mass M$_{a}$ and radius a rotates freely with initial angular velocity ω$_{a}$(0). A second drum with mass M$_{b}$ and radius b greater than b is mounted on the same axis and is at rest, although it is free to rotate. a thin layer of sand with mass M$_{s}$ is distributed on the inner surface of the smaller drum. At t=0 small perforations in the inner drum are opened. the sand starts to fly out at a constant rate λ and sticks to the outer drum. Find the subsequent angular velocities of the two drums ω$_{a}$ and ω$_{b}$. Ignore the transit time of the sand.

The Attempt at a Solution

torque on drum A = $\frac{1}{2}$(M$_{a}$ + M$_{s}$- λt)a$^{2}$dω$_{a}$/dt + $\frac{1}{2}$λa$^{2}$ω$_{a}$(t)

torque on drum B = $\frac{1}{2}$(M$_{b}$- λt)b$^{2}$dω$_{b}$/dt - $\frac{1}{2}$λb$^{2}$ω$_{b}$(t)

now applying angular momentum conservation on the system I got a relation between ω$_{a}$ and ω$_{b}$.

 

[darkxponent]

I don't think that the Torque equations are needed. all you need to do is to conserve the angular momentum.

 

[Avi Nandi]

but i can not find any more relations between ω$_{a}$ and ω$_{b}$.

 

[haruspex]

I don't think that the Torque equations are needed. all you need to do is to conserve the angular momentum.

That can't be enough. For any given distribution of sand between the drums there will be a continuum of solutions for the two angular velocities that give the same overall angular momentum.

Avi Nandi, I'm unconvinced by your expression for torque on the inner drum. If a cart is traveling along and some of the load on the cart falls off, what force does that exert on the cart?

 

[Avi Nandi]

thank you haruspex and darkxponent.