# Rotation: mass transfer and angular momentum conservation

1. Jul 5, 2013

### Avi Nandi

1. The problem statement, all variables and given/known data

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.

3. 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$_{b}$.

Last edited: Jul 5, 2013
2. Jul 5, 2013

### darkxponent

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

3. Jul 5, 2013

### Avi Nandi

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

4. Jul 5, 2013

### haruspex

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 travelling along and some of the load on the cart falls off, what force does that exert on the cart?

5. Jul 6, 2013

### Avi Nandi

thank you haruspex and darkxponent.