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## Homework Statement

A drum of mass MA and radius a rotates freely with initial angular velocity ωA(0). A second drum with mass MB and radius b > a is mounted on the same axis and is at rest, although it is free to rotate. A thin layer of sand with mass Ms 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 ωA and ωB. Ignore the transit time of the sand.

clue. If λt=MB and b=2a then ωA = ωA(0)/8

## Homework Equations

Mass of sand into drum A : ##m_a(t) = M_s -\lambda t ##

Mass of sand into drum B : ##m_b(t) = \lambda t ##

Angular momentum in the system Drum A + sand : ## L_a = (M_a + m_a) a^2 \omega_a +m_b b^2 \omega_b ##

Angular momentum in the system Drum B + sand : ## L_b = m_a a^2 \omega_a +(M_b+m_b) b^2 \omega_b ##

## The Attempt at a Solution

Hello, I have just finished reading the chapter dealing with rotational motion of rigid bodies. I am very confused for the moment. I have spent a lot of time thinking about this problem but did not manage to get to the right answer. Can you help me please ?

------ attempt :

I need two coupled equations in order to find ##\omega_b## from ##\omega_a##.

Since the motion is rotational around a fixed axis, it is natural to use angular momentum.

Furthermore, angular momentum is conserved in both systems ( Drum A + sand and Drum B + sand) because in both cases, external torque is due to :

(i) Sand weight in both drums. Rings of sand concentrate their weight on the axis of rotation so the torque due to sand weight is 0.

(ii) Radial push from the other drum (that is not in the system). A radial force has 0 torque.

So,

## L_a(t) = L_a(0) = (M_a+M_s) a^2 \omega_a(0) ##

## L_b(t) = L_b(0) = M_s a^2 \omega_a(0) ##

Subtracting second equation to first equation, I get:

## \omega_b = \frac{M_a}{M_b}\frac{a^2}{b^2} (\omega_a - \omega_a(0))##

Replacing ##\omega_b## by its expression in ##L_a##, I get:

## \begin{array}{lcr}

\omega_a = \frac{M_a+M_s+\lambda t \frac{M_a}{M_b}}{M_a+M_s+\lambda t (\frac{M_a}{M_b} - 1 ) } \omega_a(0) & &

\omega_b = \frac{M_a}{M_b}\frac{a^2}{b^2} \frac{\lambda t}{M_a+M_s+\lambda t (\frac{M_a}{M_b} - 1 ) } \omega_a(0)

\end{array}##

but it does not match with the hint given in the problem statement