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A circular turntable of mass $M$ and radius $3R$ has a semi-adhesive ring of mass $\frac{M}{2}$ with negligible thickness. On top, as shown below, three small balls of mass $m$, $2m$, and $3m$ sit in place at a distance of $R$ from the center of the turntable.

The system is accelerated from rest by a motor until it reaches a speed of $\omega_1$. As soon as it reaches this speed, the balls slip and the motor turns off.

The balls drift outward and are stopped by a thin wall of negligible mass at the edge of the turntable, at a distance of $3R$ from the center. The system reaches a new angular speed $\omega_2$ after the balls drift outward.

What is the ratio of the initial and final angular speeds $\cfrac{\omega_2}{\omega_1}$ of the system? Note: The balls are small enough to be treated as particles.​

A

$\cfrac{3M+72m}{3M+216m}$

B

$\cfrac{3M+12m}{3M+36m}$

C

$\cfrac{9M+12m}{9M+108m}$

D

$\cfrac{5M+6m}{5M+54m}$

E

$\cfrac{5M+6m}{5M+18m}$

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