Rotational velocity of the ring

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

Wrt inertial frame with origin at the pivot,
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The final angular velocity of the ring, the bug are $\vec ω_r$ and $\vec ω_b$ and the final velocity of the bug is $\vec v_b$.
Since, there is no net external torque and force is acting on the system , conservation of angular momentum and energy could be applied.
Conservation of angular momentum about the pivot gives,
0 = I_{piv} $\vec ω _r + 2 \vec R \times m \vec v_b$
assuming that $\vec v_b$ is perpendicular to $\vec R$,
This gives $v_b = \frac M m ω_r R$
Conservation of energy gives,
$\frac 1 2 m v^2 = \frac 1 2 I_{piv} {ω_r}^2 + \frac 1 2 m {v_b}^2$
Substituting the value of $v_b$ in the above eqn. gives
$ω_r = \frac {mv} {R\sqrt { M ( 2m + m) }}$
Is this correct so far?
Is $\vec v_b = \vec ω_b \times \vec R$ ?

 

[haruspex]

the final velocity of the bug

Not sure what you mean by that. Do you mean the velocity of the bug in the lab frame when the bug is halfway round?

conservation of angular momentum

Yes, but only if you are taking moments about the pivot.

and energy

The bug does work. You cannot use conservation of energy.

 

[Pushoam]

The bug does work. You cannot use conservation of energy.

I got it: The mechanical energy of the system due to bug's internal interactions gets changed. So, conservation of mechanical energy cannot be applied here.