A cockroach of mass frac{M}{2} starts moving | vedclass

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(A) Since there is no external torque acting on the system,the total angular momentum is conserved.Initially,both the disc and the cockroach are at re

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$\frac{V}{R}$

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(A) Since there is no external torque acting on the system,the total angular momentum is conserved.Initially,both the disc and the cockroach are at rest,so the initial angular momentum $L_i = 0$.Let $\omega$ be the angular velocity of the disc. The cockroach moves with velocity $V$ relative to the ground. The angular momentum of the cockroach is $L_c = mvr = (\frac{M}{2})VR$.The angular momentum of the disc is $L_d = I\omega = (\frac{MR^2}{2})\omega$.According to the conservation of angular momentum,the total final angular momentum must be zero: $L_c + L_d = 0$.$(\frac{M}{2})VR + (\frac{MR^2}{2})\omega = 0$.Solving for $\omega$: $(\frac{MR^2}{2})\omega = -(\frac{M}{2})VR$.$\omega = -\frac{V}{R}$.The magnitude of the angular velocity is $\frac{V}{R}$,and the negative sign indicates that the disc rotates in the direction opposite to the motion of the cockroach.

$A$ cockroach of mass $\frac{M}{2}$ starts moving with velocity $V$ on the circumference of a disc of mass $M$ and radius $R$. What will be the angular velocity of the disc?

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