A hemisphere of radius R and mass 4m is free | vedclass

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(A) Let $v_r$ be the velocity of the particle relative to the hemisphere and $v$ be the horizontal velocity of the hemisphere at this moment.Since the

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$\frac{5v}{R \cos 	heta}$

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(A) Let $v_r$ be the velocity of the particle relative to the hemisphere and $v$ be the horizontal velocity of the hemisphere at this moment.Since there are no external horizontal forces acting on the system,the horizontal component of the linear momentum is conserved.Initially,the system is at rest,so the total horizontal momentum is $0$.At an angular displacement $	heta$,the horizontal velocity of the particle relative to the table is $(v_r \cos 	heta - v)$ in the direction opposite to the hemisphere's motion.Applying conservation of linear momentum in the horizontal direction:$4m(-v) + m(v_r \cos 	heta - v) = 0$$-4mv + mv_r \cos 	heta - mv = 0$$mv_r \cos 	heta = 5mv$$v_r \cos 	heta = 5v$$v_r = \frac{5v}{\cos 	heta}$The angular velocity $\omega$ of the particle relative to the hemisphere is given by $\omega = \frac{v_r}{R}$.Substituting the value of $v_r$,we get $\omega = \frac{5v}{R \cos 	heta}$.

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