A hemisphere of mass 3m and radius R is free | vedclass

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Question

(A) Let the mass of the hemisphere be $M = 3m$. Since there are no external horizontal forces acting on the system,the horizontal position of the cent

Vedclass · Accepted Answer

$\frac{4v}{R \cos 	heta}$

Vedclass · Answer

(A) Let the mass of the hemisphere be $M = 3m$. Since there are no external horizontal forces acting on the system,the horizontal position of the centre of mass remains constant.Let $x$ be the displacement of the hemisphere to the right. The horizontal displacement of the particle relative to the table is $(R \sin 	heta - x)$ to the left.Equating the horizontal displacement of the centre of mass to zero:$m(R \sin 	heta - x) - M x = 0$$m R \sin 	heta = (M + m) x$Substituting $M = 3m$:$m R \sin 	heta = (3m + m) x = 4m x$$x = \frac{R \sin 	heta}{4}$Differentiating with respect to time $t$:$\frac{dx}{dt} = \frac{R \cos 	heta}{4} \frac{d	heta}{dt}$Given that the velocity of the hemisphere is $v = \frac{dx}{dt}$ and the angular velocity of the particle relative to the centre is $\omega = \frac{d	heta}{dt}$:$v = \frac{R \cos 	heta}{4} \omega$$\omega = \frac{4v}{R \cos 	heta}$

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