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Question

A force $f$ acts on the sphere towards left, so the particle experience pseudo force $f =$ ma towards right.

Initial kinetic energy of the particle

Vedclass · Accepted Answer

$\sqrt {2Rg\,(1\,\, + \,\,\sin 	heta \,\, - \,\,\cos 	heta )} $

Vedclass · Answer

A force $f$ acts on the sphere towards left, so the particle experience pseudo force $f =$ ma towards right.

Initial kinetic energy of the particle is zero.

Let the speed of the particle at point $C$ be $v$.

From work-energy theorem, $W _{ g }+ W _{ f }=\Delta K \cdot E = K \cdot E _{ f }$

where $W _{ g }$ is work done by gravity and $W _{ f }$ is work done by pseudo force.

From figure, we get $AB = R - R \cos 	heta= R (1-\cos 	heta)$

Also $AC = R \sin 	heta$

$	herefore mg ( AB )+ f ( AC )=\frac{1}{2} mv ^2$

Or $m g R(1-\cos 	heta)+(m a)(R \sin 	heta)=\frac{1}{2}{m v^2}^2$

Or $m g R(1+\sin 	heta-\cos 	heta)=\frac{1}{2}{m v^2}^2 \quad(\because a=g)$

$\Rightarrow v =\sqrt{2 gR (1+\sin 	heta-\cos 	heta)}$

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