Question 204 of 210

A ball is released from rest from point $P$ of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point $Q$ is $A$ while angular position of point $Q$ is $\alpha$ with respect to point $P$. Which of the following graphs represent the correct relation between $A$ and $\alpha$ when ball goes from $Q$ to $R$?

ABCD

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Answer: (C)

At angular position $\alpha$ as shown in the figure, applying conservation of mechanical energy
$mgR\sin \alpha =\dfrac{1}{2}m{v}^{2} \Rightarrow \dfrac{m{v}^{2}}{R}=2mg\sin \alpha$
In the radial direction,
$N-mg\sin \alpha =\dfrac{m{v}^{2}}{R} \Rightarrow N=3mg\sin \alpha$
Now the ratio of centripetal force and normal reaction is,
$A=\dfrac{\dfrac{m{v}^{2}}{R}}{N}=\dfrac{2mg\sin \alpha }{3mg\sin \alpha }=\dfrac{2}{3}$. Therefore, $A$ is constant.