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

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 posi

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 ?

Option: 1

Option: 2

Option: 3

Option: 4

Answers (1)

Let the speed of ball at pt. Q be V
By energy conservation,
$\mathrm{TE_{p}= TE_{Q}}$
$\mathrm{mgR+0= \frac{1}{2}mv^{2}+mg\left ( R-R\sin \alpha \right )}$
$\mathrm{ \frac{1}{2}mv^{2}= mgR\, \sin \alpha }$
$\mathrm{V= \sqrt{2gR\left ( \sin \alpha \right )}\rightarrow (1) }$

$\mathrm{N-mg\sin \alpha = \left ( \frac{mv^{2}}{R} \right )= F_{c} }$
$\mathrm{N-mg\sin \alpha = 2\, mg\sin \alpha \; \left ( from\; eqn\; 1 \right ) }$
$\mathrm{N= 3\, mg\sin \alpha \rightarrow (2)}$
$\mathrm{\frac{F_{c}}{N}= A= \frac{2\, mg\sin \alpha }{3\, mg\sin \alpha } = \frac{2}{3}\rightarrow (3)}$

A is independent of $\mathrm{\alpha }$
$\mathrm{\therefore }$ The correct option is (3)