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 α with respect to point P. Which of the following graphs represent the correct rela…

Question

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 α with respect to point P. Which of the following graphs represent the correct relation between A and α when ball goes from Q to R?

Accepted Answer

(3) a horizontal line at a constant positive A. Idea: work out the centripetal force and the normal reaction separately and then divide — the sin α that appears in each of them cancels. Speed at Q. The bowl is smooth, so energy is conserved. Starting from rest at the rim, the ball has dropped R sin α by the time it reaches angular position α: v²=2gR sin α. Centripetal force: F_c=(mv²)/R=2mg sin α. Normal reaction. Along the radius, the inward forces are N minus the outward component of gravity, mg sin α: N-mg sin α=(mv²)/R=2mg sin α N=3mg sin α. The ratio: A=(F_c)/N=(2mg sin α)/(3mg sin α)=2/3. Constant — independent of α, of the mass and of the radius. So the graph is a horizontal line. The reason it is constant is worth keeping: both quantities are built from the same mg sin α, one at two-thirds and the other at three-thirds of it.

Step-by-step solution

Why the other options are wrong