A spherical balll of mass `m` is the highest point in the space between two fixed , concentic sphere `A` and `B` the smaller the two sphere `A` has a radius `R` and the space between the two spheres has a width `d` . The bell has a disneter very dightly less then d . All surface are frictionless . The bell is a given a gentle push (owards the right in the figure ) The upward vertical is denoted by `theta` (shown ijn the figure)

(a)Express the total normal reaction force exerted by the sphore on the as a finction of angle `theta`
(b) Let N_(A)` and N_(B)`denote in the magnitubes of the normal reaction force on the bell evered by the sphare `A` and `B` repectively Skech the variation of `N_(A) and`N_(B)` as functions of `cos theta ` in the range `0 le theta le pi ` by drawing two separate graph in your answer book taking `cos theta ` an the horizental axas.

Answer

A, B

Explanation

The ball is moving in a circular in a motion . The necessery constant force is provided by `(mg cos theta - N) ` Therefore , ` mg sin theta - N = (mv^(2)/((R + (4)/(2)) ` ..(i)

According to energy conservation
`(1)/(2) mv^(2) = mg(( R +(4)/(2)) (1 - cos theta ) ` ...(ii)
from (i) and (ii)
`N_(A) = mg (3 cos theta -2)` ...(iii) The above equation shown that as `thrta` increase `N_(A)` decrease . At a particular value of `theta , N_(A)` well became zero and the ball will loes constant with sphere `A` . This condition can be found by pating `N_(A) = 0` in eq (iii)
`0 = mg (3 cos theta - 2) .,. theta cos^(-1) ((2)/(3)) `
The graph between `N_(A) `and `cos theta `
from equation (iii) when `theta = 0 , n_(A) = mg`
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