A sphercial ball of mass m is kept at the highest point in the space between two fixed, concentric spheres A and B (figure). The small sphere A has a radius R, and the space between the two spheres has a width d. The ball has a diameter very slightly less than d. All surfaces are frictionless. The ball is given a gentle push (towards the right figure). The angle made by the radius vector of the ball with the upward vertical is denoted by `theta` (shown in figure).

a. Express the total normal reaction force exerted by the sphere on the ball as a function of angle `theta`.
b. Let `N_A` and `N_B` denote the magnitudes of the normal reaction forces on the ball exerted by spheres A and B, respectively. Sketch the variations of `N_A` and `N_B` as functions of `cos theta` in the range `0lethetalepi` by drawing two separate graphs in your answer book, taking `cos theta` on the horizontal axes.

Answer

0,2

Explanation

The ball is moving in a circular motion. The necessary centripetal force is provided by `(mg cos theta - N)`.
Therefore `mg cos theta - N_A = (m v^2)/((R + d//2))p" "…..(i)`
According to mechanical energy conservation
`Delta K + DeltaU = 0`
`(1/2 mv^2 - 0) + (- mgh) = 0`
`1/2 m v^2 = mg (R + d/2) (1 - cos theta)" "....(iii)`
From (i) and (ii)
`N_A = mg (3 cos theta -2)" ".....(iii) `

The above equation shows that as `theta` increases `N_A` decreases. At a particular value of `theta` . `N_A` will become zero and the ball will lose contact with sphere A. This condition can be found by putting NA = 0 in eq. (III) `0 = mg (3 cos theta - 2)`
`:. theta = cos^(-1) (2/3)`
The graph between `N_A and cos theta`
From eq. (iii) when `theta = 0, N_A =mg` .
When `theta = cos^(-1)(2/3) , N_A = 0`
The graph is a straight line as shown.
When `theta = cos^(-1)(2/3)`
`N_B + mg cos theta = (m v^2)/((R + d/2))`

`1/2 m upsilon^2= mg (R + d/2) (1 - costheta)`
`(m upsilon^2)/((R + d/2)) = 2 mg (1 - cos theta)" ".....(v)`
From (iv) and (v) we get
`N_B + mg cos theta = 2 mg - 2 mg cos theta , N_B = mg (2 - 3 cos theta)`
When `cos theta = 2/3 , N_B = 0`
When `cos theta = -1, N_B = 5 mg`. Therefore the graph is as shown.