Question

A spherical ball of mass $m$ is kept at the highest point in the space between two fixed, concentric spheres $A$ and $B$ (see fig.). The smaller sphere $A$ has a radius $R$ and 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 in the figure). The angle made by the radius vector of the ball with the upward vertical is denoted by $\theta $ . What is the total normal reaction force exerted by the spheres on the ball in terms of angle $\theta $ ?

• A. $mg\left(\right.3\cos\theta -2\left.\right)$
• B. $mg\left(\right.2\cos\theta -3\left.\right)$
• C. $3mg\left(\right.2\cos\theta -1\left.\right)$
• D. $2mg\left(\right.3\cos\theta -1\left.\right)$

Answer 1

Answer: (A)

$h=\left(R + \frac{d}{2}\right)\left(1 - \text{cos} \theta \right)$

The velocity of the ball at an angle $\theta$ is
$(\nu)^{2}=2 g h=2\left(R+\frac{d}{2}\right)(1-\cos \theta) g$
Let $N$ be the normal reaction (away from the centre) at angle $\theta$.
Then, $m g \cos \theta-N=\frac{m(\nu)^{2}}{\left(R+\frac{d}{2}\right)}$
Substituting the value of $\nu^{2}$ from equation (i), we get
$m g \cos \theta-N=2 m g(1-\cos \theta)$
$N=m g(3 \cos \theta-2)$