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

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