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A spherical ball of mass $m$ is kept at the highest point in the space between two fixed, concentric spheres $A$ and $B$ (see figure). The smaller 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 then $d$ . 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 $θ$ (shown in the figure). Express the total normal reaction force exerted by the spheres on the ball as a function of angle $θ$ .

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$N = m g (3 \cos θ - 2)$

$N = m g (3 \cos θ - 3)$

$N = (3 \cos θ - 2)$

$N = m g (\cos θ - 2)$

answer is A.

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Detailed Solution

$h = (R + \frac{d}{2}) (1 - \cos θ)$

velocity of ball at angle $θ$ is

$v^{2} = 2 g h = 2 (R + \frac{d}{2}) (1 - \cos θ) g . . . (1)$

Let $N$ be the total normal reaction (away from centre) at angle $θ$ . Then

$m g \cos θ - N = \frac{m v^{2}}{(R + \frac{d}{2})}$

substituting value of $v^{2}$ from eq. $(1)$ we get

$m g \cos θ - N = 2 m g (1 - \cos θ) ∴ N = m g (3 \cos θ - 2)$

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