A tunnel is dug along a chord of the earth at a perpendicular distance $\frac{R}{2}$ From the earth's centre. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the centre of the tunnel.

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Solution

Let $d$ be the distance from center of earth to mass 'm' then,
$d^{2} = x^{2} + (\frac{R^{2}}{4}) = \frac{4 x^{2} + R^{2}}{4}$
$\Rightarrow d = (\frac{1}{2}) \sqrt{4 x^{2} + R^{2}}$
M be mass of the earth $M^{'}$ , the mass of the sphere of radius $d$
$M = (\frac{4}{3}) π R^{3} ρ$
$M^{'} = (\frac{4}{3}) π d^{3} ρ$
$\frac{M^{'}}{M} = \frac{d^{3}}{R^{3}}$

$∴$ Gravitational force on mass m due to the sphere of mass $M^{'}$ is,
$F = \frac{G M^{'} m}{d^{2}}$
$= \frac{G d^{3} M m}{R^{3} d^{2}} = \frac{G M m}{R^{3}} d$
So, Normal force exerted by the wall
$= F cos θ = \frac{G M m d}{R^{3}} \times \frac{R}{2 d} = \frac{G M m d}{2 R^{2}}$

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