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Question
A tunnel is dug along a chord of the earth at a perpendicular distance 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 distance of the particle from the centre of the Earth.
\[\text { Now }, d^2 = x^2 + \left( \frac{R^2}{4} \right) = \frac{4 x^2 + R^2}{4}\]
\[ \Rightarrow d = \left( \frac{1}{2} \right)\sqrt{4 x^2 + R^2}\]
Let M be the mass of the Earth and M' be the mass of the sphere of radius d.
Then we have :
\[M = \left( \frac{4}{3} \right)\pi R^3 \rho\]
\[ M^1 = \left( \frac{4}{3} \right)\pi d^3 \rho\]
\[ \therefore \frac{M^1}{M} = \frac{d^3}{R^3}\]
Gravitational force on the particle of mass m is given by
\[F = \frac{G M^1 m}{d^2}\]
\[ \Rightarrow F = \frac{G d^3 Mm}{R^3 d^2}\]
\[ \Rightarrow F = \frac{GMm}{R^3}d\]
∴ Normal force exerted by the wall, FN = F cos θ
\[= \frac{GMmd}{R^3} \times \frac{R}{2d} = \frac{GMm}{2 R^2}\]
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