A tunnel is dug along a chord of a non-rotating earth (hypothetical) at a distance ‘d’ as shown. A small block is released from rest in the tunnel from the surface of earth. The block comes to rest at the centre C of tunnel. If $μ$ is the coefficient of friction between block and tunnel, then (Given d=R/2)

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The work done by the friction on the block is $\frac{- 3 G M m}{4 R}$

The total work done by all the forces acting on the block is zero

$μ = \frac{1}{2}$

$μ = \frac{\sqrt{3}}{4}$

answer is D.

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Detailed Solution

Work done due to friction is given by

$W = [\frac{- G M m}{2 R^{3}} (3 R^{2} - \frac{R^{2}}{4})] - [\frac{- G M m}{R}] = \frac{- G M m}{2 R^{3}} \frac{11 R^{2}}{4} + \frac{G M m}{R} = \frac{- 3 G M m}{8 R} d W = - μ \frac{G M m}{R^{3}} \sqrt{x^{2} + R^{2}} \frac{d}{\sqrt{x^{2} + R^{2}}} d x = \frac{G M m}{R^{3}} d (d x) W = μ \frac{G M m}{R^{3}} \frac{\sqrt{3} R}{2} = - μ \frac{G M m}{4 R} \sqrt{3} \Rightarrow μ \frac{G M m}{4 R} \sqrt{3} = \frac{3 G M m}{8 R} \Rightarrow μ = \frac{\sqrt{3}}{2}$