Gravitaional potential energy

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Question

A narrow tunnel is dug along the diameter of Earth. If g be the acceleration due to gravity on the surface of earth and R be the radius of earth, Then work done in moving an object of mass m from the centre of earth to its surface is

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Solution

$\begin{array}{l} W = U_{s} - U_{c} = (- \frac{GMm}{R}) - (- \frac{3 GMm}{2 R}) = \frac{GMm}{2 R} = \frac{1}{2} m . \frac{GM}{R^{2}} R \\ \Rightarrow W = \frac{1}{2} mgR \end{array}$

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