Gravitaional potential energy

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Question

If potential energy of a body on the surface of earth is U, its potential energy at the centre of earth is

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Solution

$U = - \frac{GMm}{R}$
When body is placed at a distance $r (r \leq R)$ from the centre of earth, its potential energy is given by
$U' = - \frac{GMm}{2 R} (3 - \frac{r^{2}}{R^{2}}) (r \leq R)$
At the centre of earth, r = 0 $∴ U_{c} = - \frac{3 GMm}{2 R} = 1.5 U$

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