First slide

Gravitaional potential energy

Question

A point P is on the axis of a fixed ring of mass M and radius R, at a distance 2R from the centre O. A small particle starts from P and reaches O under the gravitational attraction only. Its speed at O will be

Moderate

A

zero
B

$\sqrt{\frac{2 G M}{R}}$
C

$\sqrt{\frac{2 G M}{R} (\sqrt{5} - 1)}$
D

$\sqrt{\frac{2 G M}{R} (1 - \frac{1}{\sqrt{5}})}$

Solution

Gravitational potential at 'P", $v_{p} = \frac{- G M}{\sqrt{5 R}}$
Gravitational potential at 'O', $v_{0} = - \frac{G M}{R}$
Using work energy theorem, $m . [\frac{G M}{R} - \frac{G M}{\sqrt{5 R}}] = \frac{1}{2} m v^{2}$
$\sqrt{\frac{2 G M}{R} [1 - \frac{1}{\sqrt{5}}]} = V$

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