question_answer

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

A) Zero

B) \[\sqrt{\frac{2Gm}{R}}\]

C) \[\sqrt{\frac{2Gm}{R}\left( \sqrt{5}-1 \right)}\]

D) \[\sqrt{\frac{2Gm}{R}\left( 1-\frac{1}{\sqrt{5}} \right)}\]

Correct Answer: D

Solution :

[d] Gravitational potential at \[P,Vp=\frac{-GM}{\sqrt{5}R}\] Gravitational potential at \[O,{{V}_{O}}=\frac{-GM}{R}\] \[W=\Delta K\Rightarrow m\left[ {{V}_{p}}-{{\text{V}}_{\text{O}}} \right]=\frac{1}{2}m{{v}^{2}}\] \[m\left[ \frac{GM}{R}-\frac{GM}{\sqrt{5}R} \right]=\frac{1}{2}m{{v}^{2}}\,\,or\,\,V=\sqrt{\frac{2GM}{R}\left( 1-\frac{1}{\sqrt{5}} \right)}\]