Question

A point $P(\sqrt{3}R,0,0)$ lies on the axis of a ring of a mass $M$ and radius $R$. The ring is located in y-z plane with its centre at origin $O$. A small particle of mass $m$ starts from $P$ and reaches $O$ under gravitational attraction only. Its speed at $O$ will be :

• A. $\sqrt{\frac{GM}{R}}$
• B. $\sqrt{\frac{Gm}{R}}$
• C. $\sqrt{\frac{GM}{\sqrt{2}R}}$
• D. $\sqrt{\frac{Gm}{\sqrt{2}R}}$

Answer 1

The correct option is A $\sqrt{\frac{GM}{R}}$

According to law of conservation of energy

$\frac{1}{2}m{V}^2=m\left(\Delta V\right)$

$\frac{1}{2}m{V}^2=m[\frac{-GM}{2R}-(-\frac{GM}{R}\left)\right]$

$\frac{1}{2}m{V}^2=m\left(\frac{GM}{2R}\right)⟹V=\sqrt{\frac{GM}{R}}$