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

• A. zero
• B. $\sqrt{\frac{2GM}{R}}$
• C. $\sqrt{\frac{2GM}{R}(\sqrt{5}-1})$
• D. $\sqrt{\frac{2GM}{R}(1-\frac{1}{\sqrt{5}})}$

Answer 1

The correct option is D $\sqrt{\frac{2GM}{R}(1-\frac{1}{\sqrt{5}})}$

$\text{Gravitational potential at point P,}{V}_P=\frac{-GM}{\sqrt{5}R}$
$\text{Gravitational potential at point O,}{V}_O=\frac{-GM}{R}$
$\text{Using work KE theorem}W=△KE$
$\Rightarrow m[V_P-V_O]=\frac{1}{2}m{V}^2$
or $m[\frac{GM}{R}-\frac{GM}{\sqrt{5}R}]=\frac{1}{2}m{V}^2$
$\text{so, speed at O,}V=\sqrt{\frac{2GM}{R}(1-\frac{1}{\sqrt{5}})}$