Question

A point p$(R\sqrt{3},0,0)$lies on the axis of a ring of 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. $\sqrt{\frac{GM}{2R}}$
• C. $\sqrt{\frac{GM}{\sqrt{2}R}}$
• D. $\sqrt{\frac{GM}{\sqrt{3}R}}$

Answer 1

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

Potential Energy due to a ring of radius $R$ at $x$ distance on its axis is given by

$U=\frac{-GMm}{\sqrt{R^2+x^2}}$

$U=\frac{-GMm}{\sqrt{R^2+({\sqrt{3}R)}^2}}$

applying conservation of energy

Loss in potential energy = gain in kinetic energy

$\frac{1}{2}m{v}^2=-\frac{GMm}{2R}+\frac{GMm}{R}$

$v=\sqrt{\frac{GM}{R}}$

hence option (a).