Question

Three identical particles each of mass $M$ move along a common circular path of radius $R$ under the mutual interaction of each other. The velocity of each particle is

• A. $\sqrt{\frac{GM}{R}\sqrt{\frac{2}{3}}}$
• B. $\sqrt{\frac{GM}{\sqrt{3}R}}$
• C. $\sqrt{\frac{GM}{3R}}$
• D. $\sqrt{\frac{2}{3}\frac{GM}{R}}$

Answer 1

The correct option is B $\sqrt{\frac{GM}{\sqrt{3}R}}$

Resultant force force : $F_{net}=2F\cos {30}^{\circ }$

where $F$ is gravitational force between any of two particles

this net force should be equal to centripetal acceleration:

$\sqrt{3}F=\frac{m{v}^2}{R}$

$\sqrt{3}\frac{G{m}^2}{a^2}=\frac{m{v}^2}{\frac{a}{\sqrt{3}}}$

$v=\sqrt{\frac{GM}{a}}=$ $\sqrt{\frac{GM}{R\sqrt{3}}}$