Question

Four particles, each of mass $M$, move along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle is

• A. $\frac{GM}{R}$
• B. $\sqrt{\frac{\left(2\sqrt{2}\right)GM}{R}}$
• C. $\sqrt{\frac{(2\sqrt{2}+1)GM}{R}}$
• D. $\sqrt{\frac{2\sqrt{2}+1}{4}\left(\frac{GM}{R}\right)}$

Answer 1

The correct option is D $\sqrt{\frac{2\sqrt{2}+1}{4}\left(\frac{GM}{R}\right)}$

Given,
Mass of each particle $=M$
Radius of rotation for each particle $=R$

Consider particle $1$ which is in circular motion about the center under the influence of gravitational forces by particles $2,3$ and $4$.


Gravitational force between two masses $m_1$ and $m_2$ placed at distance $R$ from each other is $$F_g=\frac{G{m}_1{m}_2}{R^2}$$ Using this, gravitational force of attraction between particle $1$ and $2$ and between particle $1$ and $4$ is,

$F_{21}=F_{41}=\frac{G{M}^2}{(\sqrt{2}R{)}^2}=\frac{G{M}^2}{2R^2}$

$\Rightarrow F_{21}=F_{41}=\frac{G{M}^2}{2R^2}$

Components of $F_{21}$ and $F_{41}$ along the center are

$F_{21}\cos {45}^{\circ }=\frac{G{M}^2}{2R^2}\frac{1}{\sqrt{2}}=\frac{G{M}^2}{2\sqrt{2}{R}^2}$

$F_{41}\cos {45}^{\circ }=\frac{G{M}^2}{2R^2}\frac{1}{\sqrt{2}}=\frac{G{M}^2}{2\sqrt{2}{R}^2}$

Gravitational force of attraction between the particles $1$ and $3$

$F_{31}=\frac{G{M}^2}{(2R{)}^2}=\frac{G{M}^2}{4R^2}$

Net force towards the center is

$F_{net}=F_{21}\cos {45}^{\circ }+F_{41}\cos {45}^{\circ }+F_{31}$

$\Rightarrow F_{net}=\frac{G{M}^2}{2\sqrt{2}{R}^2}+\frac{G{M}^2}{2\sqrt{2}{R}^2}+\frac{G{M}^2}{4R^2}$

$\Rightarrow F_{net}=\left(\frac{2\sqrt{2}+1}{4}\right)\frac{G{M}^2}{R^2}$

If $v$ is the velocity of particle $1$, we have

Centripetal force, $F_C=\frac{M{v}^2}{R}$

Since, the net force provides necessary centripetal force for particle $1$, we get

$F_{net}=F_C$

$\Rightarrow \left(\frac{2\sqrt{2}+1}{4}\right)\frac{G{M}^2}{R^2}=\frac{M{v}^2}{R}$

$\therefore v=\sqrt{\frac{GM}{R}\left(\frac{2\sqrt{2}+1}{4}\right)}$

Hence, option (d) is the correct answer.

Why this question: The particles in circular motion under the influence of mutual gravitation. This is a commonly asked question and this concept can be extended to any number of particles.