Question

Six stars of equal mass m are moving about the centre of mass of the system such they are always on the vertices of a regular hexagon of side length a. Their common time period will be

• A. $4\pi \sqrt{\frac{a^3}{Gm}}$
• B. $2\pi \sqrt{\frac{4\sqrt{3}{a}^3}{Gm(5\sqrt{3}+4)}}$
• C. $4\pi \sqrt{\frac{3a^3}{Gm}}$
• D. None of these

Answer 1

The correct option is B

$2\pi \sqrt{\frac{4\sqrt{3}{a}^3}{Gm(5\sqrt{3}+4)}}$

$\overrightarrow{F}=\overrightarrow{F_1}+\overrightarrow{F_2}+\overrightarrow{F_3}+\overrightarrow{F_4}+\overrightarrow{F_5}$

$|\overrightarrow{F_2}|=|\overrightarrow{F_5}|$ and $|\overrightarrow{F_2}|=|\overrightarrow{F_4}|$

$\overrightarrow{F}=\overrightarrow{F_3}+2F_{2}cos{30}^{\circ }+2F_{1}cos{60}^{\circ }$

$F_3=\frac{G{m}^2}{4a^2}$; $F_2=\frac{G{m}^2}{3a^2}$; $F_1=\frac{G{m}^2}{a^2}$

$F=\frac{G{m}^2}{a^2}(\frac{5}{4}+\frac{1}{\sqrt{3}})=m{\omega }^{2}a$

$\omega =\sqrt{\frac{Gm}{a^3}(\frac{5}{4}+\frac{1}{\sqrt{3}})}$; $T=2\pi \sqrt{\frac{4\sqrt{3}{a}^3}{Gm(5\sqrt{3}+4)}}$