Three identical spheres of mass $m$, are placed at the vertices on an equilateral triangle of length

Question

Three identical spheres of mass $m$, are placed at the vertices on an equilateral triangle of length $a$. When released, they interact only through gravitational force and collide after a time $T=4$ seconds. If the sides of the triangle are increased to length $2 a$ and also the masses of the spheres are made $2 m$, then they will collide after $\_\_\_\_$ seconds.\

Professor PC Thomas Classes · Answer

As limiting case of elliptical path is straight line therefore proportionality $T^2 \propto \frac{a^3}{M}$ will holds from
$$
\begin{aligned}
& \left(T^2=\frac{4 \pi^2}{G M} r^3ight) \
& \frac{4^2}{T^2}=\frac{a^3}{m} \cdot \frac{2 m}{(2 a)^3}=\frac{2}{8}=\frac{1}{4} \
& \frac{4}{T}=\frac{1}{2} \
& \mathrm{~T}=8 	ext { seconds }
\end{aligned}
$$