Question

Three identical spheres of mass m , are placed at the vertices of 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.

Answer 1

8

Answer 2

The time to collision $T$ is proportional to $a^{3/2}/\sqrt{m}$.

For the first case: $T_1 \propto \frac{a^{3/2}}{\sqrt{m}}$. For the second case: $T_2 \propto \frac{(2a)^{3/2}}{\sqrt{2m}} = \frac{2^{3/2} a^{3/2}}{2^{1/2} \sqrt{m}} = 2 \frac{a^{3/2}}{\sqrt{m}}$.

Thus, $T_2 = 2 T_1$. Since $T_1 = 4$ seconds, $T_2 = 2 \times 4 = 8$ seconds.