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 2a and also the masses of the spheres are made 2m, then they will collide after ______ seconds.

Hint:

Use dimensional analysis to find the relationship between the collision time and the given parameters.

Answer 1

Correct Answer: 8

$T \propto m^x G^y a^z$

$T \propto M^x \left[ M^{-1}L^3T^{-2} \right]^y [L]^z$

$T \propto M^{x - y} L^{3y + z} T^{-2y}$

$x - y = 0 \Rightarrow x = y$

$-2y = 1 \Rightarrow y = -\frac{1}{2}$

$3y + z = 0 \Rightarrow z = -3y = \frac{3}{2}$

$\Rightarrow T \propto m^{-\frac{1}{2}} G^{-\frac{1}{2}} a^{\frac{3}{2}}$

$T \propto \frac{a^{3/2}}{\sqrt{m}}$

$T = 4 \left( \frac{2a}{a} \right)^{3/2} = 8s$