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

This problem is solved by analyzing the gravitational interactions among spheres and the impact of mass and distance variations on collision duration. The initial setup involves three spheres, each with mass \( m \), positioned at the vertices of an equilateral triangle with side length \( a \). These spheres collide after \( T = 4 \) seconds. The computations required are as follows:

1. Newton's law of universal gravitation defines the force between two spheres as:
   \( F = \frac{Gm^2}{a^2} \), where \( G \) is the gravitational constant.
2. The acceleration \( a_{\text{original}} \) of one sphere, caused by the other two, is directly proportional to this force and inversely proportional to the mass \( m \), resulting in:
   \( a_{\text{original}} \propto \frac{m}{a^2} \)
3. When the masses are doubled to \( 2m \) and the side lengths are doubled to \( 2a \), the new force \( F' \) is calculated as:
   \( F' = \frac{G(2m)^2}{(2a)^2} = \frac{4Gm^2}{4a^2} = \frac{Gm^2}{a^2} \)
   This indicates that \( F' = F \), meaning the force magnitude remains constant.
4. The new acceleration \( a_{\text{new}} \) is determined by:
   \( a_{\text{new}} \propto \frac{2m}{(2a)^2} = \frac{2m}{4a^2} = \frac{m}{2a^2} \)
   Consequently, \( a_{\text{new}} = \frac{1}{2}a_{\text{original}} \).
5. The collision time is inversely proportional to the square root of the acceleration, expressed as:
   \( T_{\text{new}} = T \sqrt{\frac{a_{\text{original}}}{a_{\text{new}}}} = 4\sqrt{\frac{a^2}{\frac{a^2}{2}}} = 4\sqrt{2} \)
   This calculation yields approximately 8 seconds.

Therefore, in the modified configuration, the spheres will collide in approximately 8 seconds. This result is confirmed as it falls within the expected range of 8,8.