Three identical spheres of mass m are placed at the vertices of an equilateral triangle of side a. Starting from rest, they interact only through mutual gravitation and collide after 4 s. If the side is changed to 2a and each mass is changed to 2m, after how many seconds will they collide?
Physics · Gravitation · law of gravitation ,gravitational field intensity ,gravitational potential and potential energy
JEE Mains 2025 3 April Morning Shift

Question

Three identical spheres of mass m are placed at the vertices of an equilateral triangle of side a. Starting from rest, they interact only through mutual gravitation and collide after 4 s. If the side is changed to 2a and each mass is changed to 2m, after how many seconds will they collide?

Physics · Gravitation · law of gravitation ,gravitational field intensity ,gravitational potential and potential energy

JEE Mains 2025 3 April Morning Shift

G1 Physics · Accepted Answer

For this self-similar gravitational collapse, the characteristic time scales as T ∝ √(a³/m). This scaling follows from dimensional analysis or from the energy equation for the three-body symmetric motion. Initially, T₁ = 4 s, side = a, mass = m. In the new system, a₂ = 2a, m₂ = 2m. Therefore, T₂/T₁ = √[(a₂³/m₂)/(a³/m)] = √[(2a)³/(2m) × (m/a³)] = √(8/2) = √4 = 2. Thus, T₂ = 2T₁ = 8 s. Answer: 8.