Physics

Physics

Problem

A pair of stars rotates about a common center of mass. One of the stars has a mass MM and the other has mass mm such that M=2mM=2m. The distance between the centres of the stars is dd (dd being large compared to the size of either star). The ratio of kinetic energies of the two stars (Km/KMK_m/K_M) is (A) 1 (B) 2 (C) 4 (D) 9
Intuitive explanation

The key insight is that both stars orbit the same center of mass with the same angular velocity, but at different distances from it. Use the center-of-mass condition to find each star's orbital radius, then calculate their kinetic energies using K=12mv2K = \frac{1}{2}mv^2 with v=ωrv = \omega r. The ratio will depend on how the separation dd splits between the two orbits, which is determined entirely by their mass ratio.

Intuitive explanation

The key insight is that both stars orbit the same center of mass with the same angular velocity, but at different distances from it. Use the center-of-mass condition to find each star's orbital radius, then calculate their kinetic energies using K=12mv2K = \frac{1}{2}mv^2 with v=ωrv = \omega r. The ratio will depend on how the separation dd splits between the two orbits, which is determined entirely by their mass ratio.

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