Two stars, one with radius $r$ and the other with radius $3 r$, orbit each other so that their centers of mass are $25 r$ apart. Assume that the stars have the same uniform density. This system's center of mass is inside the larger star.

Question

Quizplus · Accepted Answer

The center of mass of a system is closer to the more massive object. Since both stars have the same density, the larger star (with radius $3r$) has a greater volume and thus a greater mass. Given the uniform density, the mass of the larger star is proportional to its volume, which is $3^3 = 27$ times the volume (and thus the mass) of the smaller star. Therefore, the center of mass, being influenced more by the larger mass, will be located closer to the larger star, and given the distances involved (with the stars' centers only $25r$ apart), it is indeed inside the larger star.

Two Stars, One with Radius rrr And the Other with Radius 3r3 r3r

Two Stars, One with Radius $r$ And the Other with Radius $3 r$