Stable ratio
Classical Mechanics
Level
2
Two stars A and B orbit about their center of mass under their gravitational attraction, forming a binary star. If the ratio of their masses is 4:3, what is the ratio of the centripetal force experienced by stars A and B?
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1 solution
In a binary star system, the two stars are kept in orbit about their center of mass by their gravitational attraction. We assume the binary star is isolated so that the influence of other forces (i.e. gravity from distant cosmological objects) is negligible.
Since the gravitational force is directed along the line separating the two stars (a central force), the centripetal acceleration of each star is given by Newton's law of gravity, i.e. a i = F c / m i = m i G r 1 2 2 m 1 m 2 Since this acceleration is symmetric with respect to the masses of the two stars, the ratio is 1:1.