Question

Three identical point masses each of mass m are located at the vertices of an equilateral triangle of side a. They revolve in a circular orbit circumscribing the triangle, maintaining their relative positions, under the influence of their gravitational force. The speed of each is

(1) \(\sqrt{\frac{GM}{a}}\)

(2) \(\sqrt{\frac{2GM}{a}}\)

(3) \(\sqrt{\frac{3GM}{a}}\)

(4) \(\frac{2Gm}{3a}\)

Answer 1

 (1) \(\sqrt{\frac{GM}{a}}\)

Let PQR be the equilateral triangle side a. Let O be centeroid of the triangle. The circular orbit circumscribing the triangle is a circle of center O and radius 

r = OP = a/√3.

Let f = Gm²/a² be gravitational force on P due to mass at Q or R. The net force on mass at

For circular orbit; 

centeripetal force = net gravitational force