Problem 8.43 Part A Exact solutions for gravitational problems involving more th

Question

Problem 8.43 Part A Exact solutions for gravitational problems involving more than two bodies are notoriously difficult. One solvable problem involves a configuration of three equal-mass objects spaced in an equilateral triangle. Forces due to their mutual gravitation cause the configuration to rotate. Suppose three identical stars, each of mass M, form a triangle of side L Find an expression for the period of their orbital motion. Express your answer in terms of the variables M, L, the constant of universal gravitation G Submit My Answers Give Up

Explanation / Answer

here,

as the configuration will rotate about the centre of mass of system

the distance of each mass of COM is

r = 2/3 * sqrt(3)* L/2

r = L/sqrt(3)

Now, for the net force acting on a particle

Fnet = 2 * F * cos(30)

Fnet = 2 * G * M^2/L^2 * cos(30)

Fnet = sqrt(3) * G * M^2/L^2

Now, let the period is T

as Fnet = centripetal force

2 * G * M^2/L^2 * cos(30) = M *(2pi/T)^2 * r

2 * G * M^2/L^2 * cos(30) = M *(2pi/T)^2 *L/sqrt(3)

3 * G*M/L^3 = 4pi^2/T^2

T = 2 pi * sqrt(L^3/(3 * G * M))

the time period is 2 pi * sqrt(L^3/(3 * G * M))

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