Two satellites A and B are revolving around a planet in coplanar circular orbits. If the radius of satellite B's orbit is 4 times the radius of satellite A's orbit, what is the ratio of their time periods ( T_A : T_B )?

Step-by-step solution

According to Kepler's third law of planetary motion, the square of the time period of revolution is proportional to the cube of the orbital radius: T^2 R^3 Taking the square root on both sides, we get: T R^ 3/2 Let the radius of satellite A's orbit be R_A = R . Then the radius of satellite B's orbit is R_B = 4R . The ratio of their time periods is: T_A T_B = ( R_A R_B )^ 3/2 = ( R 4R )^ 3/2 = ( 1 4 )^ 3/2 Since ( 1 4 )^ 1/2 = 1 2 , cubing this gives: T_A T_B = ( 1 2 )^3 = 1 8 Thus, the ratio of their time periods i