A uniform solid spherical planet is rotating about its own axis. Due to geological changes, the planet shrinks uniformly such that its mass remains constant but the length of its day decreases by 36 % . What is the percentage decrease in the radius of the planet?

Step-by-step solution

Let the initial time period be T_i and the initial radius be R_i . Since the length of the day decreases by 36 % , the final time period is: T_f = T_i - 0.36 T_i = 0.64 T_i By conservation of angular momentum, I_i _i = I_f _f . For a uniform solid sphere, I = 2 5 MR^2 . Since = 2 T , we have R_i^2 T_i = R_f^2 T_f . ( R_f R_i )^2 = T_f T_i = 0.64 Taking the square root on both sides: R_f R_i = 0.64 = 0.8 This means the final radius is 80 % of the initial radius. The percentage decrease in the radius is: R_i - R_f R_