Determine the mass of the Earth from the known period (27.4 days) & distance of the Moon (3.84x10^8m).?

Question

so i don't see how the distance from the earth to the moon relates to finding the mass of the earth. other info given is that the radius of the earth is 6.38x10^6 meters, radius of the moon is 1.74x10^6 meters, mass of the earth is 5.98x10^24 kilogram, and mass of the moon is 7.4x10^22 kg, and the universal graviatity constant is -6.67x10^-11 Nm^2/kg^2

Answer 1

Force between: earth (mass M, radius R) and moon (mass m and radius r) separated by distance 'd' (centre to centre)

Centripetal force on moon
= m v^2 / d = m (d*w)^2 / d
= m d w^2 (w=angular speed moon = 2 Pi/T) T =time period
= m d (2 Pi/T)^2 = 4 pi^2 m d / T^2 ....... (K)

This Centripetal force = force of attraction between the 2

4 pi^2 m d / T^2 = G M m /d^2
(see whether m cancels out > meaning whether moon or a football rovolves at distance d, it will have same T period)

M = 4 pi^2 d^3 / GT^2 (T in seconds)

M = 4*3.14*3.14*(3.84*10^8)^3 / [6.67x10^-11 *(2367360)^2
M in Kg = [2233.12 / 373.81] * 10^24= 5.97*10^24 as given by you