Calculate the mass of the earth from the period of the moon 27.3 d and its mean orbital radius of 3.84 x 10^8m. The universal gravitational constant is 6.673 x 10^-11 N.m^2/kg^2.
The moon revolves around the Earth once in 27.3 days and the mean orbital distance is 3.84*10^8 m.
Let the mass of the Moon be Mm and that of the Earth be Me.
The gravitational force of attraction between the Earth and the Moon is (G*Mm*Me)/r^2 where r is the distance between the two.
This force is also equal to the centripetal force of attraction given by F = Mm*r*(4pi^2)/T^2 where T is the time taken to complete one orbit.
F = Mm*r*(4pi^2)/T^2 = G*Mm*Me/r^2
=> r^3*(4pi^2)/(G*T^2) = Me
Substituting the values given:
Me = (3.84*10^8)^3*4*pi^2/(6.673 x 10^-11*(27.2*24*60*60)^2)
=> Me = (3.84*10^8)^3*4*pi^2/(6.673 x 10^-11*(27.2*24*60*60)^2)
=> 6.021*10^24 kg
The mass of the Moon is approximately 6.021*10^24 kg