Question

The mean orbital radius of the moon is r_m = 384 X 10⁶ m, its mean period is 27.3 days . Take G = 6.6726 X 10^-11 N ^. m^2. kg^-2 To 3 significant figures, what is the mass of the Earth in kilograms calculated from the above quantities?

Answer 1

Let the moon is orbiting the Earth in circular orbit of radius 384 x 106 m. The time period of the moon is 27.3 days = 27.3 x 24 x 60 x 60 s

Hence time period T = 23,58,720 s

The gravitational force between the Earth and moon is responsible for the rotation of the moon. Hence the gravitational force is equal to the centripetal force.

The gravitational force between the Earth and moon is $F_{g}=G\frac{Mm}{r^{2}}$ ; where M is mass of the Earth, m is mass of the moon; r is radius of the orbit.

F = mrw?
is centripetal force on moon. where $\omega$ is angular velocity. $\omega = 2\pi /T$

Then  $F_{c}=mr(2\pi /T)^{2}=4\pi^{2}mr/T^{2}$

Since Fg = Fc then

$G\frac{Mm}{r^{2}}=\frac{4\pi^{2}mr}{T^{2}}$ => $M=\frac{4\pi^{2}r^{3}}{GT^{2}}$ => $M=\frac{4\pi^{2}(384X10^{6})^{3}}{(6.6726X10^{-11})(2358720)^{2}}$

The mass of the Earth is M = 6.02 x 1024 kg