Estimate the mass of the Earth

Question

Suppose that the Earth is fixed and that the orbit of the Moon around the Earth is approximately circular. If the radius of the Moon's orbit is R= 384,000 km and its period is T= 27.3 days, estimate the mass of the Earth.

Answer 1

Let the mass of the moon be m and the mass of the earth be M. r=the radius of the moon's orbit.

Gravitational force between the moon and earth = GMm/r²,

where G=6.67×10^-11Nm²/kg² and r=384000000=3.84×10⁸m.

The circular motion creates a centripetal force on the moon=mv²/r, where v is the velocity in orbit.

These two forces are equal and opposite:

GMm/r²=mv²/r. The length of the orbit is 2πr and the moon takes time T (orbital period) to travel a complete orbit, therefore T=2πr/v, so v=2πr/T. The centripetal force is 4π²rm/T².

GMm/r²=4π²rm/T², M=4π²r³/(GT²), r³=5.66231×10²⁵m³, T=27.3 days=2358720 seconds, T²=5.56356×10¹² s².

M=(4π²)(5.66231×10²⁵)/(6.67×10^-11×5.56356×10¹²)=6.02386×10²⁴kg.

So the mass of the earth is 6.024×10²¹ metric tonnes approx.