The mass of the Earth can’t be calculated physically. It is too large for that, and we haven’t made a weighing scale big enough to hold the entire Earth, yet we still managed to calculate its mass. How was this possible? Is it just a wild guess, or is there another way we could measure the mass of the Earth?
To calculate the mass of the Earth, we will need to assume a few values.
Consider:
Mb → Mass of the Ball
ME → Mass of The Earth
G → Universal Gravitational Constant
RE → Radius of the Earth (i.e. the distance between the ball and the centre of Earth)
According to Newton’s Law of Universal Gravitation, the gravitational force of the Earth acting on the ball is:
\(F = G\frac{mM_E}{{r_e}^2}\)→(i)
Whereas the force of the attraction of the Earth is equal to the weight of the ball. Therefore:
F = W = mg →(ii)
Compare equation (i) and (ii); we get:
\(mg = G\frac{mM_E}{{r_e}^2}\)
Rearranging the equation:
\(M_E = \frac{g R_E^2}{G}\) →(iii)
Values of constants in equation (iii) are:
g = 10 Nkg-1
R = 6.38 * 106 m
G = 6.67 * 10-11 Nm2 Kg-2
Substituting these values in equation (iii):
\(M_E = \frac{(10\,NKg^-1) \times (6.38 \times 10^6\,m)^2}{6.67 \times 10^{-11}\,Nm^2/kg^2}\)
ME = 6.0 * 1024 kg
\(M_E = 6.0\times 10^{24}\,kg\)
Thus, the mass of the Earth is \(M_E = 6.0\times 10^{24}\,kg\)