Mass of Earth cannot be measured directly by placing it on any weighing scale. But it can be measured by an indirect method. This method utilizes the Newton's law of universal gravitation.
Derivation
Consider a body of mass m on the surface of the Earth. Let the mass of the Earth be Me and the radius of the Earth be Re.
According to the law of gravitation, the gravitational force F of the Earth acting on a body is given by:
`F=G\frac{M_em}{R_e^2}\rightarrow\left(1\right)`
But the Force with which Earth attracts a body towards its center is equal to its weight W.
`F=W=mg\rightarrow\left(2\right)`
Comparing equation (1) and (2); we get:
`mg=G\frac{M_em}{R_e^2}`
Re-arrange of above equation for Me
`M_e=\frac{gR_e^2}G`
Calculation of Mass of Earth
Mass Me of the Earth can be determined on putting the values in equation:
`M_e=\frac{gR_e^2}G`
`g=10Nkg^{-1}`
`R_e=6.38\times10^6m`
`G=6.673\times10^{-11}Nm^2kg^{-2}`
Substituting these values in above equation
`M_e=\frac{10Nkg^{-1}\times\left(6.38\times10^6m\right)^2}{6.673\times10^{-11}Nm^2kg^{-2}}`
`M_e=6.0\times10^{24}Kg`
Thus, mass of the Earth is `6.0\times10^{24}Kg`
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