Mass of Earth

STATEMENT:

Consider a body of mass $m$ is placed at the surface of the Earth. Let,

$M_E$ = mass of earth.

$R_E$ = radius of earth, Which also is distance between the center of earth and the body.

$G$ = Universal gravitational constant.

DERIVATION:

By the Newton’s law of universal gravitation, the force between the body and the earth is given by:

$\boxed{F =G\frac{mM_E}{{R_E}^2}}$ ……….. (i)

But the force with which the earth attracts a body towards its center is called weight of the body, i.e.

$F = W=mg$ …………….. (ii)

Comparing equation (i) and (ii):

$mg =G\frac{mM_E}{{R_E}^2}$

$g =G\frac{M_E}{{R_E}^2}$

By re-arranging we get:

$\boxed{M_E =g\frac{{R_E}^2}{G}}$

Here,

$g = 9.8 \thinspace m/s²$ $G = 6.67 \thinspace \text{x} \thinspace 10^{-11} \thinspace Nm^2/kg^2$

$R_E = 6.38 \thinspace \text{x} \thinspace10^6\thinspace m$

By substituting these values, we get:

$M_E =(9.8) \Large \frac{(6.38 \thinspace \text{x} \thinspace10^6)^2}{6.67 \thinspace \text{x} \thinspace 10^{-11}}$

$\boxed{M_E = 6.0 \thinspace \text{x} \thinspace 10^{24} \thinspace kg}$