# Half way at the centre of earth, what would be the weight of a body as compared to that on the surface?

The weight of a body on the earth’s surface is, $W=mg$ where, $m$ is the mass of the body and $g$ is the acceleration due to gravity.

The effective acceleration due to gravity at depth $x$ from the surface of earth is, $g’=\left(1-\frac{x}{R}\right)mg$ where, $R$ is the radius of the earth.

Let $W_x$ be the weight of the body at depth $x$. Then,

$W_x=mg’=m\left(1-\frac{x}{R}\right)g=\left(1-\frac{x}{R}\right)W$

Half way down at the centre of earth, $x=\frac{R}{2},$

$\therefore W_x=\left(1-\frac{R/2}{R}\right)W=\left(1-\frac{1}{2}\right)W=\frac{W}{2}$

Hence, the weight of the body will be half of that on the surface of the earth.