# Assuming the earth to be a sphere of uniform mass density,

Question:

Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface?

Solution:

Weight of a body of mass $m$ at the Earth's surface, $W=m g=250 \mathrm{~N}$

Body of mass $m$ is located at depth, $d=\frac{1}{2} R_{e}$

Wher

$R_{c}=$ Radius of the Earth

Acceleration due to gravity at depth $g(d)$ is given by the relation:

$g^{\prime}=\left(1-\frac{d}{R_{e}}\right) g$

$=\left(1-\frac{R_{e}}{2 \times R_{e}}\right) g=\frac{1}{2} g$

Weight of the body at depth $d$,

$W^{\prime}=m g^{\prime}$

$=m \times \frac{1}{2} g=\frac{1}{2} m g=\frac{1}{2} W$

$=\frac{1}{2} \times 250=125 \mathrm{~N}$