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 $d$,
$W^{\prime}=m g^{\prime}$
$=m \times \frac{1}{2} g=\frac{1}{2} m g=\frac{1}{2} W$
$=\frac{1}{2} \times 250=125 \mathrm{~N}$