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 ?

Answer 1

Weight of a body of mass m at the Earth’s surface, $W$ = $mg$ = $250 \,N$
Body of mass m is located at depth, $d$ = $\frac{1}{2} R_e$
Where,
$R_e$ = Radius of the Earth
Acceleration due to gravity at depth g (d) is given by the relation:
$g' \bigg(1-\frac{d}{R_e}\bigg)g$

= $\bigg(1-\frac{R_e}{ 2 \times R_e}\bigg)g = \frac{1}{2}g$

Weight of the body at depth d,
$W'$ = $mg'$
= $m \times \frac{1}{2} g$= $\frac{1}{2} mg$= $\frac{1}{2} W$

= $\frac{1}{2} \times 250$ = $125 \;\text N$