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

Let, $g_d$ be the acceleration due to gravity at depth $d$ from the ground.

We know that $\frac{g_d}{g}=(1-\frac{d}{R})$

Here, $d=\frac{R}{2}$

$\therefore \frac{g_d}{g}=(1-\frac{\frac{R}{2}}{R})=\frac{1}{2}$

If $W$ be the weight of the body on the surface of Earth and $W_d$ be the weight of the body at depth $d$

then, $\frac{W_d}{W}=\frac{m{g}_d}{mg}=\frac{g_d}{g}=\frac{1}{2}$

or $W_d=\frac{1}{2}W=\frac{1}{2}\times 250=125N$