Assuming the Earth to be a sphere of uniform mass density, how much would a body weigh half way down

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 $250	ext{ N}$ on the surface?

Professor PC Thomas Classes · Answer

The acceleration due to gravity at a depth $d$ is given by $g_d = g(1 - \frac{d}{R})$. For a depth $d = \frac{R}{2}$, the effective gravity becomes $g_d = g(1 - \frac{1}{2}) = \frac{g}{2}$. Hence, the weight is half of its value at the surface, which is $\frac{250}{2} = 125	ext{ N}$.