Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $$d = \frac{R}{2}$$ from the surface of earth, if its weight on the surface of earth is 200 N, will be : (Given $$R$$ = radius of earth)

We need to find the weight of a body at depth $$d = R/2$$ below Earth's surface, given its surface weight is 200 N.

To begin,

Assuming Earth has uniform mass density, the acceleration due to gravity at depth $$d$$ below the surface is:

$$ g_d = g\left(1 - \frac{d}{R}\right) $$

This is because, by the shell theorem, only the spherical mass at radii less than $$(R - d)$$ contributes to gravity. This inner mass is proportional to $$(R-d)^3$$, and the gravitational acceleration at distance $$(R-d)$$ from the centre is $$g_d = g(R-d)/R$$.

Next,

$$ W' = W \times \frac{g_d}{g} = W\left(1 - \frac{d}{R}\right) = 200\left(1 - \frac{R/2}{R}\right) = 200\left(1 - \frac{1}{2}\right) = 200 \times \frac{1}{2} = 100\;\text{N} $$

The weight at depth $$R/2$$ is 100 N.

The correct answer is Option 2: 100 N.