Solve the following :

Solution:

Method 1:

The gravitational field inside a sphere of radius $R$ at a distance $x$ from center is So, force on particle of mass $m$ is

$E=\frac{G M_{g} x}{R_{e}^{3}}$

So, force on particle of mass $m$ is

$F=m E$

$F=\frac{G M_{e} m}{R^{3}} x$

Method 2:

Acceleration due to gravity at depth $d$ from surface

$g_{d}=g_{s}\left(1-\frac{d}{R_{e}}\right)$

Here, $d=R_{e}-x$

$g=g_{s}\left[1-\frac{\left(R_{e}-x\right)}{R_{e}}\right\rfloor$

$g=g_{s} \frac{x}{R_{e}}$

So, force on particle of mass $m$ is $=m g$

$F=m g_{s} \frac{x}{R_{e}}$

$F=\frac{G M_{e} m}{R_{e}^{3}} x$