Question:
A tunnel is dug along a diameter of the earth. Find the force on a particle of mass m placed in the tunnel at a distance $x$ from the center.
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 $=m g$
$F=m g_{s} \frac{x}{R_{e}}$
$F=\frac{G M_{e} m}{R_{e}^{3}} x$