Q. The change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $d$ below the surface of earth. When both $d$ and $h$ are much smaller than the radius of earth, then which one of the following is correct?
A) $d = \frac{h}{2}$
B) $d = \frac{3h}{2}$
C) $d = 2h$
D) $d = h$
Explanation
(c) Acceleration due to gravity at height $h$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g_{h} = g\left( 1 - \frac{2h}{R} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots.(i)$
and depth $d$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g_{d} = g\left( 1 - \frac{d}{R} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots.(ii)$
From Eq. (i) and (ii),
$\ \ g\left( 1 - \frac{2h}{R} \right) = g\left( 1 - \frac{d}{g} \right)$
$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ 2h = d$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.