Question

Let ‘$g_h$’ and ‘$g_d$’ be the acceleration due to gravity at height ‘$h$’ above the earth’s surface and at depth ‘$d$’ below the earth’s surface respectively. If $g_h = g_d$ then the relation between ‘$h$’ and ‘$d$’ is

• A. $d= h$
• B. $d=\frac{h}{2}$
• C. $d=\frac{h}{4}$
• D. $d = 2h$

Answer 1

Answer: (D)

Acceleration due to gravity at a height h above the earth’s surface,
$g_{h}=g\left(1-\frac{2 h}{R}\right)$
Acceleration due to gravity at a depth d below the earth’s surface
$g_{d}=g\left(1-\frac{d}{R}\right)$
As $\,\,\, g_{h}=g_{d}$
$\therefore \,\,\, g\left(1-\frac{2 h}{R}\right)=g\left(1-\frac{d}{R}\right)$
or $1-\frac{2 h}{R}=1-\frac{d}{R} \Rightarrow \frac{2 h}{R}=\frac{d}{R}$
$\Rightarrow \,\,\, d=2 h $