Question

If the change in value of acceleration due to gravity at a height $h$ above the surface of the earth is same as at a depth $x$ below it, then (Assume, $x$ and $h$ being much smaller than the radius of the Earth)

• A. $x=h$
• B. $x=2h$
• C. $x=\frac{h}{2}$
• D. $x=h^2$

Answer 1

The correct option is B $x=2h$

The acceleration of due to gravity at a depth $d$ below the surface of the Earth is given by $$g_d=g(1-\frac{d}{R})...\left(1\right)$$ The acceleration due to gravity at a height $h$ above the surface of the Earth is given by $$g_h=g(1-\frac{2h}{R})...\left(2\right)$$
According to the question,
$g_h-g=g_x-g$

Substituting the values,

$g(1-\frac{2h}{R})-g=g(1-\frac{x}{R})-g$

$\Rightarrow g(-\frac{2h}{R_e})=g\left(\frac{-x}{R_e}\right)$

$\Rightarrow x=2h$

Hence, option (b) is the correct answer.

The acceleration of due to gravity at a depth $d$ below the surface of the earth is given by $$g_d=g(1-\frac{d}{R})$$Why this question? The result $g_d=(1-\frac{d}{R})$ is valid for any value of $d$, small or large.