Question

If the radius of earth of $R$ then the height h at which the value of $g$ becomes one-fourth, will be

• A. $\frac{R}{8}$
• B. $\frac{3R}{8}$
• C. $\frac{3R}{4}$
• D. $\frac{R}{2}$

Answer 1

The Correct Option is B

The value of acceleration due to gravity at a height $h$ above the surface of the earth is given by
$g^{'}=\frac{g}{\left(1+\frac{h}{R}\right)^{2}}$
where $R$ is radius of earth. When li is negligible compared to $R$, we have
$g^{'}=g\left(1+\frac{h}{R}\right)^{-2}=g\left(1-\frac{2 h}{R}\right)$
Given $ g^{'}=\frac{g}{4}$
$\frac{g}{4}=g\left(1-\frac{2 h}{R}\right)$
$\Rightarrow \frac{1}{4}=1-\frac{2 h}{R}$
$\Rightarrow\frac{2 h}{R}=\frac{3}{4}$
$\Rightarrow h=\frac{3 R}{8}$Note : The value of acceleration due to gravity decreases on going above or below the surface of earth.