Question

At what distance from the centre of earth, the value of acceleration due to gravity $g$will be half that of the surfaces ($R$ $=$ Radius of earth)

• A. $2R$
• B. $R$
• C. $1.414R$
• D. $0.414R$

Answer 1

The correct option is D

$0.414R$

Step 1. Given data

Acceleration due to gravity,$g$.

The radius of earth, $R$.

Step 2. Find height $\left(h\right)$ at which the value of acceleration due to gravity $\left(g\right)$will be half that of the surfaces :

$g\text{'}=g_h$(From figure ) gravity at some height from the earth's surface.

$g_{h=}\frac{g}{2}$

$$\begin{gathered} g_h=\frac{g}{{\left(1+\frac{h}{R}\right)}^2} \\ \frac{g}{2}=\frac{g}{{\left(1+\frac{h}{R}\right)}^2} \\ {\left(1+\frac{h}{R}\right)}^2=2 \\ {\left(R+h\right)}^2=2R^2 \\ R+h=R\sqrt{2} \\ h=R\sqrt{2}-R \\ =R(\sqrt{2}-1) \\ =(0.414)R \end{gathered}$$

Hence, option (D) is correct.