Question

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

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

Answer 1

The correct option is C

$R$

Step 1: Given data

Radius of earth is $R$

Acceleration due to gravitation is $g$

Height when $g$ becomes one-fourth is $h$

$g\text{'}=\frac{g}{4}$

Step 2: Formula Used

The acceleration due to gravity on Earth can be given as $g=\frac{GM}{R^2}$, where $G$ is the gravitational constant, $M$ is mass of Earth and $R$ is radius of Earth.

So, the acceleration due to gravity at a height of $h$ above Earth will be $g\text{'}=\frac{GM}{{\left(R+h\right)}^2}$.

Step 3: Solution

The expression for acceleration due to gravity at a height of $h$ above Earth is $g\text{'}=\frac{g}{4}$. Substitute $g=\frac{GM}{R^2}$ and $g\text{'}=\frac{GM}{{\left(R+h\right)}^2}$ in $g\text{'}=\frac{g}{4}$.

$$\begin{gathered} g\text{'}=\frac{g}{4} \\ \Rightarrow \frac{GM}{{\left(R+h\right)}^2}=\frac{GM}{4R^2} \\ \Rightarrow {\left(R+h\right)}^2=4R^2 \\ \Rightarrow {\left(\frac{R+h}{R}\right)}^2=4 \\ \Rightarrow {\left(1+\frac{h}{R}\right)}^2=4 \\ \Rightarrow 1+\frac{h}{R}=2 \\ \Rightarrow h=R \end{gathered}$$

The height $h$ at which the value of $g$ becomes one-fourth is $h=R$

Hence, the correct option is C.