Question

If the radius of the earth is $R$, 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

$\frac{3R}{8}$

Step 1: Given data

The radius of earth, $R$

Height, $h$

Acceleration due to gravity, $g$

Value of $g$ becomes one-fourth at a height h, $g\text{'}=\frac{g}{4}$

Step 2: To find

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

Step 3: Formula used

Acceleration due to gravity at a height $h$ is given by,

$g\text{'}=g\left(1-\frac{2h}{R}\right)$

Step 4: Calculation

Acceleration due to gravity at a height $h$,

$g\text{'}=g\left(1-\frac{2h}{R}\right)----------\left(1\right)$

At a height $h$,

$g\text{'}=\frac{g}{4}--------------\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$, we get

$$\begin{gathered} \frac{g}{4}=g\left(1-\frac{2h}{R}\right) \\ \Rightarrow \frac{1}{4}=1-\frac{2h}{R} \\ \Rightarrow \frac{2h}{R}=1-\frac{1}{4} \\ \Rightarrow 2h=\frac{3R}{4} \\ \Rightarrow h=\frac{3R}{8} \end{gathered}$$

Therefore, the height $h$ at which the value of $g$ becomes one-fourth will be $\frac{3R}{8}$

Hence, option B is correct.