Question

The value of acceleration due to gravity at certain height $h$ above the surface of the earth is $\frac{g}{4}$ , where $g$ is the value of acceleration due to gravity at the surface of the earth. The height $h$ is ($R$ is the radius of earth)

• A. $R$
• B. $2R$
• C. $3R$
• D. $4R$

Answer 1

The correct option is A $R$

The acceleration due to gravity at a distance $r$ from the surface of a planet is given by $$g=\frac{GM}{r^2}$$
At the surface, $$g=\frac{GM}{R^2}...\left(1\right)$$ At a height $h$, $$g_h=\frac{GM}{(R+h{)}^2}...\left(1\right)$$From equation $\left(1\right)$ and $\left(2\right)$, we have

$\Rightarrow g_h=g{\left(\frac{R}{R+h}\right)}^2$

As, $g_h=\frac{g}{4}$

$\Rightarrow \frac{g}{4}=g{\left(\frac{R}{R+h}\right)}^2$

Taking square root,

$\Rightarrow \frac{1}{2}=\frac{R}{R+h}$

$\Rightarrow h=R$

Hence, option (a) is the correct answer.

$\text{Key Concept:}$The acceleration due to gravity due to a planet or point mass $=\frac{GM}{r^2}$ $\text{Why this question?}$ Caution: The result $g^{\prime }=g(1-\frac{2h}{Re})$ can be used only when $h<<R_e$.