Question

The value of gravitational acceleration ‘g’ at a height ‘h’ above the Earth’s surface is $\dfrac{g}{4}$ then ( $R = $ radius of Earth)
A. $h = R$
B. $h = \dfrac{R}{2}$
C. $h = \dfrac{R}{3}$
D. $h = \dfrac{R}{4}$

Answer 1

Hint: The acceleration due to gravity is inversely proportional to the square of the distance between the center of the Earth and the body. For a body at Earth’s surface the acceleration due to gravity is given as: $g = \dfrac{{GM}}{{{R^2}}}$ Here, \[g\] is the acceleration due to gravity
And for a body at a height $'h'$ above the Earth’s surface is given as:
\[{g_h} = \dfrac{{GM}}{{{{\left( {R + h} \right)}^2}}}\]
Here, \[{g_h}\] is the acceleration due to gravity
\[G\] is the universal gravitational constant
\[M\] is the mass of Earth
\[R\] is the radius of Earth
\[h\] is the height of the body above the surface.
We are given that \[{g_h} = \dfrac{g}{4}\] , equate both the equations and find the value of \[h\] in terms of radius

Complete step by step answer:
The acceleration due to gravity is given as:
$g = \dfrac{{GM}}{{{R^2}}}$ --equation \[1\]
The acceleration due to gravity at some height \[h\] above the Earth’s surface is given as:
\[{g_h} = \dfrac{{GM}}{{{{\left( {R + h} \right)}^2}}}\] --equation \[2\]
We need to find the value of \[h\] such that \[{g_h} = \dfrac{g}{4}\] , as
\[{g_h} = \dfrac{g}{4}\]
From equation \[1\] and equation \[2\] , we get:
\[\dfrac{{GM}}{{{{\left( {R + h} \right)}^2}}} = \dfrac{{GM}}{{4{R^2}}}\]
\[ \Rightarrow \dfrac{1}{{{{\left( {R + h} \right)}^2}}} = \dfrac{1}{{4{R^2}}}\]
\[ \Rightarrow 4{R^2} = {\left( {R + h} \right)^2}\]
\[ \Rightarrow {\left( {2R} \right)^2} = {\left( {R + h} \right)^2}\]
Taking square root on both sides, we get:
\[ \Rightarrow 2R = R + h\]
\[ \Rightarrow h = R\]
Thus, at a height of \[h = R\] above the Earth’s surface the acceleration due to gravity will be $\dfrac{g}{4}$ .

So, the correct answer is “Option A”.

Note:
As we move above the Earth’s surface the value of acceleration due to gravity decreases.
As we move below the surface of Earth, the acceleration due to gravity increases.
The value of acceleration due to gravity at height \[h = 2R\] will be $\dfrac{g}{9}$ .
The value of acceleration due to gravity is not constant but changes as we go to different locations on the Earth’s surface.