Question

The ratio of acceleration due to gravity at the earth's surface and at a height h from the earth's surface is 4. Determine the value of h in terms of the radius of earth.

Answer 1

Acceleration due to gravity:

1. It is the acceleration gained by an object due to gravitational force.
2. It is given by $g=\frac{GM}{R^2}$where G= gravitational constant, M=mass of earth, R= radius of the earth.

Given data and calculation:

Let g' be the acceleration due to gravity at a height ‘h’. and g be the acceleration at the earth's surface.

It is given, $\frac{g}{g\text{'}}=4$

Acceleration due to gravity at height h , $g\text{'}=\frac{g}{{(1+{\displaystyle \frac{h}{R}})}^2}$

So,

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

Thus, the height at which the acceleration due to gravity becomes ${\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\text{th}}$ acceleration due to gravity on the surface is $h=R$