Question

The weight of an object reduces by a factor of 4 as it is taken from the earth's surface to a height of 6400km. Determine the earth's radius.

Answer 1

Weight:

It is the force with which an object is attracted towards the center of the earth by gravity.

It can be calculated using the formula, $W=mg$ where m= mass of the object and g= acceleration due to gravity.

Given data and assumptions:

Let W be the weight of an object on the earth's surface.

Let W' be the weight of the object when it is taken from the earth's surface to a height of h.

Height h=6400 km

So, $W\text{'}=mg\text{'};g\text{'}=\text{acceleration due to gravity at h}$

Acceleration due to gravity at height h is given by, $g\text{'}=\frac{g}{{\left(1+{\displaystyle \frac{h}{R}}\right)}^2}$; R = radius of the earth.

Applying the formula and finding radius:

Given, that the weight of an object reduces by a factor of 4 as it is taken from the earth's surface to a height, $W\text{'}=\raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

Hence, we can write

$$\begin{gathered} mg\text{'}=\raisebox{1ex}{$mg$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ g\text{'}=\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \frac{g}{{\left(1+{\displaystyle \frac{h}{R}}\right)}^2}=\frac{g}{4} \\ {\left(1+\frac{h}{R}\right)}^2=4 \\ \left(1+\frac{h}{R}\right)=2 \\ \frac{h}{R}=1 \\ h=R \end{gathered}$$

Therefore, the radius is equal to the height which is 6400 km.

Thus, the earth's radius is 6400 km