Question:
A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?
Solution:
Weight of the body, $W=63 \mathrm{~N}$
Acceleration due to gravity at height h from the Earth’s surface is given by the relation:
$g^{\prime}=\frac{g}{\left(\frac{1+h}{R_{e}}\right)^{2}}$
Where,
g = Acceleration due to gravity on the Earth’s surface
Re = Radius of the Earth
For $h=\frac{R_{e}}{2}$
$g^{\prime}=\frac{g}{\left(1+\frac{R_{e}}{2 \times R_{e}}\right)^{2}}=\frac{g}{\left(1+\frac{1}{2}\right)^{2}}=\frac{4}{9} g$
Weight of a body of mass m at height h is given as:
$W^{\prime}=m g^{\circ}$
$=m \times \frac{4}{9} g=\frac{4}{9} \times m g$
$=\frac{4}{9} W$
$=\frac{4}{9} \times 63=28 \mathrm{~N}$