Q. What is the height the weight of body will be the same as at the same depth from the surface of the earth? Radius of earth is $R$
A) $\frac{R}{2}$
B) $\sqrt{5}R - R$
C) $\frac{\sqrt{5}R - R}{2}$
D) $\frac{\sqrt{3}R - R}{2}$
Explanation
(c) $\frac{gR^{2}}{(R + h)^{2}} = g\left( 1 - \frac{h}{R} \right)$
or $\left( 1 - \frac{h}{R} \right)\left( 1 + \frac{h^{2}}{R^{2}} + \frac{2h}{R} \right) = 1$
or $\frac{h^{3}}{R^{3}} + \frac{h^{2}}{R^{2}} - \frac{h}{R} = 0$
or $\frac{h}{R}\left( \frac{h^{2}}{R^{2}} + \frac{h}{R} - 1 \right) = 0$
or $\frac{h}{R} = \frac{- 1 \pm \sqrt{1 + 4}}{2} = \frac{\sqrt{5} - 1}{2}$
or $h = \frac{\sqrt{5}R - R}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.