Question

At what depth below the surface of the earth, the value of $g$ is the same as that at a height of $5\,km$ ?

• A. 1.25 km
• B. 2.5 km
• C. 5 km
• D. 7.5 km
• E. 10km

Answer 1

Answer: (E)

Acceleration due to gravity at the surface of earth
$ {{g}_{h}}=g{{\left( 1+\frac{h}{R} \right)}^{-2}} $
$ {{g}_{h}}=g\left( g-\frac{2h}{R} \right) $
Acceleration due to gravity at depth d from the surface of the earth
$ {{g}_{d}}=g\left( 1-\frac{d}{R} \right) $
Given: $ {{g}_{h}}={{g}_{d}} $
$ \therefore $ $ \frac{2h}{R}=\frac{d}{R} $
Or $ d=2h $ Or $ d=10\,km $