Question

A hole is bored along the diameter of the earth and a stone is dropped into it. If the radius of the earth is R then the speed of the stone, when it is at the centre of the earth, is

• A. $\sqrt{Rg}/2$
• B. $\sqrt{Rg}$
• C. $\sqrt{2gR}$
• D. Zero

Answer 1

Answer: (B)

$\sqrt{Rg}$

Gravitational potential energy at the centre of the earth $P.E_{centre}=\frac{-3GMm}{2R}$

Let $v$ be the speed of stone of mass $m$ at the center of earth and

$R$ and $M$ be the radius and mass of earth respectively.

Using Energy conservation at surface and at centre of earth:

$K.E_s+P.E_s=K.E_{centre}+P.E_{centre}$

$0+\left(\frac{-GMm}{R}\right)=\frac{1}{2}m{v}^2+\left(\frac{-3GMm}{2R}\right)$

$⟹v=\sqrt{\frac{GM}{R}}=\sqrt{gR}$ (as $GM=g{R}^2$)