Question

A diametrical tunnel is dug across the earth. A ball is dropped into the tunnel from one side. The velocity of the ball when it reaches the centre of the earth is [Given: gravitational potential at the centre of the earth$=-3/\left(2\right)$GM/R)]

• A. $\sqrt{R}$
• B. $\sqrt{gR}$
• C. $\sqrt{2.5gR}$
• D. $\sqrt{7.1gR}$

Answer 1

Answer: (B)

$\sqrt{gR}$

Let mass of ball is $m$ and velocity at centre is $v$.

Total energy of the ball at surface$=-\frac{GMm}{R}+\frac{1}{2}m\times 0^2=-\frac{GMm}{R}$,

Total energy at the centre of the earth$=-\frac{3GMm}{2R}+\frac{1}{2}m{v}^2$

Total energy of the ball will be conserved.

$\Rightarrow -\frac{3GMm}{2R}+\frac{1}{2}m{v}^2=-\frac{GMm}{R}$

$\Rightarrow v=\sqrt{\frac{GM}{R}}$ (Using $g=\frac{GM}{R^2}$)

$\Rightarrow v=\sqrt{gR}$