Question

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

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

Answer 1

The correct option is B $\sqrt{gR}$

Gravitational potential at a point on the surface of Earth $=-\frac{GM}{R}$
If Earth is assumed to be a solid sphere,then the gravitational potential at the centre of earth $=-\frac{3}{2}\frac{GM}{R}$
Decrease in gravitational potential $=\frac{R}{2}\times \frac{GM}{R^2}=\frac{Rg}{2}$
Loss in potential energy $=\frac{Rg}{2}\times m$
Now,gain in kinetic energy $=$Loss in potential energy
$\frac{1}{2}m{v}^2=\frac{1}{2}mgR$
$or,v=\sqrt{gR}$