Question

A tunnel has been dug through the centre of the earth and a ball is released in it. It executes S.H.M. with the time period:

• A. 42 minutes
• B. 1 day
• C. 1 hour
• D. 84.6 minutes

Answer 1

The correct option is D 84.6 minutes

On falling deeper, the acceleration gets progressively smaller as it depends on distance from the center of the earth. Gravitational force at any point depends upon the mass inside the radius $r$.

$F=-mg\frac{r}{R_{earth})}=-kr$

Here, $F$ is the force on the body and $r$ is the distance from the centre. This is the equation of a body undergoing SHM with a time period of:

$T=2\Pi \sqrt{\frac{m{R}_{earth}}{mg}}=2\Pi \sqrt{\frac{R_{earth}}{g}}$

$R_{earth}=63.78\ast {10}^{6}m$

$T=5068s=84.6min$