Question

A straight tunnel is dug into the earth as shown in figure at a distance $b$ from its
centre. A ball of mass $m$ is dopped from one of its ends.The time it takes to reach the other end is approximately

• A. $42min$
• B. $84min$
• C. $84\frac{b}{R}min$
• D. $42\frac{b}{R}min$

Answer 1

The correct option is A $42min$

Gravitational field inside is

$g=\frac{GMr}{R^3}$

Considering at an angle theta at displacement x from mean position,

$g=\frac{GM}{R^3}\frac{x}{sin\theta }$

Taking component along the line of motion,

$a=gsin\theta =\frac{GMx}{R^3}$

This is SHM with time period as

$T=w=2\pi \sqrt{\frac{R^3}{GM}}=2\pi \sqrt{\frac{R}{g}}\approx 84min$

Thus, time taken is $t=\frac{T}{2}=42min$

Answer is option A.