Question

A tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from one end of the tunnel will execute simple harmonic motion.

Answer 1

Acceleration due to gravity inside Earth

g′ = \(\frac{GM}{R^3}x=\frac{g}{R}x\) 

Here x = distance of the point from centre of earth (x<R)

If, block of mass m is placed along the diameter inside the earth.

So, force on block

F = − \(\frac{mg}{R}x\) = −kx

∴ T = 2π\(\sqrt{\frac{m}{k}}\) = 2π\(\sqrt{\frac{m}{\frac{mg}{R}}}\) 

T = 2π\(\sqrt{\frac{R}{g}}\)