A tunnel is dug through the centre of the earth. Shwo that a body of mass m when dropped form rest from one end of the tunnel will execute simple harmonic motion.

Explanation

Consider the situation shown in the diagram.
The gravitational force on the earth in shells internal to the position of the particle.

More clearly,
let g' be the acceleration at P.
So, `g'=g(1-(d)/(R))=g((R-d)/(R))`
From figure, `R-d=y`
`rArrg'=gy/(R),`
Force on body at p, `F=-mg'=(-mg)/(R)y" ".....(i)`
`rArrFprop-y" " ["where,y is distance from the centre"]`
So, motion is SHM.
For time period, we can write Eq. (i)
As`ma=-(Mg)/(R)yrArra=-(g)/(R)y`
Comparing with `a =-omega^(2)y`
`omega^(2)=(g)/(R)`
`rArr((2pi)/(T))=(g)/(R)rArrT=2pisqrt((R)/(g))`