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

Explanation

Suppose a body of mass m reaches at point P with depth d from the surface of earth and P point at distance r from the centre of earth. So there is no force exerted on body of mass m from external part at the distance r from earth but force is exerted only by a mass of earth of radius r.

Acceleration due to gravity at depth d from the surface of earth
`g. = g(1-(d)/(R ))= g((R-d)/(R ))`
Let `R- d = y`,
`g. = (gy)/(R )`
Force on a body of mass m at point P,
`F= -mg." "` (force is considered negative toward the centre)
`F= -(mg)/(R )*y" ""........."(1)`
`therefore F propto -y`
Now, `ma= -(mg)/(R )y" "` (From equation (1))
`therefore a= -(g)/(R )y" ""........."(2)`
comparing this equation `a= -omega^(2)y` with (2) (acceleration of SHM oscillator)
`omega^(2) = (g)/(R )`
`therefore omega = sqrt((g)/(R ))`
`therefore (2pi)/(T)= sqrt((g)/(R ))`
`therefore T= 2pi sqrt((R )/(g))`.