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.

A tunnel is dug through the centre of the earth. Show that a body of m

1.	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.
Answer» SOLUTION :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))`.