Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance `R//2` from the earth's centre where R is the radius of the earth. The wall of the tunnel is frictionless.
(a) Find the gravitational force exerted by the earth on a particle of mass m placed in the tunnel at a distance x from the centre of the tunnel.
(b) Find the component of this force along the tunnel and perpendicular to the tunnel.
(c) Find the normal force exerted by the wall on the particle.
(d) Find the resultant force on the particles.
(e) Show that the motion of the particle in the tunnel is simple harmonic and find the time period.

Explanation

(a)
The mass of sphere of radius r
`M^'=(M)/(4/3piR^3)*4/3pir^3=(Mr^3)/(R^3)`
M: mass of earth
`F=(GM^'m)/(r^2)=(GMm)/(R^3)r=(GMm)/(R^3)sqrt(x^2+R^2/4)`
(b)
Component of force
(i) along the tunnel
`Fcostheta=Fx/r=(GMm)/(R^3)x`
(ii) `_|_^(ar)` to the tunnel
`Fsintheta=F(R//2)/(r)=(GMm)/(2R^2)`
(c)
`N=Fsintheta=(GMm)/(2R^2)`
(d) Resultant force on particle `=Fcostheta=(GMm)/(R^3)x`
(e) Restoring force `F^'=-Fcostheta=-(GMm)/(R^3)x`
`a=(F^')/(m)=-(GM)/(R^3)x=-omega^2x`
`T=2pisqrt((R^3)/(GM))`