(a) A tunnel is dug along a diameter of the earth. Find the force on the a particle of mass `m` placed in the tunnel at a distance `x` from the centre.
(b) A tunnel is dug along a chord of the earth at a perpendicular distance `R//2` from the earth's centre. the wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass `m` when it is at a distance `x` from the centre of the tunnel.

Explanation

(a)
Gravitational field at `A`
`E=(GM_(e)x)/(R_(e)^(3))`, along `AO`

`r=sqrt((R/2)^(2)+x^(2))`
Gravitational field at distance `r` from centre
`E=(GM_(e)r)/(R^(3))`
Force on `m`
`F=mE=(GM_(e)mr)/(R^(3))`

Force by wall
`N=F cos theta=(GM_(e)mr)/(R^(3)) (R//2)/r=(GM_(e)m)/(2R^(2))`