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 particle. (e) Show that the motion of the particle in the tunnel is simple harmonic and find the time period.
(a) 
(b) 
(c) 
(d) 
(e) 
(a) Let ‘l’ be the distance from the center of the earth to the particle at distance ‘x’
Mass of the earth with reduced radius ‘l’
mass of the earth with radius R
The gravitational force on the particle will be
------->(1)
From the triangle in the figure
Substituting in equation (1)
(b) Let the component of force along tunnel 
’
------→ (
) [from eqn (1)]
Let the perpendicular component be 
[from figure 
]
-------→(
(c) the walls exert the same amount of force which the particle exerts on them
[
]
(d) Resultant force on particle is the same force that is acting along the tunnel
[from 
]
(e) For a body to be in S.H.M 
Here 
------→(e)
Here 
is constant
