Assume that a tunnel is dug across the earth (radius =R) passing through its centre. Find the time a particle takes to cover the length of the tunnel if (a) it is projected into the tunnel with a speed of √gR (b) it is released from a height R above the tunnel (c) it is thrown vertically upward along the length of tunnel with a speed of √gR.
in each case
a) Here we need to calculate the time period of the oscillation of a particle and center of the earth acts as mean position.
Let position of particle = 
Velocity = 
Acceleration = 
Time period 
Velocity 
[ Let ‘A’ be the amplitude ]
Let 
be the time when particle is at distance ‘R’
From equation 
Let 
be the time when particle is at distance ‘-R’
b) when body is dropped from a height ‘R’. the final velocity of the body on reaching the ground is ‘v’
we know 
We know 
This is same velocity as in case (a). therefore time taken to cover the length of tunnel will be same
c) when it is projected upwards with velocity 
it reaches the surface with the same velocity 
.
From there it enters tunnel with same velocity as in case (a) and hence time taken to cover the tunnel will be 
