Q. 39

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 :

When the body is made to fall through the tunnel it falls with some acceleration, and on reaching the centre it has some velocity to move ahead. the velocity vector points away from the centre. But gravity of the earth tries to pull it back due to which it experiences force in opposite direction of motion. All happens when it crosses the centre for the first time.


And acceleration due to gravity inside the earth is given as,


,


where d=depth from the surface of the earth


and R=radius of the earth


net force


Now let , which the distance from the centre of the earth.



we see that is proportional to -


Hence it is Simple Harmonic Motion.


Comparing the above equation with ,where k is the spring constant,


We get,



And time period


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