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  • 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.

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.


 

Answers (1)

We know that,

Acceleration due to gravity inside the earth = g’

g^{'} = g \left (1 - \frac{d}{R} \right )

    = = g \left (R - \frac{d}{R} \right )

Now, R-d = y

Thus, g^{'} = \frac{g.y}{R}

On both of them, force at depth ‘d’ will be –

F = -mg^{'}

  \frac{-mg. y}{R}

F \propto (-y)

Thus, in the tunnel, the motion of the body is SHM.

We can say,

ma = -mg^{'}

a =\frac{ -g}{R }y

\omega ^{2}y =\frac{ -g}{R}y

Thus, \frac{2\pi }{T} =\sqrt{ \frac{g}{R}}

Or, T =2\pi\sqrt{ \frac{R}{g}}.

Posted by

infoexpert24

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