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Question
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.
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Solution
Consider the situation shown in the diagram.
The gravitational force on the particle at a distance r from the centre of the earth arises entirely from that portion of matter of the earth in shells internal to the position of the particle. The external shells exert no force on the particle.
More clearly,
Let g' be the acceleration at P.
So, `g^' = g(1 - d/R) = g((R - d)/R)`
From the figure, `R - d = y`
⇒ `g^' = g y/R^'`
Force on the body at p,
F = `- mg^' = (- mg)/R y` .......(i)
⇒ F ∝ – y ......[Where y is the distance from the centre]
So, motion is S.H.M.
For time period, we can write equation (i)
As ma = `- (Mg)/Ry`
⇒ `a = - g/R y`
Comparing with a = `- ω^2y`
`ω^2 = g/R`
⇒ `((2pi)/T) = g/R`
⇒ T = `2pi sqrt(R/g)`
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