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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Payal Gupta | Contributor-Level 10

4 months ago

This is a long answer type question as classified in NCERT Exemplar

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 positiin of the particles . the external shells exert no force on the particle.

Let g’ be the acceleration at P

So g’ =g (1-d/R)=g (R-d/R)

R-d=y

g’=gy/R’

F=-mg’= -mgy/R

F $\propto - y$

Ma=-Mgy/R, a = -gy/R

Comparing a=-w²y

w²=g/R

T=2 $π \sqrt[]{\frac{R}{g}}$

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