Question

Assume the earth to be perfect sphere of uniform density. If a body is dropped at one end of a tunnel dug along a diameter of the earth (remember that inside the tunnel the force on the body is – k times the displacement from the centre, k being a constant), it (body) will

Answer 1

Answer:

If a body is dropped at one end of a tunnel dug along a diameter of the earth, the body will do Simple Harmonic Motion

Explanation:

The tunnel is along a diameter and the body is released from the surface. If the body at any time is at a distance y from the centre of the earth, the restoring force will act on the ball due to gravitation between body and earth.

Acceleration of the particle at the distance y from the centre of earth.

$\sf{a=\dfrac{-GMy}{R^3}}\\\\\sf{a=\dfrac{-(GM)y}{R^3}}$

Futhermore as $\sf{g=\dfrac{GM}{R^2}}$

$\sf{a=\dfrac{-(gR^2)y}{R^3}}\\\\\sf{a=\dfrac{-gy}{R}}$

Comparing with $\sf{a=\omega^2y}$

$\sf{\omega^2=\dfrac{g}{R}}\\\implies\sf{\omega=\sqrt{\dfrac{g}{R}}}$

$\sf{Time\:Period(T)=\dfrac{2\pi}{\omega}}$

$\implies\sf{T=2\pi\sqrt{\dfrac{R}{g}}=84.6}$

$\implies$ Time period of a body Passing through a tunnel through earth does Simple Harmonic Motion with Time Period = 84.6