Question

Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is

• A. simple harmonic
• B. parabolic
• C. on a straight line
• D. periodic

Answer 1

Answer: (C), (D)

The correct options are
C

on a straight line

D

periodic

This should be verified.

If the particle is released from height h the force on it when it is out of the tunnel is

$F=\frac{G{M}_{e}m}{(R+h{)}^2}=ma$

$a=\frac{GM}{(R+h{)}^2}$

you mean, this ain't no SHM equation!

but while it comes inside the tunnel, let's say x distance from center

$M^{\prime }=\frac{M{X}^3}{R^3}$

$F_x=\frac{-GMm}{R^3}x$

$a_x=\frac{-GM}{R^3}x$

yeah, that's more like it.

So clearly the motion is not SHM

The particle reaches Q with velocity v as inside the tunnel its doing SHM.

After coming out the particle goes straight up and free falls under gravity taking time $\frac{2v}{g}$

So the total time period is fixed and the path is a straight line, not parabola.

Hence options $C\&D$ are correct