A tunnel is bored through the earth along a chord that does not pass through the center. A particle is dropped into the tunnel. What is the nature of the motion of the particle?

Solution

1. Inside a uniform Earth, the gravitational force on a particle at a distance $r$ from the center is $\vec{F}=-\frac{GMm}{R^3}\vec{r}$.
2. Let $x$ be the displacement of the particle from the midpoint of the tunnel chord.
3. The component of the gravitational force along the tunnel is $F_x=-\frac{GMm}{R^3}x$, which is directly proportional to the displacement $x$.
4. Since the restoring force follows the form $F=-kx$, the particle undergoes simple harmonic motion.
5. Hence the answer is (A).