Consider a tunnel excavated along a diameter of the earth. A...

When the particle is released from a height $h$ directly above the tunnel, it falls towards the center of the earth due to gravity.

Assuming the earth is non-rotating, the initial velocity is zero, and the gravitational force is purely radial. Thus, the particle moves along a straight line passing through the center of the earth.

For $r > R$ (outside the earth), the gravitational force is $F = -\dfrac{GMm}{r^2}$.

For $r < R$ (inside the earth), the gravitational force is $F = -\dfrac{GMmr}{R^3}$.

Since the force is always directed towards the mean position (center of the earth) and the system is conservative, the particle will cross the center, reach a distance $h$ on the opposite side, and return. Hence, the motion is periodic.

However, because the restoring force is not proportional to the displacement for the entire journey (specifically for $r > R$), the motion is not simple harmonic.

Thus, the motion is on a straight line and periodic.