# Why isn't this motion simple harmonic?

If a tunnel is dug along a diameter of the Earth, and a ball is dropped in the tunnel from a certain height $h$ above the surface of Earth, what should the motion of the ball be as seen from the Earth?

The answer says it'll be along a straight line and periodic, but not simple harmonic. I understand the former. But why will it not be simple harmonic?

From what I could make out, the mean position would be the center of the Earth, with the force of gravity acting on the ball. Speed would be maximum at the mean point (center of the Earth), and minimum at the extreme positions (surface of the Earth).

• – John Rennie Oct 30 '17 at 5:59
• You might find this question and associated answers helpful - better than the duplicate that John marked, I think. – Floris Oct 30 '17 at 6:17
• Is the ball dropped from above the tunnel (surface of the earth), or while it's already inside? If you start outside the tunnel, you don't have harmonic motion; if you start inside, it will be. The derivation of the latter is in my linked answer. Please let us know if that doesn't resolve your question. – Floris Oct 30 '17 at 22:03
• @Floris, it's being dropped from a height h, i.e. outside the tunnel. Why won't it have harmonic motion? – Shreya Oct 31 '17 at 3:01

While it is outside the earth, the force goes as $\frac{1}{r^2}$; once inside, the force is proportional to $r$. To have simple harmonic motion the force needs to be proportional to displacement for the entire range of motion - clearly it won't be if you drop it from height $h>0$. Just think about it - dropping from a very great height, it would take a long time - as height goes to infinity so does the time. But simple harmonic motion is characterized by the fact that the period is independent of the amplitude...