# Simple harmonic motion versus oscillations

I want to see whether certain oscillations in my daily life, such as the oscillation of violin strings when plucked, are simple harmonic motion or not. Can we identify whether an oscillation is simple harmonic motion or just an oscillation by observing it?

I don't truly understand the difference between the two - mathematically, we know that acceleration should be proportional to negative displacement for simple harmonic motion. Am I right when I say that an oscillation, such as violin strings oscillating when plucked, cannot be identified to be either a regular oscillation or simple harmonic motion until its motion is precisely tracked and analysed?

• From my reading of this: en.m.wikipedia.org/wiki/Simple_harmonic_motion, SHO is an idealisation and not really applicable in everyday situations, as energy loss is always involved. So timing the motion and checking if it's periodic, which for macro world it won't be, should be enough. In the micro world, as far as I know, SHO can exist, for example in vibrational modes of molecules. But hopefully you will get an answer that sorts us both out. – user81619 Nov 7 '15 at 0:04

The pure simple harmonic motion is in real life very very rare. There are some cases which are really close (e.g. for engineering purposes). That might be:

• Small-amplitude oscillation of a mass on a spring (small enough for spring nonlinearities not to be pronounced) or other kinds of these simple or moreless model oscillators.
• Tuning fork. Strictly speaking it has more oscillatory modes but it is hard to effectively excite more than the one.
• Speaker membrane when playing a pure sine tone (that's easy to be generated).
• For the last but not least case, there is a practical possibility to excite just a harmonic oscillation of the damped system with just pure harmonic driving force. That is what it's done e.g. when the room acoustics is examined using sine sweep tones.

For your example of a violin string, you can immediately determine that it is not simple harmonic motion by listening to it. Simple harmonic motion is a pure tone of a single frequency. Violins don't sound like that so you immediately know there are harmonics and it therefore is not a simple harmonic oscillator. As some other people have mentioned, a tuning fork produces a very pure tone and so it's a very good approximation to a simple harmonic oscillator.

Only if an object moves according to $x(t)=A \sin (\omega t)$ do we call it simple harmonic motion.

If, for example, the body moves according to $x(t)=A \sin (\omega t) + 0.2A \sin (2 \omega t)$ then we say its motion is periodic, but it is not a simple harmonic motion.

The points on the strings of musical instruments do not have simple harmonic motions in general but a complicated movement of the type: $x(t)=A \sin (\omega t) + 0.2A \sin (2 \omega t) + 0.05A \sin (3 \omega t)+ ...$. Because the amplitude of $A \sin (\omega t)$ is quite high in comparison with the other terms, $x(t)$ can be approximated by $A \sin (\omega t)$ and this is the pure musical note. This high amplitude term, $A \sin (\omega t)$, explains why we can identify a musical note no matter what musical instrument plays it.

I would like to foil the statement that "plucking a violin string will not result in simple harmonic motion." This is absolutely true in practice, but brings up an important point the OP might want to understand.

ANY time periodic motion is

1. driven by a restoring force and
2. has a small amplitude

The motion will be approximately SHO. The smaller the amplitude, the closer to SHM.

What defines a small amplitude? I can't think of a generic definition except to say that it is the regime where the restoring force is linear with distance to within your ability or desire to measure it.

So, for any given physical system, you perform experiments to find the range of displacement $\Delta x$ such that the restoring force is $F = -k\Delta x$ to within whatever degree of experimental accuracy you desire. Within that range of displacements (assuming you can find one), when you cause the system to oscillate about a point of equilibrium, the resulting periodic motion will be SHM, to within some degree of measurement accuracy of your clocks and rulers.

So, if you can experimentally find the linear regime of a violin string, and give it a gentle pluck, in that case it would vibrate with SHM.

Don't spend a lot of time trying to get this to work, though :)