Written by Travis M. Moore

Last edited Sep-2019

Simple harmonic motion (SHM) is just that: simple! A tuning fork exhibits this kind of motion when struck. Let's examine in more detail what the tines of a tuning fork are actually doing when they vibrate.

It's clear from Figure 1 that SHM describes a back and forth motion, but
we can be more specific. When the tuning fork is at rest, the tines are
completely vertical (let's call this p0 for position 0).
This is called the point of **equilibrium**.
When struck, the tines first move toward each other, which means they
move out of equilibrium by a certain amount. The tines will move inward
as far as the force that was applied to it will allow, but eventually
the energy is not enough to bend the tines any further and they stop
(p1). The tines, of course, do not stay
bent toward the center. There is a **restoring force** that makes
the tines move back toward equilibrium. This comes from the
**elasticity** of the metal springing back after being struck.
The tines make their way back to vertical. However, there is still
enough energy from hitting the fork that the tines continue past
p0 and move outward (p2). From here the restoring force
takes over again and springs the tine back to equilibrium. At this
point we have described one complete **cycle** of SHM: p0 →
p1 → p0 → p2 → p0.

A handy way to quantify the speed of an object in SHM is by the
number of cycles that occur per second, or cps. For example, the tines
might complete 100 cps when struck. In other words, cps expresses the
frequency with which the tines are vibrating. The formal term for this
is, you guessed it, *frequency*. You are likely familiar with
the official unit of frequency, the Hertz (Hz).

Now we have a way to quantify the number of cycles the tines move every second, which also tells us how many times the air molecules were pushed together (condensation) and pulled apart (rarefaction). We can now describe the frequency of a sound wave! Slower vibrations are lower frequency, and sound low-pitched (e.g., 250 Hz). Faster vibrations complete more cps and sound high-pitched (6000 Hz). We can even define the range of human hearing at this point. Humans can hear frequencies as low as 20 Hz and as high as 20,000 Hz.

We've discussed the "simple" part of SHM (i.e., back-and-forth movement across a point of equilibrium), but what about the "harmonic" part? If you're thinking that harmony has to do with music (e.g., to sing in harmony), you're right! Musical instruments produce sound by using a variety of strings and cavities that move back and forth. An example is the vibration of a guitar string after it is plucked. Different instruments have different sounds because every instrument produces a different combination of simple harmonic waves.

The motion of an object in SHM can also be referred to as
**sinusoidal** motion. SHM can be described using a circle:
around-and-around rather than back-and-forth. If an object is moving
in a circle, we can quantify exactly where it is by calculating the
sine, cosine and tangent. It is not a coincidence that plotting
circles uses the sine and circular motion is called sinusoidal (the
adjectival form of sine).

Next Topic: Characterizing Sine Waves