# Simple Harmonic Motion

Written by Travis M. Moore
Last edited Sep-2019

### Simple Harmonic Motion

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.

### Sinusoidal Motion

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).

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