Simple harmonic motion (SHM) is the simplest kind of periodic motion, which is motion that repeats itself over regular intervals. Although other kinds of periodic motion exist, SHM is the most basic and the easiest to deal with. That doesn't mean this unit is going to be easy, though!

This unit does not introduce any new fundamental quantities unlike the previous few, but instead focuses heavily on applying everything we've learned so far in order to analyze a type of complex motion. We don't just have things moving in straight lines or circles anymore. They're now moving back and forth!

First, let's talk about what SHM looks like. SHM is a type of oscillation, which involves some object moving back and forth in a regular pattern. What distinguishes SHM from other kinds of oscillation is that it is very "smooth" and "regular". The exact reason for this will be revealed later.

Since we introduced the idea of periodic motion, we should also introduce the idea of aperiodic motion. In simple terms, this is any motion that does not repeat itself in a regular pattern. Sometimes, periodicity is used to represent that the motion repeats itself over a regular interval, while oscillatory motion refers to motion that repeats itself both in a regular pattern and interval. This usually involves something moving back and forth around a central position.

Why exactly do we call simple harmonic motion "harmonic"? Well, if you are a music student you might have some idea. In music, a harmonic is a wave that is a multiple of a fundamental frequency. I'm used to string instruments, so I'll explain it that way. Basically, a harmonic is a wave that covers the entire string smoothly. The key thing here is that this wave is sinusoidal. If you know what a sine wave looks like, you'll know that it's smooth and regular.

Indeed, SHM is characterized by some quantity relating to the object oscillating in a sinusoidal way with respect to time. Usually, this means position, velocity, acceleration, or something similar, but the exact definition is more general than that! What is the exact definition? Well, it has to do with something that we've touched on briefly before: the restoring force.

The mathematical definition of simple harmonic motion is, well, pretty simple. The first condition is that the force has to be a restoring force that points back towards the equilibrium. The second is that in general, the acceleration has to be directly proportional to the displacement from the origin and oppositely directed.

$$ a \propto - x $$
This means that the acceleration is always directed towards the equilibrium position, and the farther away you are from the equilibrium position, the greater the acceleration is. Sound familiar? It should! Something we learned about previously follows this exact relationship!

The precise definition doesn't necessarily involve acceleration and position. Instead, it has to do with the second derivative of the quantitiy in question. We can call this arbitrary quantity $x$ for simplicity. In general, for simple harmonic motion to occurr, we need to satisfy the following condition:

$$ \frac{d^2x}{dt^2} \propto - x $$

The negative sign indicates the restoring nature of SHM.

The other thing that describes simple harmonic motion is the angular frequency, which is a measure of how fast the oscillation occurs. It also takes the letter $\omega$, much like angular velocity. They even have the same units of $\textrm{rad/s}$. They are not the same thing, however! The angular frequency describes oscillatory motion, while angular velocity describes rotational motion. That being said, they are somewhat related, which I'll get to later.

The angular frequency is different for each kind of simple harmonic oscillator, but it is always related to the period (time to complete one cycle) of the motion. The period is denoted by $T$ and is related to the angular frequency by the following equation:

$$ T = \frac{2\pi}{\omega} $$
This relationship will be important later on when we talk about the different kinds of simple harmonic oscillators. You should recall that we briefly talked about period earlier on, in our circular motion unit. Well, we also have frequency, which we also talked about there. It's simply the reciprocal of the period, and in this case it would take the form:

$$ f = \frac{1}{T} = \frac{\omega}{2\pi} $$
With this information, we can also now finish the differential equation that describes simple harmonic motion. We can write it as:

$$ \frac{d^2x}{dt^2} = - \omega^2 x $$
This will be useful later on when we actually calculate things about simple harmonic oscillators. Of course, there's a simpler version as well that doesn't use differential equations:

$$ a = - \omega^2 x $$
Now that we know about the angular frequency, we can now more accurately write the mathematical definition for simple harmonic motion as:

$$ a = - \omega^2 x $$
That's all the discussion of the math behind SHM. However, we still need to talk about why angular frequency shares the same letter and unit as angular velocity. In fact, the relationship goes deeper than just the two quantities. They connect simple harmonic motion and circular motion.

Consider an object uniformly moving in a horizontal circle. If we look at the object, it has the same velocity throughout all points in time. However, what's not the same is the distribution of that velocity into its components!

Consider only the y-component in this case. We will look at where the y-position of the ball is at each point in time as it traverses the circle.

The colored circles in the diagram correspond to the positions of the object on the circle as well as the positions on the sine wave.

This can tell us that if we only look at one dimension of circular motion, we have motion that varies sinusoidally with time. This is why angular frequency and angular velocity are so closely interconnected: circular motion is actually just SHM that occurs simultaneously in two different dimensions! This is also why you might have noticed the formula for the period to be somewhat familiar.

In fact, we can draw even more mathematical parallels between the two. We said that the period for SHM is given by $ T = \frac{2\pi}{\omega}$. For circular motion, we can find the angular velocity with the equation $ \omega = v/r $. This means that applying the SHM formula gives us:

$$ T = \dfrac {2\pi r}{v} $$
The above result is what we stated to be true during our circular motion lesson!

We also can mathematically show that the motion for each dimension of circular motion is sinusoidal in nature. We begin by writing each component of the velocity as a function of $\theta$, measured from the positive x-axis, being sure to take direction into account. We can say that the particle has a speed of $v_0$.

$$ v_x = - v_0 \sin \theta $$ $$ v_y = v_0 \cos \theta $$
Why is $v_x$ negative? Well, that's because it's directed leftwards whenever the angle is less than $180 \degree$ and directed rightwards when it's between $180 \degree$ and $360 \degree$. This is the opposite of the behavior that $\sin \theta$ produces, so we need a negative sign.

Moreover, because the motion is uniform, we know that $\theta = \omega t$, indicating that the velocities are sinusoidal functions with respect to time. We can re-write the formulas as:

$$ v_x = - v_0 \sin (\omega t) $$ $$ v_y = v_0 \cos (\omega t) $$
If we show that the velocity varies with time sinusoidally and averages to zero over the entire period of time, we can assume that the position varies sinusoidally with time as well. (It has to average to zero, else the object doesn't repeat its motion.) However, with a little calculus, we can actually find the exact expressions for each. We simply integrate each expression.

$$ x = r \cos (\omega t) $$ $$ y = r \sin (\omega t) $$
This means that the position of the object in circular motion is sinusoidal with respect to time, which is exactly what we wanted to show. See, circular motion and SHM are very closely related. Some would say circular motion is just SHM in disguise, but I think they're different enough to warrant different classifications.

This is all there is behind SHM! It's a relatively simple definition that hinges on ideas we've talked about before. Now, we'll move to talking about the various different specific kinds of SHM, starting with the most simple. While we've only really talked about forces and kinematics here, do note that we are going to delve into other areas such as energy with each specific type of SHM.