Simple Harmonic Motion

If you’ve ever taken a physics class on waves, the first type of mathematical wave you learn is the one due to what is called simple harmonic motion. The idea is pretty simple, so I’ll go through a rough derivation here.

First, we imagine what the situation looks like. It’s usually shown as a mass attached to a spring moving along an idealized (read: frictionless) surface, as shown below:

Idealized situation with only the force of the spring coming into play.

As we can see, there is only one force acting on it: the force of the spring, $\textbf{F}_s$. This force is proportional to the displacement $x$ of the mass attached to the spring and is known as Hooke’s Law. Therefore, if we write out the forces that act on this body using Newton’s second law, we get (along the x direction):

Once we’re at this point, we need to figure out what the solution to this second order differential equation is. I won’t go into full detail here, but you can use a bunch of different methods to solve for the function x we are looking for. You won’t need to solve the equation each time, since doing it once will give us the pattern we need in order to describe simple harmonic motion for all situations.

I’ll use the method of writing down our characteristic equation and solving.

We give the quantity $\sqrt{\frac{k}{m}}$ another variable name: $\omega$. This is called the angular frequency, and is measured in radians per second.

From here, our solution is given by:

For most applications that we use in physics, B ends up being zero. Additionally, $\phi$, which is just the phase constant and adjusts where the function begins, is often zero too. This means we are left with a rather simple equation that accurately describes how an idealized version of a mass moving on a spring would work.

But I don’t want to stop there. Instead, I want to show you another way of looking at how we describe simple harmonic motion. In particular, look at the form of our solution to the differential equation. It’s a combination of trigonometric functions, which we know can describe something else: a circle.

Simple Harmonic Motion through Uniform Circular Motion

At first glance, these two ideas might seem to have nothing to do with each other. Circular motion goes round and round while our simple harmonic motion situation only goes up and down or side to side. What can they possibly have in common?

(Before I dive into it, I just want to note that this is one of the coolest things I saw in my waves and optics class when I had to take it.)

First, we will build up our experiment before I show you the mathematics of it. Imagine you have a circular disc that is rotating at a constant rate on a table. Now, attach a small object somewhere along the circumference of the disk so that it spins with the disk. This means that the object on the disk is exhibiting uniform circular motion. Next, place a screen behind the rotating disk. Finally, we shine a light edge on to the disk so that its shadow is displayed on the screen behind it.

What do you think you will see? What will the object that is rotating on the disk look like on the screen?

The answer isn’t that the object will move from one side to the other at a constant rate. Instead, it will move faster in the centre and slower on the ends before it turns around. Effectively, it will look just like simple harmonic motion!

Because I know you are just craving to see what this looks like outside of your mind’s eye, here you go:

(A brief note: I did not make this myself. I found it on the Desmos site, and I unfortunately could not find the author. If you know who made it or if it is yours, please reach out to me and I’ll add your name to it. Also, press the “play” button on the ‘a’ in the calculator for it to start.)

The red function you see moving from left to right is the function given by simple harmonic motion that we saw above. It’s moving simply to show how the function lines up with the circular motion.

In the centre, we can see a point on the circle moving round and round through uniform circular motion. At the same time, there are two vertical lines in yellow acting as “screens” on each side. Notice the horizontal line connecting the point on the circle with the point on the screen. This is exactly what would happen if we shone light onto an object and have projected it onto the screen.

That’s a very nice way of viewing it physically, but what about mathematically? What is that screen? In essence, it’s the projection of the circle on the vertical axis. Imagine squishing the circle such that it creates a vertical line. That’s the role of the screen. Put another way, if we define the circle parametrically as $(cos\theta, sin\theta)$, we get the projection on the vertical axis by only looking at what happens to the y-component of the parametric curve and setting the x-component to zero.

And now, we have two ways of looking at simple harmonic motion. We can either view it as the physical motion, or as a projection of uniform circular motion onto a straight line (and since we’re dealing with a circle, you could project the circle onto any line you wanted).

Hopefully, this gives you a better idea of the basics of simple harmonic motion and how it relates to uniform circular motion. In the future, I’m going to go into detail on what happens when we add more forces to the system, since physical systems are never (or rarely ever) idealized.

Struggle

One of the unfortunate side effects of having a curriculum and set schedule in mathematics is that one never gets to think about concepts for too long. Instead, the goal of a class is to simply throw a bunch of ideas to students and let them “ponder” the ideas on their own time. This is seriously backwards, and it’s at the heart of what is wrong with a lot of mathematics education today.

Think back to your time as a student in mathematics. When was the last time you were presented with a problem, worked on it, and then had to stop and mull over the idea for a few days? If you’re education has been anything like my own, the answer is nearly never. That’s simply not how school works. You never hear the teacher give you a problem that you can think about and come to a solution on. What happens is that a teacher will explain the problem, and then jump right into the solution, with no explanation in between. And that’s if you’re lucky. Sometimes, an answer will be given to you without there even being a question!

I’m willing to bet that if you’ve ever been asked a mathematics puzzle, you usually think about it for a few seconds, decide that you may as well just listen to the explanation, and jump right ahead. It’s what I do, because I know I will understand the explanation, and I don’t want to pause the video or stop reading because thinking about the problem will be difficult.

However, I’ve realized that this is a way to make me think that I know a lot about mathematics, but it’s a sort of pseudo-knowledge. After all, why do we engage in mathematics if not to try and solve problems and puzzles? Once the answer is given, there’s no more fun to it anymore. It’s just an algorithm or step-by-step process to be applied.

That is why I’m in favour of learning how to struggle in mathematics more. And I don’t mean the kind of struggling that occurs because you don’t know what formula to use. I see this so often when I tutor students, and I’m starting to harbour the belief that it’s because schools are just presenting them with a vast amount of information without any use for it. They don’t know what to do in problems because they don’t even understand what the problems are about. The reason: they have no link in their minds between the material they learn in class and the problems presented to them. And since it doesn’t match up, they are at a loss for what to do.

In place of just giving them a bunch of theory on different concepts that they will have to remember how to apply an algorithm, we should have them struggling on the problems that led to the various equations and concepts that are learned. By doing this, it gives the student an idea what it is they are learning about. These aren’t just abstract symbols on a piece of paper. They are ideas about objects, and not giving students this information and not allowing them to struggle on these sorts of problems before giving them the answer as class notes is doing more harm than is worth the productivity boost.

Starting Simple

If you want to get a concept across, the best thing to do is to start simple. Learning can be difficult and many parts initially might not make sense, so it’s important to make the “jump” to that knowledge in a way that we can follow. If not, it will simply be too difficult to make that conceptual leap.

This is incredibly important when being introduced to a formula of some kind. The thing about formulas is that they can be used with little understanding of what is actually going on. It’s definitely possible to do well in a physics or chemistry or even mathematics class by only knowing formulas without actually knowing what is happening within the formulas.

For example, I was recently learning about the concept of curl and divergence in multivariable calculus. Without actually knowing the idea behind curl and divergence, I could simply used the formulas $2dcurl(\textbf{v})=\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}$ and $div(\textbf{v})=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}$ respectively without any problems. Since I know how to compute partial derivatives, finding the value for curl and divergence is relatively trivial.

However, the deeper understanding comes from when a formula is explained. I hate having formulas simply presented to me. What do they mean? How did certain terms come into play for calculating this certain thing? These are questions that interest me, because they give a reason for a formula. Without a reason, I’m just blindly computing.

This is why I was pleasantly surprised to view these videos from Khan Academy, because the author (Grant Sanderson) always gave intuitions before presenting the formulas. This way, there wasn’t befuddlement as to why a certain formula was supposed to represent curl or divergence. Instead, it made sense because of the intuitions and explanations beforehand.

How did he do this? By constructing the simplest case possible, and building the formula up from there. Because he approached the formula like this, it made sense as to why a certain operation was being done. Consequently, the explanation was easy to follow, and the formula wasn’t surprising once it was presented.

Let’s face it: mathematics can be difficult and abstract at times, making it difficult to make sense of formulas. Often, it can seem like formulas pop out from nowhere, which make them look mysterious and exotic. However, by exploring the simplest cases possible for the situation and slowly working up the intuitions, it makes it much easier to understand the origin of a formula.

This idea can just as well be extended for ideas in physics or other sciences, and I firmly believe it should be. I’ve been in classes in which time didn’t allow for us to derive various formulas, so they were simply presented to us as correct. While efficient, this made it difficult to understand how each component of a formula worked to calculate the thing we were searching for.

Therefore, whenever you get a chance to understand how a formula came about, take it. Do not simply absorb the final answer. In the long run, it will be more productive to figure out how a formula works instead of just knowing it.

The Short Route

If you think about what students learn in mathematics at an early age, it isn’t too difficult to realize why many kids find it useless.

To begin, students learn about arithmetic and patterns, which is relatively useful. They also learn about money and time, which is practical. However, they then start to learn about algebra, which starts to make things more complicated. Suddenly, variables as well as constants are flying all over the place, and keeping track of them can be a pain.

Because of this, teachers will often give students easy examples in order to “show them the ropes”. Many times, the question will be in the form of a word problem, with the student having to write down two equations and then find the solution. These aren’t particularly difficult problems, but they often confuse students because of their wording. Worse, a student can sometimes solve the problem without resorting to algebra and solving it the “long” way.

When I go through such problems with students I tutor, they often look at me with an expression that asks, why do I have to do this? And frankly, I don’t have a good reason for that particular problem.

The issue I see is that students are only being introduced to problems that are trivial to solve, which means they don’t get to see the full power of mathematics. It’s like watching a world-class archer shoot from only ten metres away. Sure, their shooting will probably be impeccable, but you kind of expected that anyway. In order to really be impressed, the archer will have to shoot from their competition position. This will show just how good the archer is.

Likewise, making students solve questions that are relatively trivial means they will only see mathematics as a tool that works, but not one that is super powerful. If instead we gave difficult or tricky questions to students, they would end up seeing just how useful mathematics can be compared to mentally “finding” the answer.

For myself, calculus and the three-dimensional coordinate system are what really demonstrate the power of mathematics. The former lets us precisely analyze the behaviour of curves, while the latter lets us understand how different curves and vectors compare and contrast when placed in the same space. After using these two tools a lot, I can somewhat visualize them in my head, but it’s quite taxing on my mind and is much easier on a computer. Therefore, doing problems that require more robust tools than our minds to doing mathematics shows how useful mathematics is.

This is why I fear many students don’t “get” mathematics in secondary school. A lot of it seems to be arduous for no reason, sort of like trying to get a computer to do one “simple” thing that ends up taking hours to figure out what the write code is. The students don’t see the incredible utility of mathematics because the teachers don’t give problems that are complicated. Instead, they favour giving a bunch of problems to get the “muscle memory” of that type of problem working. This gets students good at solving a problem like this, but they will be in trouble once a more tricky problem comes along.

As educators, we should be worried about how young students perceive mathematics. It should be thought of as a really great tool to analyzing situations. Since we cannot necessarily give them advance material, at least telling them about what this “simple” formula or concept is used for can help them understand that mathematics isn’t just a fancy and long route to getting an answer. In fact, the idea is that it should be the most efficient for difficult problems.

To illustrate this last idea, I want to bring up an example that Dan Meyer recently posted to his site about billiard balls. I’ve written about his problem before, but essentially its goal is to show how mathematics is the best way to figure out where a billiard ball will go after it hits the bumper.

However, instead of just asking where it will go after it hits one bumper, he asks where it will go after hitting multiple bumpers. Suddenly, the problem goes from something that could easily be solved in one’s mind (and therefore make the mathematics inefficient and useless), to where doing the mathematics is basically the only way to accurately solve the problem. In this situation, Dan Meyer shows how simply extending the problem can prove the utility of mathematics.

If we want students to not lose interest in mathematics in secondary school, we cannot always give easy examples to them. This makes mathematics look like the long route to take, which no one will actually do in life when facing a problem. Instead, we need to switch mathematics to being the short route, which in turn will show how useful mathematics really is.