Variations on Simple Harmonic Motion

In the last post, we looked at the basics of simple harmonic motion, and how the equations is described because of the spring force being applied to the system.

However, we can generalize this much further. In our initial analysis, we assumed that there a frictionless surface for the mass to slide on. Obviously, that doesn’t happen, so we need to add in some frictional force. For simplicity, we will assume that the frictional force is proportional to the speed of the mass, and so will come into the form $-bv$, where $b$ is just a constant that will be specified in the initial conditions.

Additionally, we had the setup such that we pulled the mass and then let it go for as long as we liked. In reality though, we could supply an external force, which (along the axis of motion) we will assume is another sinusoidal function that is given by: $F_0 sin(\omega_d t)$.

Careful observers will notice that $\omega_d$ is not the same as the one we had in the last post. This is because the angular frequency will change due to the periodic applied force. An easy way to imagine this is if you’re pushing a swing. Initially, you push the swing as it is swinging back to the centre, and you do this one time each period. Imagine that period is two seconds. Then, if you start making your pushing motion every second instead of every two seconds, the swing won’t keep on going like it normally was. Instead, the swing wants to keep swinging at a period of two seconds while you’re pushing every second. Eventually, you will win out and the swing will swing at the rate of once per second.

Let’s now dig into the mathematics of this motion. Last time, we had the following equation:

Now, we add our frictional force and the external applied force to get:

Using the same method as last time, we can solve the homogeneous part of the differential equation by writing out characteristic equation:

This gives us a host of conditions depending on our values of the discriminant in the above equation.

If $\gamma^2-4\omega_0^2 = 0$, then $r = \frac{-\gamma}{2}$, which is a repeated root.

For a repeated root, we must add a factor of $t$ into one of our solutions, making the solution:

If $\gamma^2-4\omega_0^2 \gt 0$, then $r_1 = \frac{-\gamma + \sqrt{\gamma^2-4\omega_0^2}}{2}$ and $r_2 = \frac{-\gamma – \sqrt{\gamma^2-4\omega_0^2}}{2}$.

The solution is therefore:

Finally, if $\gamma^2-4\omega_0^2 \lt 0$, we get a system that has complex roots given by $r = \frac{-\gamma \pm \sqrt{4\omega_0^2-\gamma^2}i}{2}$. Note how the terms inside the square root are switched around, which happens because we factored out a $-1$ and wrote $\sqrt{-1}=i$.

This solution for the characteristic equation has the form $r = \lambda \pm \mu i$, so the solution is:

So those are your three cases, but this doesn’t even take into account the *external* force. If you remember from above, we were trying to solve the homogeneous portion of our differential equation, which is when the equation is equal to zero. We now need to tackle the right-hand side of the equation.

To do this, I like to use the method of undetermined coefficients. To me, this method is one that involves a little bit of skill at times, but is useful because it’s all about guessing. Essentially, we want to try and guess the form of the solution given that the external force is a sinusoidal function. Thankfully, this knowledge is of great help, because we know the derivatives of sines and cosines flip back and forth and become negative, possibly cancelling out in the final solution. Therefore, I propose a solution of the following nature:

Note that I’ve only chosen $\omega_d$ because it matches the non-homogeneous part.

Taking derivatives, we get:

Substituting our equations for $x(t), x’(t), x”(t)$ into the differential equation gives us:

We then can group the expression into sines and cosines.

Cosines:

And sines:

Solving for these coefficients through substitution gives use these two lovely expressions:

Most of the time, you’re not going to actually use these equations to solve the problem. It’s often easier to solve the characteristic equation and make sure the initial conditions are satisfied instead of just blindly using these formulas. You can also rearrange them to use slightly different variables, but the end product is the same. What’s interesting to note about this kind of forced oscillation is that you may notice that the homogeneous solution dies off over time. This means that the natural frequency $\omega_0$ is only relevant in the beginning. After a certain amount of time, it’s effects die off and the non-homogeneous part of the solution becomes more important. This coincides with our expectations. If you keep pumping a system with a different frequency, it will eventually match the one you’re pumping with, regardless of what it started with.

So that’s probably enough to keep your mind thinking for a while. These systems can be quite complex, and there’s a whole host of situations that can come out of it. I’ll leave it here though, and we’ll look at a slightly different wave topic next time, which is interference.

The Least Memorization Possible

I’ve found that there are two general groups of people when it comes to subjects like mathematics and physics. There are those who memorize, and those who internalize the material. Both can bring understanding to the student, but they are much different.

One of my mathematics professors illustrated this when he said, “As a mathematician, I like to do the least memorization possible.”

At first, this struck me as a little ironic, because a staple of mathematics exams is just remembering the truckloads of formulas for various situations. I know that at least in my classes in university, we get no formulas, no regular expressions (such as the trigonometric identities), no unit circle, or anything else. Everything needs to stay in our head, which means we have to memorize some things.

However, the deeper point I think he was trying to get at was that mathematics isn’t about remembering formulas and knowing when to use them. Sure, that’s what happens when we work on these ideas for a long time and get used to doing them, but the point is that these steps and procedures we take shouldn’t necessarily feel foreign. At the very least, they need to be logical and consistent. Doing a double integration by parts with say components $x^2$ and $e^x$ by choosing the former as $u$ and the latter as $dv$ but then doing the opposite after the first integration by parts isn’t logical.

I try to keep this in mind when working on both improving my skills in a mathematics or physics class and while trying to tie everything together for the end of the semester. I don’t want to remember a thousand different formulas. Instead, I want to remember the intuitive and powerful principles that I learned throughout the semester and be able to apply them when I get to problems, without necessarily memorizing everything.

However, I do want to point out one final thing: a lot of the teachers are being a bit disingenuous when telling you, “It’s not the end of the world if you don’t remember a formula. You can easily re-derive it.” Sure, that’s true and you can do that, but most students do not have the time to re-derive a formula and then answer the question that was troubling them on a test. This is particularly true if the test has a short duration (as mine were) or if there are multiple questions in which you have to do this. Given enough time, I’m sure I could get the formulas I needed, but that kind of time isn’t typically available during tests. That’s why it bugs me when teachers say this, because it’s true but not practical.

In a broader sense though, there’s something nice about being able to remember a few principles and working from there. I’m not saying you have to reinvent calculus for your test, but it might not hurt to try and “compress” the number of things you need to remember into more broad categories that can adapt to your specific situation.

Last-Minute Stuffing

Since I just finished up my exams for the semester, I’ve noticed a funny trend that most students seem to engage in. I call it “last-minute stuffing”, and it just refers to the minutes before a test where students quiz themselves and try to remember a bunch of information that they memorized.

At my university, the exams are mostly held in our sports complex, since there’s a bunch of room. Right before the exams, all the students taking the exam are bunched in the hallway outside, and this is where I hear the muttering and recitation going on. I can almost hear the desperation in some of their voices as they try to explain a concept to their friends but can’t quite get it, and how they try to memorize the way to do certain problems.

I don’t know if it’s just because I’m faking it to seem more in control than I really am, but this has almost never been something I did. The only time I remember doing this is when I was trying to remember a the different scenarios for trig substitution when I was about to enter a test that wasn’t integral calculus, but differential equations, where it could pop up. The reason I did that was because we never really worked with trig substitution in a while, and I needed to remember at least the first step.

Barring that, I won’t engage in this type of last-minute rush. The reason is simple: it’s not necessary. I reminded my friends of this as they too looked around with me and saw a bunch of people trying to recite things about the test. I said, “What you look at in the next five minutes isn’t going to change your mark to a drastic degree (if at all). The fact is that we’ve looked at the subject for three months, so all the work that we’ve done has prepared us for the test.”

That’s what I find people seem to forget during exam season. Tests aren’t aced by studying a bunch at the last minute (at least, I’d say that this isn’t the case for most people). Rather, it’s through learning and working throughout the semester.

You’ve done the work all semester, so now you just have to show that.

Confidence

When I look at a student who knows what they’re doing while solving a physics or mathematics problem versus someone who has no clue what they are doing, there’s an enormous difference in their confidence. The former can usually zero-in on the objective of the problem and knows the strategy needed to tackle it while the latter will try to remember useful formulas or try to do something more or less random.

This is relevant with almost all the students I tutor. I can tell the difference almost right off, because the person that doesn’t know what they are doing is usually unsure of the appropriate strategy and seems lost. This is only further exemplified when they pick the correct strategy to answer the question, but then ask me if what they are doing is correct.

I try to rarely give encouragement like this. The reason is simple: actually solving a problem (unless it’s a real hairy differential equation or integration) is relatively straightforward. It’s the logic of going from one step to the next that is more difficult, and the point of learning. Therefore, by answering the student’s question when they have to make a decision of what to do, I am taking away the most difficult part of the problem.

Instead, I do my best to let them choose the path to follow. Consequently, they get to figure out what works for them and what doesn’t make sense to do. If I just told them which steps to do, the crux of most questions would be gone.

As a tutor, my goal is to give the students I help the confidence they need to figure out the steps to a problem on their own. That is also why I try to stop them from referring to a memory aid or their notes during every problem, because I want them to feel comfortable with saying, “This is the next step. Not because my notes say so, but because it’s the next logical thing to do if we want to answer the question.” In my eyes, my job is only to do this, and definitely not to give them the procedure during each question. There’s a time for refining a procedure for a certain kind of problem later, but when they are first having problems it tends to be due to something conceptual.

When I think back to my own experiences in tests where I’ve felt confident on some questions and not so much on others, the reason I did not feel confident at some points was because I didn’t know what kind of steps to take. But when I was confident, it was due to practicing many times and internalizing the process. That’s confidence, and it’s what I want the students I tutor to have when I’m done working with them.

If you want to work on a particular quality for a student, work on their confidence, and the rest should follow.