Above My Level

As of this moment, I’m in my second year of undergraduate studies in physics. Therefore, I’ve seen a bunch of classical topics (electricity and magnetism, mechanics, wave motion, elementary astrophysics) while only seeing a snippet of “modern” physics (special relativity, mostly in the form of time dilation). I haven’t taken any classes on quantum mechanics, so I’m definitely a novice there.

On the mathematics side, I’ve taken the usual classes of calculus, linear algebra, and differential equations. In a sense, I’m still learning the techniques before actually applying them to topics such as topology.

As you can see, I’m just starting to really dip my feet into my mathematics and physics training.

However, this hasn’t stopped me from pushing my boundaries and learning about other topics. I follow a variety of mathematics and physics blogs that I try and keep up with while reading. Additionally, I watch many video series which are part of the new swath of educational videos, such as PhysicsGirl, Infinite Series, 3Blue1Brown and PBS Spacetime (probably the best channel that I’ve found online). I also read books here and there that push my comfort zone in terms of physics and mathematics.

The point is that I don’t back down from being completely bamboozled. In fact, I want to find those places where I’m baffled, because they point to new opportunities to learn, which is always exciting. I also think that while I may not understand everything I read or watch, it can spark my curiosity to learn about it some more. Sometimes, it’s good to just have your head explode from not understanding.

At the same time, don’t just ignore what you can’t understand and move on. Instead, think about the problem and what’s fuzzy in your mind, and you’ll usually be able to form a plan to figure out what you need to learn. From there, you can give yourself a boost in your education without necessarily waiting for the appropriate classes at school. This is even more relevant when studying topics that aren’t part of your field, since self-study is the only opportunity you will usually get.

Take advantage of the wealth of information available to you (much of it free), and learn. It’s probably the best investment of time you can make.

Participation

When I was in secondary school, I really hated going to French class (I still wouldn’t particularly enjoy it). It wasn’t that the teacher was horrible or anything. Instead, it was simply because I didn’t like taking language classes other than English. I really excelled in English, and the huge chasm in my abilities in French versus English weren’t something I liked being reminded of. In English, not only was I well read and could write, I could also speak well. Conversely, speaking was my weakest link in French. It’s frustrating being able to know all the words that you want to say in your head (and even being able to think them), but not be able to actually express them. As such, I didn’t participate in class at all, preferring to only listen.

The result of this was that I came close to getting a horrible mark in the class for never participating. Since I didn’t want that, I changed my habits about halfway through the term and participated as needed. I even found that participating made me more engaged in the class, rather than just waiting for it to end.

The thing I see over and over again in a science or mathematics class now that I am out of secondary school and into higher education is the lack of participation in class. And I know it’s not because the students aren’t listening or are bored. Rather, no one participates because we are scared to be wrong. Almost every time a professor asks a question in my classes, they are met with silence, because we never want to answer a question and be wrong. I’m definitely a part of this group, and I struggle with this all the time.

Rationally, there’s no better place to be wrong than when giving answers to teachers. If the teacher is doing their job correctly, a mistake in an answer is an opportunity to expand someone’s mind and teach them a way to think. Done wrong, it makes a student feel unwelcome to share their answers again, most likely closing the door on future participation. I feel this all the time in class, and it’s why I don’t answer all that often. However, I’d almost wager that I am one of the most frequent contributors, showing just how infrequent we participate in class.

Being on the other end of the situation, I know that it’s not a fun feeling when no one wants to answer the question you pose in class. As such, I struggle not answering in class. After all, it makes no difference if I get something wrong in class. Who cares if others think I’m an idiot for asking the question? Who cares if it’s obvious to everyone else, but not me?

Personally, I’ve always struggled with this to some degree. I’m not actually shy, but I’m always so obsessed with being “good” at subjects that I don’t want to appear like I’m a novice. Therefore, I try to hide my lack of knowledge by doing the “rough work” at home, so I can appear to be knowledgeable during class. The truth, of course, is that the class is there for me to learn, not to know everything before we start.

This is why I encourage participation in mathematics and science class. We need to stop being worried about what others will think of our hypotheses and ideas. Yes, people will judge them, but who really cares? On an intellectual level, I know that asking questions and giving answers in class (even if I’m not sure about them) is a way to help me, not hinder me. As such, it’s one of my personal goals to work on, and I hope you can reflect on this as well. Do you participate in school, even when you aren’t sure about your answer? Do you ask questions, no matter how dumb? If the answer is “no”, I encourage you to work on this with me.

Participation in class is a key component to learning. Do what most people aren’t, and use it.

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.