by Bradley Knockel
A physics course is made more challenging due to its mathematical nature. In addition to having to understand the difficult concepts, the required math may seem like an unwelcome obstacle. The math and the concepts are complementary. We cannot fully understand the concepts without being able to understand their mathematical nature (and we cannot be proficient at solving mathematical physics problems without understanding the concepts).
A conceptual physics-101 course avoids much of the math, which highly limits the amount of physics that can be learned. People cannot move beyond a basic understanding of the concepts if they cannot mathematically apply them. Luckily, the goal of a conceptual physics course is only a basic understanding, which can be a great foundation for future physics, engineering, science, or technology courses.
While I hope to have expressed that the math is very important, I also want to now emphasize that math without the concepts is not a good use of your time. Not only will your grade in the course suffer by skipping the concepts that guide problem solving, but you will miss the primary goal of taking an introductory physics course: to understand the fundamental concepts of how the universe works. Unlike just memorizing mathematical steps, these concepts will forever enhance the way we understand the world and will help us learn new technical things faster and deeper. Seeing the world as someone who understands physics is a great thing. Mathematics is the language of the universe, so we learn it because it helps us understand the concepts.
Before taking a physics course that uses algebra and trigonometry (and maybe calculus), it is ideal to have as much math preparation as possible. Let's explore how math is used in physics.
Math education sometimes focuses on procedures to solve certain types of problems. For example, when adding fractions, we give the fractions a common denominator, add the numerators, and then simplify the resulting fraction. Techniques like this are useful in a physics course, but understanding when to apply these procedures when not being directly told to add the fractions can be more difficult. We must not only have all of math at our fingertips ready to be recalled, but we must be able to recognize when to recall it.
The procedural way introductory math is sometimes taught in math classes can be acceptable for fields like engineering and chemistry. The math in these areas typically does not require deep understanding, so these fields can get away with focusing on procedures. Physics uses math differently than these fields (and people who are good chemists and engineers and math teachers also know how to use math differently).
Physics requires a greater fluency with math than what is sometimes taught, though the word problems we were given in math classes are a great start to actually applying math (and math is just starting to be taught in a better way here in the United States). Let's do an example word problem. Which set of numbers contains numbers that have increased the most: {2, 3} or {25, 30}. The answer depends on the physical context, so we cannot answer it unless we are given more information! If we are told that these are mile markers on the road designating start and finish, than the first set is a 1-mile trip and the second set is a 5-mile trip, so the second set has a larger increase. However, if these are cost per pound of various types of food, then we need the ratio instead of using subtraction. The first set increases to 3/2 = 150% of its original price, while the second set only increases to 30/25 = 120%, so the first set has a greater increase. We cannot always apply the same procedure, so we must develop a feel for how to use math such as when to take differences and when to take ratios. We have to carefully read and understand the problems to know what to do. The ultimate goal of a physics class is to teach you how to do this.
Math in a physics class is a language. The physical situation (mile markers for example) provides the context that tells us how to proceed (subtracting the numbers for example). Learning to use this language requires practice and experience. Many people who are good at math are not good at applying math in a physics class. They only know the grammar of the language and not how to use the grammar to express ideas, and their progress in a physics class is limited. A math class and using math as a language are different skills, and both are important. After much work, maybe physics will finally make math seem useful to you and help you understand and enjoy it!
We need to become good at seeing applications of mathematical equations. I sometimes tell students that, if they truly understand the formulas and know how to apply them, they will get most of the answers correct on the tests, even the questions that do not require calculations! Let's look at this example question: ants with the greatest linear speed on a turntable are located (a) near the inside, (b) near the outside, or (c) anywhere because they have the same speed. We first need to recognize the relevance of the formula
v = r ω
which, in English without using shorthand, is
linear speed = distance from center of rotation × angular rotation.
This formula is relevant because we know things or want to know things about each of the numbers that appear in it. We must remember that the angular rotation (ω) is the same for both ants since they both go around the same number of times per second. The distance from the center of rotation (r) is the only other thing that linear speed depends on. Since r is larger on the outside, then linear speed (v) is also larger on the outside, so the answer is (b). In fact, if looking at a video of ants on a turntable, the ones on the outside are moving the fastest! But we could know this without having ever observed it because we used the formula that describes the motion. In fact, many things in physics are invisible such as fields and forces, so we have to learn how to rely on math alone. This process of thinking using the formula might seem tricky, but this type of thinking is used over and over again!
Not only do we need to be fluent enough with math to apply it, but, if we are advanced enough, we can also develop the formulas based on applications we see. Doing this will help us understand the formulas in the way needed to apply them. For our example, we want to know the formula for the fundamental frequency of a guitar string. The first step is to think about what the frequency depends on. We have experienced that the greater we turn up the tension (T), the larger/higher the frequency (f), so
f ∝ T
where ∝ means is proportional to. That is, a larger T gives a larger f. What else? Well, the strings with higher frequency are thinner with less mass per length (μ), which we can say in a mathematical language as
f ∝ 1 / μ
Since the μ is in the denominator, a larger μ gives a smaller f. That's all the proportionality symbol means! The above mathematical statement is the same thing as us saying that thinner wires have higher frequency. Lastly, when we put our fingers on the frets, the frequency changes because we change the length (L) of the strings. Higher frequencies come from shorter strings, which can be written mathematically as
f ∝ 1 / L
Putting it all together, we now have
f ∝ T / (μ L)
But there's a problem! Frequency is in hertz (which are 1/seconds), but tension is in newtons (which are kilograms meters / seconds2), mass per length is in kilograms / meters, and length is in meters, so, after some arithmetic, we get that T / (μ L) has units of meters / seconds2, which is not the same as the units of frequency! We must get the correct units! If we square L, we now have T / (μ L2) with units of 1/seconds2, which is closer. To get the desired units of 1/seconds, we must take a square root giving us the final answer of
f ∝ √( T / (μ L2) )
We did it! We can now use this to answer many questions. Maybe we want to know how doubling the length (L → 2) of a wire affects the frequency (f). So we use what we just learned
f ∝ √( 1 / L2 )
f ∝ 1 / L
f → 1 / 2
So the frequency is halved (that is, reduced by an octave), which is what any guitar player would tell you. That is, if it were 600 hertz, it will now be 300 hertz. Great! But it turns out that the correct formula has a 4 in it that we couldn't know using our method:
f = √( T / (4 μ L2) )
Notice that, without this final formula, we got all the proportional relationships correct which let us answer many questions. The exact formula is needed to get correct numbers, but the physics and the understanding that we need is in the proportional relationships that we have been talking about this whole time. If you followed all of this, then you now understand frequency from a string in a way that physics requires you to.
Often, we want a number as an answer. For example, we want to know precisely how high a roller coaster must start so that the roller coaster train makes it around a vertical loop without falling off. For simplicity, you are asked to solve the problem ignoring air resistance (which means that a real roller coaster would have to start higher than the answer we find). It's not enough for the train to be still be moving at the top of the loop; we need it to be moving fast enough to prevent it from falling out of the circular loop. You are told that the loop has a diameter of 20 meters and that you specifically want to know the vertical distance between the top of the loop and the starting height of the roller coaster. You are told nothing about the mass (which means that the mass does not affect the answer, which is good for the theme park owners since they do not have to redesign their roller coaster depending on the weight of the passengers). To do the problem, we apply the centripetal force formula to the top of the loop to calculate the required velocity needed to stay in the loop with gravity pulling down (keeping in mind that the radius in the formula is 10 meters, not the full 20-meter diameter). But why did we solve for velocity? Well, velocity was the only thing in the formula that we didn't know (except for the mass, which ended up mathematically canceling), so we could solve for it. We then remember, from a few chapters back, how to use energy to relate our velocity to the starting height, so we can solve for the starting height. This problem is a lengthy exercise in thinking about which formulas could be used and then doing the necessary algebra, but we can do it!
Problems like this require algebra to solve for the variables we need in addition to knowing how to figure out which formulas are helpful. Again, you will be using math differently than in a math class. You not only have to be able to remember or relearn the math, but you have to build new mathematical skills. Most problems will be word problems, so you must not only understand the math but how to apply it. There will be many symbols instead of simply x (v for velocity, m for mass, g for the acceleration of gravity, r for the radius of the circular path, and h for the height we are solving for), and some symbols such as m often mathematically cancel so we don't need to know them. You will have to decide which of these symbols are relevant and what to do with them. Physics will teach you the approach to solve complex problems, and practicing this method is important.
Like anything mathematical, physics is cumulative. This means that it keeps building on previous material, so you cannot forget the old material if you want to understand the new. You simply cannot fall behind. Vectors, algebra, trig, scientific notation, and units (and sometimes calculus) will be the first things you learn, and you can never forget this. From there, you will learn the math of forces, energy, waves, etc., which cannot be forgotten (until you are completely done with physics, which perhaps is never as long as you are physically alive!). We do not have to be an expert at one thing before moving on because we cannot really become an expert at anything until we try to build on it, but we need to understand the basics of each idea before moving on, which might mean postponing your physics class until you take a few more math classes.
Most students take their first couple physics courses with expectations that will never be met. Many students (you?) think that remembering the procedure and answers is the path to success and a good grade. It is not. We want you to understand the ideas and how to apply the math based on that understanding to create your own procedures. Understanding is more difficult than just learning a procedure, and applying our understanding is even more difficult. Solving lots of problems is an important part of developing your understanding and ability to apply your understanding. Physics classes take work, but the payoff is learning a deeper more careful way of thinking.
Many students walk into a physics class believing that the course will reinforce their common sense. Notice that I have not mentioned common sense anywhere here! Instead, I have used a mathematical way of thinking about the world. Why? Because the physical world obeys mathematical laws and certainly does not obey what our common sense tells us. Did you know that the force a bug exerts on your windshield when it splatters is always the same as the force the windshield exerts on the bug? Not being able to rely on common sense makes physics challenging, but it is also what makes physics so exciting. I love finding out when something that seems obviously one way is actually the opposite way, and I love then trying to understand where my common sense went wrong. Common sense is very important, but it has its limits and can be honed to be more correct.