Here is a super cool veritasium video about the Magnus Effect.
And, a confusing video about the physics of turning a bike.
When I first learned about significant figures, back in high school, they seemed totally arbitrary and annoying to me. Now, though, I know they’re incredibly important. The rules seem silly at first, but actually they make sense. So, if they seem dumb, please, please stay with me.
Ok, here we go. Let’s say I give you a ruler that only has inches marked on it. (So, none of the little tick marks in between, just 1 inch, 2 inches, etc.) Now, I ask you to measure the height of a cup I have. What answer might you give?
Really, you can only say for sure that it’s close to 4 than it is to 5 or 3. You might be able to guess that it’s halfway between 4 and 5, so you could say 4.5 pretty reasonably, but you definitely couldn’t say 4.56712. Because how would you know that?
In science, we’re measuring stuff all the time, and not only do we need to know what we’ve measured, but we also need to know how precisely we’ve measured it. Like, is the acceleration due to gravity really 9.8, or were we only able to measure it to one decimal place? (Actually, it’s different in different places, and even points in slightly different directions if you’re near a big mountain or a pile of gold.)
For example, I was out in the woods one day as a search and rescue volunteer, looking for crime scene evidence, and I was pacing out distances. If you’ve never paced out a distance, when most people take two steps they go a pretty constant distance, maybe 4 feet, so you can use pacing to measure how far you’ve gone. Well, I was out in the woods with the other members ESAR, and we were pacing along looking for evidence, and I found a piece of trash. I reported it to my teach lead and he asked me how far I was from the place we started. I have a 4.5 ft pace, and had gone 5 and a half paces, so I did the math in my head and then, without thinking, said something like 24.75 ft. The team lead was laughed and was like “oh yeah? wow.” Because, here I am being incredibly imprecise, wading through bushes, and I’m telling him that the piece of trash I’ve found is exactly 24 ft and 9 inches away. But, that’s what my math told me.
So, the problem is, when we do math with imprecise numbers, we sometimes get things that look more precise than they are.
Like, if I take 1 and divide it by 3. I get 0.333333333333333333333333333333333333333…
The first two numbers (1 and 3) are integers, I have no idea if they’re 1.4, 1.6, 3.1, or what, but when I divide them, I get something that looks like I’ve measured it out to the infinite decimal place. Silly.
So, we have to watch out for situations where math spits out numbers that don’t mean anything.
After that first 3, the other 3’s don’t mean anything at all. 1 divided by 3 might be 0.32 or 0.34447. Because, I don’t know whether the 1 was really a 1 or whether it was actually 1.1 but I didn’t have good enough tools to measure that carefully.
Does this make sense? The key here is that 4 is different from 4.000000000 because if I just say 4, then that might actually be 4.1 or 3.9 or 4.12321111.
Because really, in the real world we don’t often have things that are precise, we only have measurements (unless I say something like “I have 2 dogs.” That’s probably exact. But, I’m 5′ 4″ tall, but I have no idea how many nanometers that is.
I hope that wasn’t boring. I realized partway through that maybe no one cares about sig figs, except that you’re made to calculate them. If you just wanted to learn how to calculate them, you can look through this short powerpoint I made that summarizes all the rules and shows some examples. Feel free to steal it and use it however you like (I mean, within reason):
In physics, math suddenly becomes something useful. It’s now a tool that will tell you how quickly something gets somewhere, or how high it will go if you shoot it out of a cannon. This is awesome, but it can also be hard, because if your math isn’t solid then when you’re in the middle of a tough physics problem you might get distracted as you try to figure out how to solve for the root of a quadratic. By the time you figure it out you’ve forgotten why you were solving for it. (Also, this happens to everyone, so it’s not bad, we just want to minimize it as much as possible.) Hopefully, the math has been practiced enough that it’s somewhat second nature, so you can focus on the reasoning behind the math you’re doing.
Luckily, there isn’t too much math you need to know, it’s not too fancy. Here’s a quick quiz you can take to see if you’re ready:
Work through the problems, and if there are some you can’t get that’s no problem, it just means there is some stuff you might want to review before you start your class. Here is the answer key so you can check your answers:
Next, for each question you missed, here’s a homework assignment for you:
Over the next week I’ll be writing some posts about good things to practice and know before you start taking a physics class. This includes making sure your math is solid, and getting a head start on the basics before school starts. Each day I’ll post a short practice assignment. You can post answers here in the comments, or email them to me with questions.
Here’s what I’ll be covering:
1. Introduction (ie. this post 🙂 )
2. Are your math skills ready for physics?
3. What are significant digits and why are they significant?
4. Why does everybody think physics is hard? (An introduction to problem solving.)
5. SI Units and the Metric System
6. When should you start studying for the AP test, and what can you do at the beginning of the year to make that easier?
7. A head start on 1-dimensional kinematics (usually the first thing you learn).
In each of these, I’ll be giving links to resources (using them will be part of the homework) so that when school starts you’ll already have a ton of places to go to get extra information (plus you can always ask me questions here or on my facebook page.)
The photoelectric effect is the basis for solar panels. It’s really famous because it’s also some of the first evidence we had that light was a particle (this of course became extremely confusing when the double slit experiment gave us evidence that light was a wave, but that’s for another time.)
First, remember that materials are made of atoms, which are made of protons and neutrons and orbited by electrons:
Different materials hold their electrons more or less tightly. Metals happen to hold their electrons pretty loosely, like kids in a neighborhood where all the kids just run around wherever they want:
These kids wander around wherever they feel like it, but they generally stay inside the metal (in this case copper). However, they can get knocked out of the metal by light (photons).
This is kind of like a game of red rover (I hated this game so much when I was a kid.) Let’s imagine that the kids line up at the edge of their town, and they play red rover with the neighboring kids, who just happen to be photons (light). The kids are there just hanging out in the metal, and the photon comes and tries to knock them out.
If the photon is weak enough, nothing happens.
In fact, if the photons are too weak, it doesn’t matter how many of them hit the metal, no electrons are knocked off.
In photon-terms, a weak photon is one without a lot of energy. The energy of a photon is its frequency, which is also its color. For example, red is a lower frequency than blue. And, infra-red is lower frequency than red. Ultra-violet is higher frequency that violet. Radio waves are lower frequency than infra-red, and x-rays are higher frequency than ultra-violet. (All of these are just frequencies of light that we can’t see.)
So, if the light hitting the electrons gets more energy, let’s say its violet now (like violent!)
Now, the more photons that hit the metal, the more electrons will be kicked off.
Different metals hold their electrons more or less tightly, so different metals require different energies of photons before electrons will get kicked off. This is called the “work function” of the metal, and it’s often denoted with the Greek letter phi:
Sometimes the incoming photon has more energy than it needs to kick off an electron. If that’s the case, then the leftover energy becomes kinetic energy of the electron. (i.e. the stronger the kid from the neighboring town is, the faster you’ll be going when he tosses you out of the line of electrons.) Here’s the fancy equation for that, if you’re interested:
But, we said earlier that the energy of a photon depends on its frequency. It turns out we can calculate the energy of the photon by taking its frequency (in Hertz) and multiplying it by planck’s constant (6.6 x 10-34). This gives us the energy in Joules. So, another way to write the above equation is this:
For example, the work function of copper is 4.7 electron volts, which is 7.53e-19 Joules. This works out to a frequency of 1.14e15 Hz, which is a wavelength of 260 nanometers of light. This is higher than the visible spectrum, it’s just into the ultraviolet range, so you need ultraviolet light to knock electrons off of copper (or gamma rays 🙂
The really cool thing is that this is evidence that light is a particle. Because, if you hook up some wires to the piece of copper, and you hook those wires up to a detector that makes a sound every time there’s some current (ie. every time an electron gets kicked off- current is just moving electrons) it would make a sound like rain on a tin roof.
Thoughts? Questions? Comments? Concerns?
Every year I have a couple of students who are taking calculus-based physics at the same time as calculus. Traditionally, you take calculus first, and then you take physics with calculus. The calculus used in these introductory physics classes is usually simple enough that it’s not impossible to take them both at the same time, but it’s definitely a challenge. I wouldn’t recommend it if you can avoid it, but if you can’t there are some things you can do over the summer that will make next year go much more smoothly:
Lastly, once school starts, don’t hesitate to ask for help, either from your teacher, a tutor, or me.
(And a girl. Not that I was starting out as a girl. I had been a girl my whole life, but I was just starting out being a girl physics major.)
It’s not supposed to be easy
I had this idea that if you were smart things were easy. You either knew something or you didn’t. You either got it or you didn’t. My first few physics classes reinforced this. I had already taken high school physics, so the intro classes weren’t so bad. Then I got to my first real physics class, Mathematical Physics, taught by an old Russian professor. I assumed it would be easy. Then the first midterm came. Three problems. I had no idea what to do for any of them. I had no idea how they even related to anything we’d done in the class so far.
Suddenly I felt like I was out of my depth. I went to go talk to the professor. I had skipped class a couple of times and I think he knew that because he wasn’t that keen on helping me. What he said was “Maybe you’re just not going to get it.” After that I panicked. I felt deep down that I just wasn’t smart enough to study physics, even though it was something I had loved since I was a kid. I thought maybe I had just hit the limit of my natural intelligence.
I hadn’t. I had simply hit the limit of the place where I had learned things before. In my experience, 90% of the things that look like talent are actually practice. When you see someone with a perfect golf swing or a genius math brain, you’re only seeing the result, you’re not seeing the great coach they had, the hours they spent practicing, or all the times when they were a little girl sitting in the passenger seat of their dad’s truck talking about the differences between even and odd numbers.
I didn’t know this. I thought I had just hit the ceiling of what I was capable of. I wish I could go back and tell myself that there is no ceiling. There is no capable, not capable, or highly capable.
I work with a lot of students, and I can see some of them falling into this trap. They encounter a difficult problem and they start thinking they’re dumb. They’re wrong.
It’s hard because it is made of ambiguous problems. A lot like life. You learn some tools, some equations, some concepts, and then they set you loose on problems. At first, you will have no idea what to do. This is not bad. This is a feature of real problems. (See my post on problem solving for ways of tackling these things.)
In order to be successful in physics, you have to make the mind shift from “I’m not smart enough” to “this is hard.”
It’s okay. It’s okay if it’s hard. It’s okay if you look at a problem and have no idea what to do. It’s not supposed to be easy.
Unlike in most of your previous schooling, you’re not just learning one thing at a time and being asked to do something repetitively. You’re being asked to do something crazy hard, which is, stumble around until the pieces fit together. Or maybe they won’t, and you’ll ask for help. But not from someone who will tell you that maybe you won’t ever understand. Because if you keep trying you will.
Some people pretend they know more than they do
Another thing that left me feeling stupid was how much my classmates seemed to know. For some reason, the reputation that physics has for being difficult brings out some pretty intense egos. Be careful not to assume, just because someone is talking about something fancy-sounding that you’ve never heard of, that they know more than you do.
In this same first physics class, I met several people. We studied together. They talked a lot about things in ways that made me feel like they knew what they were talking about. Again, I felt like an idiot who didn’t belong.
Then we got our grades back. I did the worst I had ever done in a class, but it was still a 3.1. Those same people who I thought knew everything got 2.5’s or worse.
Never think someone knows more than you do just because they’re speaking authoritatively. They might just be covering insecurity with arrogance.
Ask for help
The next quarter I started my next physics class. I was completely panicking. Sure I would fail. Sure I wasn’t good enough. I had to take mathematical physics II.
Then, randomly, I did the best two things I’ve ever done for myself in the context of my physics degree.
1. I introduced myself to some other girls in that class.
2. I went to the professor after class, told him I was worried I was unprepared and that I wouldn’t do well.
The first thing immediately diminished my anxiety. Suddenly, I had people to commiserate with, to study with, to hang out with and laugh with. It was amazing and great and so much fun and the opposite of isolating. They told me later that they were so glad I had introduced myself, because they were feeling terrible and scared, too. It’s hard to do that, because sometimes other people look arrogant when they’re scared. Or they look closed and unfriendly when they’re scared. But, if you just introduce yourself it can be amazing. Having allies is probably the most important thing when you’re trying to do something hard.
The second thing was really good for my grade. Luckily, this professor was an extremely, extremely, nice guy. He laughed a little when he heard what my previous grade was. He told me it would be ok. He gave me a book. This book was basically a more detailed version of the textbook we’d used the previous quarter. It was waaaay better. (Incidentally, I do not recommend you use any textbook by Mary Boas. She leaves out waaay too many steps when she solves math equations.)
In that class I got a 3.9, and I made some awesome friends.
At some point in your life you will (hopefully) reach a place that will feel like the limit of your ability. It is not. It is the end of the mapped territory. It is the end of the things you already know how to do. It is a place where you will need to make a large step up in order to continue. Maybe physics will be this for you, maybe it won’t. Maybe it will be easy for you. If so, that’s great, but I hope you eventually find yourself in that place where you think you might just not be good enough.
For learning the basics, hands on activities can be great.
Here is one of the coolest things I’ve ever seen for teaching electric circuits:
Of course, the Phet Simulations are also great for inciting interest and sparking curiosity.
Now, for some resources for the more advanced:
Electrodynamics (More about E and B fields than Circuits, but great.)
The Art of Electronics (Basically the bible for electric circuits.)
Enjoy! And, if you’re somewhere in the middle, or would like a tutorial on problem solving, and the basic laws of circuits like kirchhoff’s and ohm’s laws, you can check out my book. It’s been either #1 or #2 in electricity principles, and electromagnetism, and AP test guides for months now. If you do check it out I would very much appreciate an honest review (whether you like it or hate it.)
Here’s a fun video of some physics-based tricks. (Called 10 Amazing Bets you will Always Win, by Quirkology)
A couple of them (moving liquid from a bottle to a glass and separating two glasses) use air pressure creatively.
Physics can also be used to make some super-cool art. Have you seen the Cloud Gate in Chicago? On the outside it’s an amazing panorama of the city skyline, on the inside it’s a crazy multidimensional universe. That’s the magic of curved mirrors.
Vsauce on YouTube has a really interesting video about curved mirrors.
Externalize your Thinking
In a physics problem, there is usually more information than you can comfortably keep in your head at once. You need a good diagram, a good list, a good way of keeping track of information. The diagram/list/table/picture should represent what you know up until this point.
If you’re having trouble solving a problem, ask yourself this question: “Is everything I know represented in my diagram?”
It’s important to put even seemingly self-evident stuff into your diagram, because sometimes what’s holding you back is that you haven’t seen all the implications of what you know is true. Put the obvious stuff in your diagrams because then you’ll see the less-obvious implications.
Problem solving is a process of taking one small step at a time. With a really good problem, you probably won’t be able to see the solution, or even how to get to the solution, until you are nearly there. Instead, you just keep asking yourself “what can I try now?”
You just keep adding things to your diagram, trying things out, getting more information, until patterns start to emerge.
Solve a Simpler Problem
A good example of this is what xkcd calls “The Hardest Logic Puzzle in the World.” I won’t summarize it here- you can go read it and come back.
Back? Ok. This problem is really really hard. First, do step one and figure out a good way to draw the problem, maybe just with L’s for blue-eyed people and R’s for Brown-eyed people.
Then, step two, simplify the problem. Thinking about what a hundred people know (about themselves, and about every other person) is a bit of a stretch. Instead, try this problem with just one islander. Seriously. Just one.
If the guru says I see at least one blue eyed person, then that one guy knows he has blue eyes, and he leaves. Yay! That was easier. At this point, we’ve basically solved the problem. All we have to do is figure out what happens when there are more people.
Try it with two islanders. Then three. Then four. Then extrapolate to 50 once you see the pattern. Easy! Jusy kidding, it’s not easy. But figuring out a simpler problem to solve makes it waaay easier. I won’t deprive you of the logical joy of this problem by telling you the answer.
You could probably simulate being tutored by me pretty well just by asking yourself “what is one thing I can figure out at this point?”
I love asking students this, because it shows them that they already have all the knowledge they need to solve a problem, and the capability to do so, they just need the willingness to try. (Some students of course need some explanations, but you’d be surprised how often students who come to me failing physics and feeling completely confused actually do just great after a couple of meetings and all I said the whole time was “what’s one thing you could figure out?” I exaggerate a little, but not all that much.)
Physics is hard not because there is a lot you have to learn, but because applying the small amount of learned information takes courage. It takes the courage to try something and see what happens and trust yourself that if you try enough things you’ll eventually find your way to the answer.
The problems are often so complex that it’s impossible, until you become an expert, to map out the path to the solution. Instead, you have to just figure out one thing, then another, then another, until a way to find what you’re looking for jumps out at you.
Can you solve for the time? Can you find the total force? Can you find the mass?
(Mass by the way, is often something that drops out. If you find yourself confused because “they don’t give you the mass” then just use m for mass and be happily surprised when it magically cancels out a few steps later.)
Just do whatever you can do, write it all down, be as organized as possible, and you will find the answer… eventually… most of the time.
Use Difference Reduction
One tried and true method for solving problems in any context, including physics, is called difference reduction. In this method, you simply look at what you have and look at what you want and ask yourself, what’s one small change that will make what I have look more like what I want?
I’m starting to feel a little bit like a life coach now, but hey, real life problems are problems too.
Sometimes to move forward you must go back
Have you heard the riddle about the farmer crossing the river with the fox, the goose, and the bag of beans? There are tons of versions of this riddle, but here’s one:
A farmer has a fox, a goose, and a bag of beans and he wants to cross a river. His boat will only hold one item at a time in addition to the farmer. If he leaves the fox with the goose the fox will eat the goose, and if he leaves the goose with the beans the goose will eat the beans. How can he do this?
I’m going to give you a partial spoiler here in about two lines, so if you want to try this on your own now is a good time.
The key to answering this problem is that you have to bring the goose back to the wrong side of the river at one point. The reason this riddle is hard is that it involves moving away from your goal state. You already have the goose on the right side, why would you bring it back?
But, in many contexts, taking a step back is the best way forward. Maybe you move home after college. This feels like the worst step back in the world, but you save money and buy a house and a car much sooner than you would have otherwise. Maybe you’ve been working on a physics problem for three hours and you feel like you’re almost there but you just can’t get the last little bit. This might mean that you’ve actually gone down the wrong path and if you keep on doing exactly what you’ve been doing you’ll never get there and what you really need to do is start over. Before you start over, though, you should:
Take a Break
There is this thing called neuronal activation. Certain neurons in your brain light up, and they have their own associations with them, which makes you more likely to think of things in one way rather than in another. Sometimes if you’re having trouble with a problem it’s because you’re thinking of it in one specific way that isn’t letting you see the route to the answer.
If you take a break you can come back fresh, with different neural pathways activated, which means you might just see the answer right away. (There is some disagreement about whether this is because of the “fresh perspective” or because your subconscious was working on the problem without you. Either way it works.)