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?
At my favorite coffee shop this week the barista asked me if I had any advice for helping her daughter with her math homework, and helping her daughter with her math facts. I sort of talked her ear off until she had to help the next person. Today it occurred to me that other people might have the same question, and also I had a lot more to say about that.
So many students come to me in 7th or 8th grade struggling because they don’t know their math facts well. If they don’t know their math facts (ie 7×7=49) then when they’re trying to do things like long division or multiplication or even algebra things like foiling, every time they run into something they need to multiply they have to stop and figure it out by counting on their fingers or adding up a whole list of 6’s, and it completely distracts them from the problem they’re working on. Lots of times it looks like they’re struggling with what they’re learning, but really it’s just that they haven’t memorized their math facts well enough.
However, I also meet students in high school who hate math (and therefore science and most of the really profitable careers) because someone tried to force them to learn their math facts in a way that made them feel really ashamed and frustrated and bitter.
So, it’s a tight rope, and it needs to be done, but it needs to be done carefully. Ideally, learning math facts can be fun.
A great place to start is with computer games like In Search of Spot or Number Munchers.
Flash cards can be great, if you can make it fun. (i.e. positive. Praise them for getting facts correct, don’t say anything if they get it incorrect. Say things like ‘whoa that was fast!’ when they answer a fact more quickly than usual.) 5-10 minutes each day is ideal, don’t make it too long or it stops being fun.
I like these flash cards.
You can also get infinitely many free math drills here.
Weird New Ways of Solving Problems
A lot of time students come in with a familiar type of problem, but their teacher is having them solve it in a way I’ve never seen before. Long division methods change all the time. So do factoring methods. This is really hard, because you want to help and explain things, and the way you’ve always done things is so familiar. But, learning multiple methods is really confusing for kids. It’s almost always better to try to teach them the method they’re learning in class. Because their teacher will build off that later.
This, of course, means that if you’re going to help them with their homework you have to first try to understand this new method yourself. The first thing you need to know is that it’s probably more similar than it looks at first, and if you think about it for a while and look at slides/notes from the teacher you can usually figure it out. There are also lots of online resources that will explain these new methods. My two favorites are Khan Academy and Purple Math.
Also, if you run into something you’re stuck on, post a comment here or email me and I can try to help.
In general, especially for younger students, the absolute most important thing is that it’s fun for them, and that they start to feel like math is something they can be good at. That idea will get them through many future math classes and will help them be successful eventually, no matter how hard it is.