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):
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.)
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.)
There’s something kind of awesome that happens when you realize that things you’ve learned in the classroom might actually have real-world applications. Like, when you go to a foreign country and can sort of communicate with the locals because you took Spanish 101.
I was watching Shark Tank recently, and these guys came on with this thing they invented. I don’t know anything for sure about how it works, but I could make some educated guesses, and I think that looking at their invention is a really interesting way to introduce thermodynamics.
I couldn’t find the clip of them on shark tank, but here’s their kickstarter campaign video, which according to youtube raised $306k in 37 days!
Anyway, it’s a pretty genius invention and I kind of want one. They’re these big metal things shaped like coffee beans and you put them in your coffee. They immediately cool it to the perfect (still hot) temperature for drinking, and then keep your coffee at that temperature for several hours.
Being a coffee drinker, extreme appreciater, addict, myself, I was like “wow, that’s genius.”
Here’s my guess at how it works: The inside is some metal that has its melting point at a good temperature for coffee. When you pour coffee over it, the excess heat goes into melting the metal. (This phase change absorbs energy, (called Joules, hence the name Coffee Joulies.)
Then, as the coffee starts to be cooled by the outside air, the metal inside starts to phase change back, solidifying and transferring heat back to your coffee. (Phase changes can absorb tons of energy, and they will stay at a constant temperature while they do so.)
We could even guess at what metal or material might be inside. It would be something with a melting point around the perfect coffee drinking temperature.
The National Coffee Association says this temperature is 180-185 Fahrenheit, which is 82-85 Celsius. But, drinks.seriouseats.com says 110-120 is ideal, which is 43-49 C. Coffee Joulies says they keep your coffee at exactly 140, or 60 C.
Then we can check the melting points on the periodic table. Sodium (Na) is 98 C and Potassium (K) is 63 C, Rubidium is 39 C. Potassium is basically right on, so it could conceivably be made of that.
I just checked their website for more clues. They call it PCM (for potassium? Maybe it’s a mixture of potassium and something else.) Other than that they don’t say what it is, only that it’s natural, edible, and already found in food (potassium again?)
Anyway, fun mystery, and an awesome invention! And, a really awesome example of how you can use physics to invent useful things.
One of my favorite things about physics is thinking about real world applications. I think the king of this was probably the physicist Richard Feynman. (From the awesome book Surely You’re Joking, Mr. Feynman.)
You can find him on youtube- there are videos of him talking about these weird puzzles, or just interesting things like how rubber bands work. My favorite: Why does a mirror swap left and right but not up and down?
Another favorite: how fire works.