# Preparing to Take Physics: 3. What are significant figures and why are they significant?

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?

4 inches?

4.5 inches?

4.56712 inches?

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):

Significant Figures

# What to do to prepare for taking physics: 1. Introduction

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 🙂 )

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 like Red Rover (except I like it a lot better)

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.