Frame of Reference & Coordinate System

Did I move? (preview)

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Relative to the car, we're not moving. If I choose a billboard outside as the frame, we are moving. Same two people! With a new frame, your motion differs. Whether you move depends not on you, but on the frame you choose.

In this lesson: why concepts matter

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This video is a Basics lesson. We'll cover a very important concept: the reference frame. Many students dislike concepts. Learning one is annoying. Why learn them? In physics, especially with concepts, the point is to solve problems. Every concept is a solution to a problem, and behind it is something elegant. Take the particle from Lesson 1. It lets us be lazy: we drop the details that don't matter, and keep only the core, which solves the problem and saves us effort. It's useful. Today's reference frame is just as important for studying kinematics. Without reference frames, no kinematics problem can be studied, because you can't tell if an object is moving at all. Now, you may ask: Can't I just see whether it's moving? One look with my own eyes!

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Isn't that obvious? If you only think about it simply, at first it does seem obvious. But think it over: is it really so? Let me give you two examples.

Two stories: the subway and the milk tea shop

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A couple of years ago, the year the pandemic broke out, a friend of mine in Shanghai invited me and Qi Xiaoyu, the one in the black mask, to come visit Shanghai. We flew from Beijing to Shanghai, then took the subway from the airport We agreed to meet at a subway station. "Buddy, I'm close! Where are you?" "Almost at this stop?" "Yes." "Which car?" "Car 15." "Okay, stay on the train. Don't move. I'm here too. I'll come find you. Then we eat. Don't get off. Don't move." I heard that and stayed perfectly still. Very obedient. I even fell asleep and got caught on camera. How embarrassing! I was so tired. So, following his order, I wasn't moving. He meant don't move. I didn't. But now, did I move or not? I really didn't move. I sat in the car, but the train was racing forward, right?

2:49
So riding the train as it raced ahead, I was moving, and very fast. So did I move or not? Not so obvious anymore, is it? Did I move? Hard to say. A bit complex: the train moves too. Let me use a simpler one. After Shanghai, we flew back to Beijing. At the airport, you go through security first. Then there's a boarding gate, and a long walk to get there. On both sides are little shops, drink stores, duty-free shops, and so on, like a busy little street. After security, my stomach hurt, so I went to the toilet. When I came out, Qi Xiaoyu was gone. I asked, "Where are you?" She said she was in a milk tea shop. Qi Xiaoyu can't walk past milk tea. She'd drink eight cups a day. So I said, "Stay in the shop. Don't move. I'll come find you." She said okay.

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At the shop, she looked like this. I took a photo. See, she stayed in the shop, not moving. Now, did she move or not? You'd surely say, "Of course not!" But think about it again. She's on Earth, right? Is Earth rotating? It also orbits the Sun. So is Earth moving? Yes, it is. Earth moves. Did she move with it? She's moving too! So did she move? My head will explode! Did she move or not? This is turning into philosophy. So, without a reference frame, we can't define whether an object is moving. That is the frame's value. Now let's look at the concept itself.

Definition: what is a reference frame?

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The reference frame. It's also called "reference system." "Reference" means to refer to. A frame is an object or system chosen to study motion and assumed at rest. Its physical meaning: to decide, in the problem you study, whether an object is moving or at rest. Take the first example. In that situation, he told me not to move on the subway. Though neither of us said it, we both took the train car as the frame. So I wasn't moving. Ask if I moved? I can't say. But if you ask, "Relative to the car, did you move?" I'll say, "I didn't." A reference frame was assumed here. In the second example, my frame was Earth, or rather the ground. So Qi Xiaoyu didn't move, and I found her.

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That's the practical value of a frame. Only after defining a reference frame can I say if an object moves or not. So the motion and rest we mean are not absolute motion and rest. Whenever we say an object moves, we mean it moves relative to a frame. But sometimes the frame is so obvious, like the ground, that we just leave it out. But it's actually still there. We should say this clearly.

Three key points about reference frames

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A reference frame has three key points. First, the frame itself is also moving. We know everything is always moving, and rest is relative. The frame moves too. With the car as the frame: the car itself moves relative to the ground. So a frame isn't at rest. It's just that in my problem, I treat it as at rest, an assumed rest. Remember: the frame itself is also moving. Second, a frame can be chosen freely. But in high school, the default frame is the ground. Look at any high school problem: a ball on an incline, a ball moving straight on the ground. Sentence one never names a frame. That would be too wordy. But the default is the ground. If a problem names no frame, it uses the ground. Third, with different frames, an object's state of motion differs.

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If I choose the train car as the frame, the two of us are not moving. If I choose a billboard outside as the frame, we are moving. Same two people! With a new frame, your motion differs. Whether you move depends not on you, but on the frame you choose. Those are the three key points. In middle school, you learned something called a "reference object." Many ask: "How is a reference object different from a reference frame?" Let me tell you: they're the same. No difference at all. In high school, we just gave it a fancier name. Reference object = reference frame. Enough about its meaning. In short, remember: The core of a reference frame is to help us decide whether an object is moving or still. This is very important.

The trick: whoever isn't moving is the frame (Example 1)

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In Chinese exams, how is a frame tested? The most common type is this one: They give you a line of poetry or a sentence, and then ask, which object is the reference frame? Many students have no idea. Not hard, yet they can't do it. Let me teach you how to do these. Again, the right way to study physics: When doing problems, you must get out a notebook, reminder or mistake book, or good-problems book. Note down the problems you found good. How? Write down the problem, and what you drew from it: useful insights and methods. For this type of problem, one sentence can nail the core idea. That sentence is your takeaway. I wrote it below. For "what is the frame" problems, it's simple: just look at what isn't moving.

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Whatever isn't moving is the frame. For example: "The bridge flows, the water doesn't flow." Which frame does "the bridge flows" use? Think about it. In that sentence, first find the verbs. The verbs are "flows" and "doesn't flow." Who is each verb's subject? First "flows": the subject is the bridge. "Doesn't flow": subject is water. So the bridge is moving, and the water is not. Whatever isn't moving is the frame. Here that's the water, so the answer is A. That simple. Just see who isn't moving. It's reading comprehension and grammar: find the verb and its subject. Whatever moves is not the frame; whatever doesn't move is. That's this problem. Let's do another.

Example 2: sun, moon, raft and mountains

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Which statements are correct? A: "The Sun rises in the east" uses Earth as the frame. What's the subject here? What's the verb? "Rises." And its subject? The Sun. "In the east" just modifies it. The Sun rises, in the east. So is the Sun moving? Yes, because "rises" is the verb. It's like watching a sunrise: the Sun appears from behind a mountain and rises up. The mountain isn't moving, it's a mountain on Earth. So the frame is Earth: Earth is still, the Sun moves. So A is correct. B: "The moon glides through the clouds." The verb is "glides through." And its subject? The moon. The moon glides through the clouds, so the clouds are the frame.

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It's like seeing a big cloud at night, and the moon slowly comes out of it. The cloud isn't moving, so it's the frame. Or reverse it: if one thing moves, the other is the frame. In B, the moon moves, and there are only two: moon and clouds. Clouds are the frame. C: "The little raft floats downriver, the green mountains walk on both banks" use the same frame, right? Look at the first line: the verb is "floats," the subject is the raft. The raft moves, so what's still is the bank. So line one uses the bank as the frame. Second line: the verb is "walk," and the subject is the mountains. The mountains move, relative to whom? Surely the people on the raft. To them, the mountains are moving. It's like riding a bus and watching poles move.

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So line two uses the people as the frame. The frames differ, so C is wrong. D: the frame must be an object at rest relative to the ground. Must it? We just said a frame can be chosen freely, so it doesn't have to be at rest relative to the ground. So D is wrong too. The answer is A and B. That's it for these two problems. Given a sentence or a line of poetry, you now know how to find the frame. That should be no problem.

Heliocentric vs geocentric: choose the simpler frame

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Next: a frame can be chosen freely. Here's something really interesting: The geocentric and heliocentric models. At first, people generally believed Earth was the center of the universe. Later, Copernicus and Bruno insisted that the Sun is the center. Most people later accepted heliocentrism. But by today's knowledge, both are wrong, because neither Earth nor the Sun is the center of the universe. So why did people accept heliocentrism? Because it's simpler. What's the difference between them? The only difference is the frame. Heliocentrism: the Sun is the frame. The other planets of the solar system move like this: all concentric circles. Simple. But with Earth as the frame, their motion gets very complicated. So people accepted heliocentrism because, with the Sun as the frame, things are simpler.

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So when choosing a frame, follow one principle: simplify the problem. Do whatever is simpler. Don't make things hard for yourself. Different frames can make a problem differ greatly in complexity. Pick the simpler one.

Coordinate systems: a frame with a ruler

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Once we have a reference frame, we can tell if an object is moving. But sometimes that isn't enough. Take the next example. I don't know about you, but as a kid I watched the Chinese cartoon Big Head Son and Small Head Dad. I always wondered: isn't Big Head's father Small Head Dad? Why did the author draw Big Head looking so much like Uncle Wang? No idea, and I dare not think too hard. Sometimes Uncle Wang can't find Big Head and asks, "Big Head, where are you?" Big Head has a few ways to answer. Either "I'm at school," or "I'm two meters behind you," or "I messaged you. Check it," or "I'm at 116° E, 40° N." See what he's doing? He's describing his own position. In some problems, locating an object is really important. We need to know its position, and a plain reference frame can't do that.

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What now? It's easy. We just upgrade the reference frame into a coordinate system. Problem solved. It's called a "coordinate system." "Coordinate" means position numbers. How do we picture it? Think of two big rulers stuck together at a right angle. A coordinate system is a frame too. It has all the properties of a frame: an object we declare to be still. If an object is still relative to it, it counts as still. If it moves relative to it, it's moving. That's all. But it has something a plain frame lacks: it easily gives an object's coordinates. Take point C. Its coordinates? You know from math. Math uses coordinates a lot. It's (3, 2). But note one big difference between physics coordinates and math. In physics, coordinates must have units, because physics is a natural science whose quantities have real meaning.

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So include units. For example, the coordinates are (3 m, 2 m). Remember the coordinate system: two big rulers, bang, stuck together. Sometimes it's two-dimensional. If you've read The Three-Body Problem (a Chinese sci-fi novel), you know different beings live in different dimensions. Some live in 3D space, like us humans. You can walk forward, back, left, right, jump up, squat down. That's 3D space. To describe where you are, you need a three-dimensional system. Roughly like this: X, Y, and Z. A 2D being only moves on paper, on a plane. Then a 2D system is enough. This diagram is a 2D coordinate system. An even simpler being is one-dimensional. What's a 1D being?

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Think of an ant in a tube. It can only go left or right, along one straight line. It can turn back, but only along one line. That's a 1D being. To describe a 1D being's position, we only need a simple 1D system.

1D coordinates and Example 3

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Why stress 1D systems? Because we'll study straight-line motion, and it uses them all the time. A 1D system is a straight line with an arrow: an axis. Then you pick an origin, and you can give coordinates by position. In 1D, a point's coordinate is one number. For example, A's coordinate is 3 m and B's is −2 m. Note: 1D needs just one number, a 2D system two, a 3D system three. A 1D system needs just one. Now a problem on 1D coordinate systems. A ball falls from 5 m above the ground, bounces off the ground, and is caught 1 m above the ground. The origin is 2 m directly below the release point. Downward is the positive direction. Find the coordinates of the release point, landing point and catch point. Let's see. This is the ground. The ball falls from 5 m above it.

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Counting one, two, three, four, five, this point is the release point. Then it bounces off the ground and is caught 1 m above it. So this point is the catch point. The catch point. The origin is 2 m below the release point. So this is point O. Downward is positive. Now we have an origin and a direction, so we can write each point's coordinate. Should this point be −1? Positive points down, With O as origin, everything above it is negative. This is −2, this is 1, this is 2. Now, the release point. It should be −2 m. Second, landing point. It's this point, right? One, two, three: so it's 3 m. Third, the catch point is this one: one, two, so 2 m.

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So the answers are −2 m, 3 m and 2 m. Choose A. I won't go into more detail. Just one thing to stress: Coordinates must have units, whether 1D, 2D or 3D. Units are a must. Very important.

Summary

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This page is the key point of the lesson. Take a screenshot. Reference frame: an object or system chosen to study motion, assumed still Note the three key points. For frame problems, to find the frame, look at what isn't moving. Whatever isn't moving is the frame. What's a coordinate system? An upgraded reference frame: a frame with a scale. That simple. It's also a frame. Finally, physics coordinates need units. That's very important.