Basic Physics 3A. Lecture 10.

So, where are we and where are we going? We spent the first three weeks learning how to describe motion. And we realized there are basically three parts to your motion as you move around the world. The first part is your position, where you are. And your position can change, so we like to really focus on your displacement, which is your change in position. So, these are really one issue. And for the special case of constant acceleration, we actually had a way to compute your displacement given your initial velocity and your acceleration. And so this was a very key part of the first two weeks. And this equation highlights for us that our displacement's a vector, our velocity's a vector, and our acceleration is a vector because direction is very important when it comes to where you move to. If I'm here and I move 10 meters this way, it's very different than being here and moving 10 meters this way. I end up in two different places. So, direction matters. That was piece one of describing our motion. Piece two was our velocity, which is our displacement in a time interval. Intuitively, what our velocity tells us is where and how fast we are moving. The where being the direction. I'm moving that way, I'm moving that way, I'm moving that way. My velocity tells me the direction I'm moving in and how fast is my speed, how fast am I doing it, how quickly is my position changing. And the instantaneous velocity was a derivative of position with respect to time. Once again, we had a special case of constant acceleration, in which case our final velocity was related to our initial velocity by adding a times t. And again, one of our main equations that we spent a lot of time on. So, those were the first two pieces of motion. The final piece of motion was our acceleration itself, which was our change in velocity in a time interval. And we worried about cases of constant acceleration, but we also could have acceleration that was changing in time, and the definition of the instantaneous acceleration was the derivative of velocity with respect to time. And so, if we had some velocity that had a complicated function of time, and we wanted to know the acceleration as a function of time, we would take the derivative and things like that. And those were our four pieces of motion. Now, the nice thing about this is, if you go back and look over the last three weeks, the reality is, with just a little bit of extra thought and care, your intuition about describing motion is pretty good. If I were to ask you, again, I said this being in class, if I was to ask you suddenly driving down the freeway, I saw a sign that our next exit is four miles away, and you're doing 60 miles an hour, with a calculator, you could figure out how long it would take to get to the exit. And you, most people, without even thinking about it, are doing these sort of calculations all the time. Do I need to go, I'm running a little late, do I need to go faster? How much faster do I need to go? And you're also doing it in the context of, how much faster can I get away with in not getting a speeding ticket? So you're doing these complex sort of calculations, and you have an idea of what the acceleration is, because you can kind of feel it when your car accelerates. When you turn a corner, your direction changes, and you feel that. What we're gonna do next is describe the causes of motion. And that is something where our intuition is not as good. And so you're really gonna have to pay close attention. So the way I do this is today and Wednesday is a lot of conceptual clicker questions, a lot of, we'll do some simple, simple demos today, but I've got a bunch of demos for Wednesday to try and make it clear to you where your intuition is good and where it's gonna fail you and why. So you can be better prepared for doing the problems associated with the causes of motion. So we just start by asking the question, what causes motion? And in particular, are causes of motion related to position, velocity, or acceleration? And one of the interesting things that we'll look at is we have position, x, we have velocity, which is dx dt, and we have acceleration, which is dv dt, which equals the second derivative of x with respect to time, because we can plug into there. And the question is, why did I stop at acceleration? Why don't I have a fourth thing? Why don't I care about the change in acceleration in time and come up with a new thing with a new name? The denomination, right? Because I'm still looking to get named, something named after me. Haven't achieved that yet. I'm going for the Denon effect. I think that's got a nice ring to it. So this is where we're going. And the problem we have, just so you're aware of it right up front, is you live in a world of friction, air resistance, and gravity, and these are invisible forces. And you don't see them, you don't think about them, and that tends to mess with your intuition. And that's kind of what we're gonna try and break over the next two days. So there won't be a whole lot of example problems. There'll be some, but that's where we're gonna focus. So we're gonna start right away with a clicker question. So if I take a book and I slide it on a table, it clearly stops, right? I take the book, I push the book, the book stops. And our question is, we're asking about now the causes of that motion, right? I can describe it as an initial velocity, a final velocity, it moves some distance, and it had an acceleration, the opposite direction of the velocity, and that's why it slowed down. But that's not really a cause. That's just still a description of all the different parts. So it didn't stop because it's just really tired and it wants to be at rest. If you're not pushing on something, it can't keep moving. It actually uses up all its energy. So when you push it, when you give it that push, you give it a bunch of energy, and then it uses up that energy to move, and so it eventually has to stop. Or more than one of the above or none of the above. I guess that should be object wants to be at rest. Don't not pick it just because the English is bad. Not saying you should pick it, but I'm a physics professor. We spell fun, P-H-U-N, so you can see. Few people got some bonus points there, they smiled. Let's see where we are. Most people went with none of the above or enough. Let's see what I went with. Look at that, I went with none of the above. Now, but I want to point out something very important. In this context, even those of you who went with none of the above, very good. You thought about it carefully. Pressure of the exam, you may not want, you may not be so lucky. This one, this one, and this one were all views of physics before Galileo. Before, actually before Newton even. I should be more careful. And the reason being, without a really good understanding of a world without friction, and without thinking of friction as a force, there are various ideas people had that objects had a natural state they wanted to be in, and that natural state was at rest. That was where everything wanted to be. So in the absence of anything happening to it, you move to your natural state, which is being at rest. That is related, and the same idea is, if it wants to be at rest, it's not gonna move unless you're pushing on it. So to keep it moving, I have to keep pushing on it. And that's pretty much your everyday experience. If you want to keep something moving, you have to keep pushing on it. The third one is another way of maybe using fancier language to express those ideas, is that there's some energy associated with moving, and as I move, I'm using up that energy. So if I start out with some initial amount of energy, just the mere act of moving uses that. And that's natural from our own intuition of what we do. As you move, you get tired. You feel like you're using up your energy. So be warned. Even if you were very, very good, and in the context of this question, pick this. That's because you took time to think about it. It was clearly a physics class, right? This is clearly the right physics answer. In the pressure of a test, your brain might take over. And your brain probably, like almost all human beings, prefers these answers. And so you gotta watch that and pay attention to it. Any questions on that? So if I were to now ask you the question, it's not a clicker question. You don't have to get your clickers out. But if I just ask you the question, what causes velocity? What causes an object to have some velocity? So you see an object moving, say, at 10 meters per second. Just moves by you at a nice 10 meters per second. What is causing that velocity? Talk to the people around you for just 30 seconds here. What is causing that velocity as the object moves past you? Okay, anyone wanna share what they think causes that velocity? Yes. Okay, a force, a push or pull, possibly. Anyone else wanna share something? Yes. Newton's third law, possibly. Anyone else wanna share something? Yes. Okay, change in acceleration that causes velocity, okay. Nothing, okay. Any other thoughts? Let's put this in like red. Notice, just as a random sample, most people went with the wrong answer when phrased that way, even though you probably were among the people who got this answer right. And that's what I'm saying. Your brain gets in the way. There is, as far as we can tell, nothing causes velocity. Things cause a change in velocity. If at some point the object that went flying past you in the past was not moving at 10 meters per second, was moving at a different speed, say at rest, then yes, there needed to be a force, a push or a pull, or some interaction through Newton's third law to cause it to change from zero to 10. But that was in the past. I was asking you about as it moved past you at a constant 10 meters per second, flying nice and straight, what was causing that? Nothing. Nothing causes that velocity. Now, one way to possibly see this is to think of a standard occurrence. How many of you have flown on an airplane? And how many of you on the airplane has thrown something to a friend in the back of the airplane? Excellent, and you've lived to tell the tale. Good. Right, so if you're in an airplane, so let's imagine, which way do I have my airplane going? So here's my airplane, right? And you're sitting in here, there you are, and you're gonna throw something backwards. And my airplane is going 145 meters per second in that direction. And I'm gonna take towards the back of the plane to be positive. And relative to the airplane, oh, look at this, we're sneaking in a practice problem for the test, ooh. Um, suppose relative to the airplane, you throw the can of soda, can of soda at five meters per second, a nice gentle toss, right? You will measure the change in velocity of that can of soda, delta V, to be V final minus V initial, which is five minus zero, because it was in your hand at rest, which will be five meters per second. Now, to me, on the ground watching it, because you're flying in Wonder Woman's invisible jet, what I see is a can of soda with an initial velocity of negative 145 meters per second, because it's flying with the plane to the left, and a final velocity of negative 140 meters per second, because it's got the five meters per second relative to the plane to the back, and I add that to the 145, and I get my minus 140. So I measure a change in velocity for the can of V final minus V initial, equals minus 140 minus a negative 145, which is a positive five meters per second. Notice a couple of things about this. First, in these situations, if we were to think of a cause of the velocity, that would be very weird and hard to come up with, because you are saying it's going at five meters per second. I am saying it's going at 140 meters per second, a negative 140 meters per second. So we disagree completely on what the velocity is, so thinking about a cause of that velocity is hard to imagine. However, we completely agree, and we will always agree on the change in velocity, as long as we are moving at constant velocity relative to each other. So we may not agree on the absolute value, but we will agree on the change in the value. Yes? After your friend catches it, doesn't it go back? Well, after my friend catches it, it goes back to 145, but my friend and I decide it goes back to zero. We see the same change again. It'll be a negative five meters per second at that point. Does that make sense? So after the catch, right, notice once it's caught, from my point of view on the airplane, I see it go from five meters per second to zero meters per second, which is a change in velocity of negative five meters per second. And outside the plane, I see it go from negative 140 back to negative 145, which will also be a negative five meter per second change. So even at that catching moment, we see the same thing in terms of the change in velocity. We again disagree on the velocity. You sitting on the plane says it's going zero, and me sitting outside the plane says it's going negative 145. Like I said, today and Wednesday is all about these conceptual things. This leads us to ask the following clicker question, which is gonna be actually a series of clicker questions. What I'm gonna have you do is click A will mean yes and B will be no. And we're gonna go through it really quickly. I'm gonna start and stop each one, and it's gonna be position. Is it or is it not independent of frame? Meaning a inertial frame, we're going at constant velocity. Do we agree on the position or not? Displacement, velocity, change in velocity, and so on down the line. So first for position, does it depend on the frame or not? Yes or no? We're just gonna take about 15 seconds, so click in quickly. And done. Okay, for this one, we had roughly a 50-50 split. Keep that in mind. Close it. Now do displacement, yes or no? Remember, you're asking yes, it's independent of the frame. No means it depends on the frame. And if you answered backwards for the first one, it probably just flips it 50-50 the other way. But a yes means you think it is independent of the frame. A no means you think it depends on which frame it's in. So this one, we're still kind of 50-50. Interesting. Now we're gonna do velocity. Does this, is it independent of the frame? That would be a yes. Does it depend on the frame? That would be a no. This is better. I would like to point out, the example I just did with the airplane was specifically this. So just letting you know, B is the correct answer there. Now let's do change in velocity. And this one we saw was independent. So A is correct. Now the change in velocity we saw was independent. What about the acceleration? Not quite 50-50. Not quite as overwhelming as the others. But a leaning towards independent. Let's do, and now what I mean by time here is the time it took the event to occur. So in this case, it might be the time it took for the can to go from me to my friend. Very overwhelmingly in favor of independent. And then the final one is whether or not an interaction occurs. So if I throw it, that's an interaction. I'm throwing the can. My friend catches it, that's an interaction. So does the person on the ground and in the plane agree if two things interact? And remember, we're assuming invisible jets so you can actually see inside. And for this one, again, slightly overwhelming in favor. So now let's look at the actual answers. The ones in red, these are independent of the frame. These depend on the frame. And this was actually pretty much where we started the class a few weeks ago. And think about it. If I have my ruler and my origin system right here and I'm measuring positions and I'm moving, I'm gonna say, oh, you were at zero, oh, now you're at minus 10. But if you have your frame of reference and you're sitting there, you're like, no, I'm just at zero, I'm at zero, I'm at zero. And notice from a displacement point of view, you've stayed at zero. So your displacement was zero. I said you went from zero to minus 10. You had a displacement of minus 10. We're gonna disagree on position. We're gonna disagree on displacement. We're gonna disagree on velocity. Once you get up to the level of change in velocity, the airplane example shows you, now we start agreeing. And since acceleration is a change in velocity over a time interval, and as you all said, we all have the same clocks. Now, if we do special relativity, if you move fast enough, then this part's not true. Then time does matter, but we're not doing special relativity. And our level of speeds in our world, my clock and your clock is the same. So if you start your clock and I start mine, and I get over here and I say, it's been 10 seconds, you'll shout back, yep, 10 seconds. We agree on the time, even though I said you just went backwards 10 meters, and you're like, no, I didn't. So we agree on the time. If we agree on the change in velocity and the time, math tells us we have to agree on the acceleration. If I divide two numbers that are the same and you divide two numbers that are the same, unless one of us types it into the calculator wrong, we get the same answer. So we agree on acceleration. We agree on change in velocity. We agree on time. And the final piece, we agree on interactions. That's just like watching and paying attention. If I throw the soda can, it doesn't matter if I throw it while I'm moving or if I throw it while I'm standing still, you and I will both say I threw the soda can. If I catch a soda can, we both say I caught it. So this tells us that in some sense, these are the two things that need to be connected from a physics point of view, right? Not interactions and velocity, because if one doesn't depend on the frame and one does, how can you connect them? Yes. Nope. Nope, notice. Let's go back to the airplane. Notice in both cases, whoops. In both cases, we get a positive five meters per second, even though I'm throwing the can backwards from the plane that's moving that way. Trust me, you can go through every combination and once you subtract the two velocities, because the only difference between you and me is a constant velocity, that will always subtract out. And what's left will be the same. Yeah. For the interactions, isn't there an interaction between the plane and moving them at 100 per second? Not at all, because that's all a constant velocity. They're all just moving at 145 meters per second. If the plane ever changes speed, then there has to be an interaction between the plane and the people to get the people to also change speed or you fly out the back end. Make sense? And we'll do more of that. Yes. So when you say the acceleration. Yes. If there's no air resistance, because that adds interactions, that changes things, right? Once there's air resistance, that's a force. If I try and throw the plane back and forth, the can back and forth, if the can interacts with the air, it gets whipped out of the way. But if we're doing that in a vacuum, we'll just throw it back and forth perfectly fine. Oh, no, you have to keep in mind, when we talk about an interaction and a change of velocity from two different frames, we're talking about being allowed to see. It's not a physical line of sight blocking. You don't want to think about it that way. Physics frames are always completely transparent and you can see through everything. That's kind of the weird thing about physics frames of reference. So yeah, the person on the ground would describe it as a velocity, at an angle back and forth would be the velocities, but the changes in velocities would all be whatever they are. Other questions? And so now we get, we're gonna come back to it again and write it as Newton's second law later on, but we get the central piece really of physics 3A, interactions cause accelerations. And we now then turn this into physics language and we call interactions forces. So forces cause accelerations. And I really encourage you to really keep in mind the equivalence between the word force and interaction. Force is the strict physics word we like to use. In contemporary modern physics, we actually don't use it much. We actually use the word interaction. I find interaction is a much more intuitively useful word because we tend to think of forces as literally the visible things we can see, the pushes and pulls. Interactions are much more obvious because if I use the word interaction, even if you know nothing about friction as a force, you can tell that the book is interacting with the table. It's in contact with it, something's going on. There is an interaction here that I have to understand. When an object moves through the air, you interact with the air and you have to decide whether or not air resistance is small enough that I can ignore it and then I can ignore that force and interaction or do I have to take it into account? And that's why from an intuition point of view, I really prefer the word interactions. The strict physics word we use for that is force. Any questions on that? So we see our first what I would call intuition problem and it goes back to what I listed before. Friction exists, air resistance exists, gravity exists. There's lots of interactions you don't necessarily see and there are forces and they do cause changes in velocity or accelerations. So you have to keep that in mind. Now, to understand forces, we really end up defining it using three laws and these are Newton's three laws. We're gonna go through them somewhat conceptually today. I don't know that we'll get through all three. Then we'll add in some good demos on Wednesday to deal with some of them again and then we really start doing problems with them next week. So let's ask ourselves another clicker question. Talk this one over with your neighbor because you do kind of have to picture it a little bit. But if we look at it from above, let me draw it for you, right? You have your car, say it's driving that way and there's a little ball on the ground and you make a sudden right turn. Okay, so you go that way. And the question is, does the ball just stay in place? And we're talking relative to you in the car. You're watching the ball, not someone outside. Does the ball tend to roll to the right in the car, the left in the car, forward, backwards, or just hang out in place? That's what I'm asking with this question, okay? So looking from above, that's the picture you wanna have. Okay, 10 more seconds. Five, four, three, two, one. Let's see where we are. Excellent, most people have it going to the left. And why does it go to the left? Anyone wanna say why it goes to the left? Yes. Perfect, both of you got it right in parallel. The ball wants to just keep moving forward because there is no force on the ball. And as the car goes to the right, the ball keeps moving forward, which looks to you like it's going to the left. And this goes to your question earlier about the interaction with the airplane. It's very important to keep in mind. And this is what we call the difference between an inertial and a non-inertial reference frame. So inertial means constant velocity. We will talk about non-inertial reference frames. These are accelerating. We don't do much with these in this class, but those are ones in which an object will suddenly appear to move even though there's no force on it because it's actually going straight and you're the one that's accelerating. And so when you feel an acceleration in your car, when you speed up and the seat hits you, that's because until something interacts with you, you don't change your velocity. When you hit the gas pedal, the wheels interact with the road, that accelerates the car, but unless there's a seat to hit you, you don't get accelerated. And that is why you wear a seat belt because when the car suddenly stops, what do you do? You keep going through that windshield unless something stops you. You'd rather have the seat belt stop you than the windshield. That's just a general rule of physics. You experience this. What makes any amusement park ride fun is that they put it in a non-inertial reference frame. You're in a roller coaster in a loop-de-loop. A loop-de-loop is an acceleration. It makes you feel like you're going faster. You're in the one where the room spins and the floor drops out and you don't fall down. You don't fall down because you're in a non-inertial reference frame. You're spinning around in a circle. So most amusement park rides, excuse me, are based on that. But this is basically the first law. You stay in uniform motion, which might be zero velocity, keep in mind, unless you interact with something. And the first law basically defines the fact that these sort of frames exist. So you very rarely use the first law actually to solve a problem. It's much more the definition, the starting point of the definition for forces. It's critical because it lets you keep in mind the difference between these two types of frames of reference and motion. So you are not allowed to use Newton's three laws if you're in an accelerating reference frame. That's basically all it says. Any questions on that? So the second law we already mentioned, but we'll now make it mathematical. The second law is that the net or total force is equal to the mass times the net or total acceleration. So we've introduced the mass. There are really three ways to think about mass. This context is what we call the inertial mass, which is your resistance to accelerations. Your mass tells you, for a given force, how big will your acceleration be? If you have a little mass, you get a big acceleration. If you have a big mass, you really resist accelerations, you resist forces, so you get a small acceleration. In chemistry, it's basically the amount of stuff you have. And later on, we'll see it sets the level of gravity as a force. Turns out, they didn't have to be true. One of the amazing things about nature is to the best of our ability to measure, the thing that sets gravity as a force and tells you how much stuff you have is also the thing that resists acceleration. So there's one concept, mass, that has three different definitions. Notice, and this is the key thing, I'm gonna rewrite F equals ma again. And I'm gonna use colors now. F equals m times acceleration. And this equals sign means cause, not the same thing. What is the definition of acceleration? Instantaneous acceleration is what? You need to know it by Friday. Yeah, let's even be more fancy and go with the derivative. That is the definition of acceleration. That is what it is. Forces, we will find lots of different forces. We will find spring forces. We will find gravity forces. We will find friction. We will find pushes and pulls. They will all have their own definition. Those will be the definitions of forces. The amazing thing about the world is if you add up all of these and you take into account the mass, it will be the same numerically as all of these, which you are defined by a completely different way. These forces cause acceleration. They are not the same thing as acceleration. That is very critical as we move on. One of the biggest mistakes is to mix that up, add up a whole bunch of forces and accelerations, set them equal to the mass times some accelerations, and you accidentally have accelerations on both sides of the equation that cancel out. Then you get the problem wrong. So you need to keep this in mind. Notice we have other places we can get acceleration from. If things are constant, acceleration, for instance, we have v final squared equals v naught squared plus 2a dot delta x, the other equation I have to sneak in as a review for Friday. Don't forget this. Notice this is a way to find the acceleration. Once you've found it, by its definition, you can use it to tell things about the forces. And we're gonna close here with what I call intuition problem number two. Force is a vector and we're talking net force. So let's look at the quiz that we had before the class and just review a couple of these as we finish up. So Newton's Three Laws, what were examples of it? This was just to point out that there's some common things we say for fun as part of Newton's Laws that really aren't. And the big one that I tricked some of you with was what goes up must come down. Just because it goes up, it doesn't have to come down. Rockets make it to the moon. They don't come back down. Now, I was very proud of you. Number two is a very classic problem. I can tell you right now, some version of this will show up on a test in this class. Not this Friday, because it's on forces. This is a classic Newton's Third Law. I know I haven't mentioned it yet. But two things always interact with an equal and opposite force. And so the magnitude here is 10 Newtons. Most of you got that in the time of the quiz. Many of you will miss this when it shows up on the exam. So keep this in mind. Here was the one that's most related to what we just did. An object is moving past you in a straight line with a constant speed. What can you conclude? Again, we went over this right at the beginning. There is no constant force in the direction of motion. Constant speed in a straight line. There is no net force. That's the third answer here. What you cannot conclude, though, is that there are no forces acting on it. Because it's the net force. And two vectors in an opposite direction of the same magnitude can cancel. Well, do cancel. So that is the final piece that's a big intuition problem. So we're adding vectors. And vectors can cancel. So that's what we're gonna have to watch for. We're gonna start Wednesday with the problem I posted for today. Obviously, we didn't get to it. Have some demos. And finish this conceptual introduction. See you Wednesday.