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# PHYS 200: Fundamentals of Physics I

## Lecture 3

## - Newton's Laws of Motion

### Overview

This lecture introduces Newton’s Laws of Motion. The First Law on inertia states that every object will remain in a state of rest or uniform motion in a straight line unless acted upon by an external force. The Second Law (*F = ma*) relates the cause (the force *F*) to the acceleration. Several different forces are discussed in the context of this law. The lecture ends with the Third Law which states that action and reaction are equal and opposite.

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html## Fundamentals of Physics I## PHYS 200 - Lecture 3 - Newton's Laws of Motion## Chapter 1. Review of Vectors [00:00:00]
So, summary of last time. If you live in two dimensions or more, you’ve got to use vectors to describe most things. The typical vector is called I mentioned something of increasing importance only later, which is that you are free to pick another set of axes, not in the traditional I gave you a law of transformation of the components; namely, if the vector has components Then, I gave you one other very important example of a particle moving in the r. Furthermore, as time increases, the angle, ωt, is increasing in this fashion. Omega is called the angle of velocity. I related it to the time period, which is the time it takes to go around a full circle, by saying once you’ve done a full circle, ωt better be 2π.So this new quantity The most important result from last time was that if you took this ω, sin ωt; second time it will become -ω cos ^{2}ωt. In other words, it will become -ω times itself. Same thing there. The final result is the acceleration is -^{2}ω times the position. That means the acceleration is pointing towards the center of the circle and it has a magnitude ^{2}a. When I draw something without an arrow, I’m talking about the magnitude. It is just ω. I have shown you yesterday that the speed of the particle as it goes around the circle is this (^{2}rωr).Again, you should make sure you know how to derive this. You can do it any way you like. You can take one full circle and realize the distance traveled is 2 r. This is called the centripetal acceleration. This is the acceleration directed toward the center. I told you these are very important results. You’ve got to get this in your head. Whenever you see a particle moving in a circle, even if it’s at a constant speed, it has an acceleration, v over ^{2}r directed towards the center.This formula doesn’t tell you which way it’s pointing, because it’s a scaler; it’s not a vector equation. If you want to write it as a vector equation, you want to write it as r minus–I want to say that it’s pointing in the direction toward the center. So sometimes what we do is we introduce a little vector here called e. I’ll tell you more about it later. _{r}e is a vector at each point of length one pointing radially away from the center. It’s like the unit vector _{r}I. Unit vector I points away from the origin in the x direction. J points away from the origin in the y direction. e is not a fixed vector. At each point, _{r}e is a different vector pointing in the radial direction of length one. The advantage of introducing that guy is that if you like, I can now write an equation for the acceleration as a vector. The magnitude is _{r}v over ^{2}r. The direction is -e. So _{r}e is a new entity I’ve introduced for convenience. It plays a big role in gravitation, in the Coulomb interaction. It’s good to have a vector pointing in the radial direction of length one. That’s what it is. That’s really the heart of what I did last time. Then we did some projectile problems. You shoot something, you should know when it will land, where it will land, with what speed it will land, how high it will go. I assume that those problems are not that difficult and I’ve given you a lot of practice._{r}## Chapter 2. Introduction to Newton’s Laws of Motion, 1st Law and Inertial Frames [00:09:29]Now I’m going to move to the really important and central topic. I guess you can guess what that is. We’re going to talk about Newton’s laws. This is a big day in your life. This is when you learn the laws in terms of which you can understand and explain a large number of phenomena. In fact, until we do electricity and magnetism the next semester, everything’s going to be based on just the laws of Newton. It’s really amazing that somebody could condense that much information into a few, namely three, different laws. That’s what we’re going to talk about. Let’s start. Your reaction may be that you’ve seen Newton’s laws, you applied them in school. I’ve got to tell you that I realized fairly late in life they are more subtle than I imagined the first time. It’s one thing to plug in all the numbers and say, “I know Newton’s laws and I know how they work.” But as you get older and you have a lot of spare time, you think about what you are doing, which is something I have the luxury of doing right now, and I realized this is more tricky. I want to share some of that with you so you can fast forward and get the understanding it took me much longer to get. That’s what I’m going to emphasize, more than just plugging in the numbers. Of course, we have to also know how to plug in the numbers so we can pass all the tests, but it’s good to understand the nature of the edifice set up by Newton. First statement by Newton–I don’t feel like writing it down. It’s too long and everybody knows what the law is. It’s called the Law of Inertia. Let me just say it and talk about it. The Law of Inertia says that, “If a body has no forces acting on it, then it will remain at rest if it was at rest to begin with, or if it had a velocity to begin with, it will maintain that velocity.” One way to say it is, every body will continue to remain in a state of rest or uniform motion in a straight line. That’s another way of saying maintaining velocity if it’s not acted upon by a force. What makes the law surprising is that if I only gave you half the law, namely every body will remain at rest if it’s not acted upon by a force, you will say, “That’s fine. I accept that, because here’s something. You leave it there, it doesn’t move. It’s not a big surprise.” People were used to that from the time of Aristotle. But Aristotle used to think that if you want something to move, there has to be some agency making it move. That agency you could call force. The great discovery that Galileo and Newton made is that you don’t need a force for a body to move at constant velocity. It’s very clear you don’t need a force if something is doing nothing, just sitting there. The fact that you don’t need a force for it to move forever at a given speed in a given direction, that’s not obvious, because in daily life you don’t see that. In daily life, everything seems to come to rest unless you push it or you pull it or you exert some kind of force. But we all know that the reason things come to a halt when you push them is, there eventually is some friction or drag or something bringing them to rest. Somehow, if you could manufacture a really smooth frictionless surface, that if you took a hockey puck or something and an air cushion and you give it a push, in some idealized world, it’ll travel forever. So it’s hard to realize that in the terrestrial situation. But Galileo already managed to find examples where things would roll on for a very, very long time. Nowadays, if you go to outer space, you can check for yourself that if you throw something out, it just goes on forever without your intervention. It’s in the nature of things to go at a constant velocity. They don’t need your help to do that. You have to be careful that this first law of Newton is not valid for everybody. In fact, I’ll give an example in your own life where you will find that this law doesn’t work. Here is the situation. You go on an airplane and then after the usual delays, the plane begins to accelerate down the runway. At that instant, if you leave anything on the floor, you know it’s no longer yours. It’s going to slide down and the guy in the last row is going to collect everything. Why is that? Because we find in that plane, when objects are left at what you think is at rest with no external agency acting on them, they all slide backwards towards the rear end of the plane. That happens during takeoff. That doesn’t happen in flight, but it happens during takeoff. That is an example of a person for whom the Law of Inertia does not work. This is something you guys may not have realized. Newton’s laws are not for everybody. You have to be what’s called an “inertial observer.” If you’re an inertial observer, then in your system of reference, objects left at rest will remain at rest. The plane that’s ready to take off or is taking off is not such a system. The Earth seems to be a pretty good inertial system, because on the ground, you leave something, it stays there. It depends on what you leave. If you leave your iPod, it’s not going to stay there for very long. But then you can trace it to some external forces, which are carrying your iPod. But if you don’t do anything, things stay. Here is the main point. The point of Newton is, two things in the Law of Inertia, which one may think is trivial. First, free velocity, constant velocity can be obtained for free without doing anything. There are people for whom this is true. For example, in outer space, you’ve got an astronaut. You send something, you’ll find it goes on forever. Here’s another thing. If you find one inertial observer, namely one person for whom this Law of Inertia works, I can manufacture for you an infinite number of other people for whom this is true. Who are these other people? Do you know what I’m talking about? If I give you one observer for whom the Law of Inertia is true, I say that others for whom is also true. Yes?
In particular, the things that I say are at rest, you will say are moving backwards at the velocity that you have relative to me. Things that I say are going at 50 miles per hour you may say are going at 80. But 50 is a constant and 80 is a constant. Therefore, it’s not that there’s only one fortunate family of inertial observers. There’s infinite number of them, but they’re all moving relative to each other at constant velocity. If the Earth is an inertial frame of reference, if you go in a train relative to the Earth at constant velocity, you’re also inertial. But if you go on a plane which is accelerating, you’re no longer inertial. That’s the main point. The point is that there are inertial frames of reference. You must know the Earth is not precisely inertial. The Earth has an acceleration. Can you tell me why I’m sure the Earth has an acceleration? Yes?
r, and r is 93 million miles, you will find the acceleration is small enough for us to ignore. But there are effects of the Earth’s acceleration, which we’ll demonstrate. The Focault pendulum is one example where you can see that the Earth is rotating around its own axis. Then, the fact that the Earth is going around the Sun. All of them mean it’s really not inertial, but it’s approximately inertial. But if you go to outer space nowadays, you can find truly inertial frames of reference.That’s the first law. The first law, if you want, if you want to say, “Okay what’s the summary of all of this?” The summary is that constant velocity doesn’t require anything. The reason it looks like a tautology, because you look around, nothing seems to have its velocity forever. Then you say, “Oh, that’s because there’s a force acting on it.” It looks like a tautology because you’re never able to show me something that moves forever at a constant velocity, because every time you don’t find such a thing, I give an excuse, namely, a force is acting. But it’s not a big con, because you can set up experiments in free space far from everything, where objects will, in fact, maintain their velocity forever. That’s a possibility. It’s a useful concept on the Earth, because Earth is approximately inertial. ## Chapter 3. Second Law and Measurements as Conventions [00:19:51]Now, we have come to the second law, which is “the law.” This is the law that we all memorize and learn. It says that, “If a body has an acceleration, then you need a force and the relation of the force to acceleration is this thing: That’s where I want to tell you that it’s actually more complicated than that. Let’s really look at this equation. Take yourself back to 1600-whatever, whenever Newton was inventing these laws. You don’t know any of these laws. You have an intuitive definition of force. You sort of know what force is. Somebody pushes you or pulls you. That’s a force. Suddenly, you are told there is a law. Are you better off in any way? “Can you do anything with this law?” is what I’m asking you. What can you do with this law? I give you Newton’s law and say, “Good luck.” What will you do? What does it help you predict? Can you even tell if it’s true? Here’s a body that’s moving, right? I want you to tell me, is Newton right? How are we going to check that? Well, you want to measure the left-hand side and you want to measure the right-hand side. If they’re equal, maybe you will say the law is working. What can you measure in this equation?
That’s the meaning of the limit in calculus. You take
a is equal to _{E}m over _{E}m, which is the one kilogram mass. What we needed was some mechanism of exerting some fixed force. We didn’t have to know its magnitude. But the acceleration it produces on the elephant and on the mass, are in an inverse ratio of their masses. If you knew this was one kilogram, then the acceleration of the elephant, which will be some tiny number, maybe 100^{th} of what this guy did; the mass of the elephant is then 100 kilograms.Note there are, again, subtleties even here. If you think harder, you can get worried about other things. For example, how do I know that when I pull the spring the first time for the mass, it exerted the same force when I pulled the spring the second time for the elephant? After all, springs wear out. That’s why you change your shock absorbers in your car. After a while, they don’t do the same thing. First, we got to make sure the spring exerts a fixed force every time. You can say, “How am I going to check that? I don’t have the definition of force yet.” But we do know the following. If I pull the one kilogram mass and I let it go, it does something, some acceleration. Then, I pull it again by the same amount and let it go; I do it 10 times. If every time I get the same acceleration, I’m convinced this is a reliable spring that is somehow producing the same force under the same condition. On the eleventh time, I pull the mass. I will put the elephant in. With some degree of confidence, I’m working with a reliable spring and then I will get the mass of the elephant. Why is it so important? It’s important for you guys to know that everything you write down in the notebook or blackboard as a symbol is actually a measured quantity. You should know at all times how you measure anything. If you don’t know how to measure anything, you are doing algebra and trigonometry. You are not doing physics. This also tells you that the mass of an object has nothing to do with gravitation. Mass of an object is how much it hates to accelerate in response to a force. Newton tells you forces cause acceleration. But the acceleration is not the same on different objects. Certain objects resist it more than others. They are said to have a bigger mass. We can be precise about how much bigger by saying, “If the acceleration of a body to a given force is ten times that of a one kilogram mass, then this mass is one-tenth of one kilogram.” This is how masses can be tabulated using a spring. ## Chapter 4. Nature of Forces and Their Relationship to Second Law [00:37:13]Imagine then from now on, we can find the mass of any object, right? We know now with the same spring, by this comparison, we will find. All objects now can be attributed a mass. Then we may, from this equation, say a certain force is acting in a given situation by multiplying the You got to understand what the minus sign is doing here. This is the force exerted by the spring on the mass. It says, if you pull it to the right, so that I want you to think for a second about two equations. One equation says
In every context in which I place a body, I’ll have to know what are the forces acting on it. I’ve got to find them by experimenting, by putting other bodies and seeing how they react and then finding out what’s the force that acts on a body when it’s placed in this or that situation. Once you’ve got that, then you come back. In the case of a spring, this is the law that you will deduce. If it’s something else, you will have to deduce another law. For example, we know that if a body is near the surface of the Earth, the force of gravity and that object seems to be By the way, that’s a very remarkable property of the gravitational force–the cancellation of the two This was known for a long time, but it took Mr. Einstein to figure out why nature is behaving in that fashion. If I have some time, I’ll tell you later. But there are two qualities which happen to be equal. One is inertial mass, which is how much you hate your velocity to change, how hard you resist acceleration. That exists far from planets, far from everything. Other is gravitational mass, which is the measure of how much you’re attracted to the Earth. There’s no reason why these two attributes had to be proportional, but they are proportional and they are equal by choice of units and you can ask, “Is this just an accident or is it part of a big picture?” It turns out, it’s part of a big picture and all of general relativity is based on this one great equivalence of two quantities which are very different attributes. Why should the amount by which you’re attracted to the Earth be also a measure of how much you hate acceleration? Two different features, right? But they happen to be the same. Anyway, what physicists do is they put bodies in various circumstances and they deduce various forces. This is the force of gravity. This is the force of the spring. Here’s another force you might find. You put a chunk of wood on a table and you try to move it at constant speed. Then you find that you have to apply minimum force. We are moving at constant velocity. That means the force you’re applying is cancelled by another force, which has got to be the force of friction. So force of friction is yet another force. Then, there are other forces. You guys know there is the electrical force. If you bring a plus charge near a plus charge, if my body Remember, Newton said Here’s a simple example of a complete Newtonian problem. A mass is attached to a spring. It is pulled by a certain amount You can formulate another problem. Later on, we know about gravity. Newton finds out there’s a force of gravity acting on everything. Here’s the Sun. Here’s your planet. At this instant, the planet may be moving at that speed. Then the acceleration of the planet is the force of gravity between the planet and the Sun, which Newton will tell you is directed towards the Sun and it depends on how far you are. Depending on how far the planet is from the Sun and where it’s located, you will get the left-hand side. That’s another law. That’s the Law of Universal Gravitation. Then again, you will find the evolution of the planetary motion, because the rate of change of the position is connected to the position. Again, go to the math guys and say, “What’s the answer to that?” and they tell you the answer, which will be some elliptical motion. Okay. By the way, Mr. Newton did not have math guys he could go to. Not only did he formulate laws of gravitation, he also invented calculus and he also learned how to solve the differential equation for calculus. He probably felt that nobody around was doing any work, because all the thing was given to this one person. It’s really amazing that what Newton did in the case of gravity was to find the expression for this. A few years earlier he had also gotten this law. By putting the two together, out comes the elliptical motion of the planets. We’ll come back to that, but you have to understand the structure of Newtonian mechanics. Generally, any mechanics will require knowledge of the force. Now, I’m going to add one more amendment. You don’t have to write in your notebook, but you’ve got to remember. Maybe you’ll but a little ## Chapter 5. Newton’s Third Law [00:50:50]Now, I’ll give you the third law. The third law says that if there are two bodies, called one and two, force of one on two is minus the force of the second on the first one. This is the thing about action and reaction. All the laws that anybody knows have this property. What does it require to be a successful mechanic, to do all the mechanics problems? You got to be good at writing down the forces acting on a body. That’s what it’s all going to boil down to. Here is my advice to you. Do not forget the existing forces and do not make up your own forces. I’ve seen both happen. Right now at this point in our course, whenever you have a problem where there is some body and someone says, “Write all the forces on it”, what you have to do is very simple. Every force, with one exception, can be seen as a force due to direct contact with the body. Either a rope is pulling, a rope is pushing it, you are pushing it, you are pulling it. That’s a contact on the body. If nothing is touching the body, there are no forces on it, with one exception which is, of course, gravity. Gravity is one force that acts on a body without the source of the force actually touching it. That’s it. Do not draw any more forces. People do draw other forces. When a body is going around a circle, they say that’s some centrifugal force acting. There is no such thing. Be careful. Whenever there is a force, it can be traced back to a tangible material cause, which is all the time a force of contact, with the exception of gravity. Okay, so with that, if you write the right forces, you will be just fine. You will be able to solve all the problems we have in mechanics. I’m going to now start doing simple problems in mechanics. They will start out simple and, as usual, they will get progressively more difficult. Let’s start with our first triumph will be motion in 1D. Here is some object, it’s 5 kilograms and I apply 10 Newtons. Someone says, “What’s the acceleration?” Everyone knows it is 10 over 5 equals 2. Now we know how we got all the numbers that go into the very question. How do we know 10 Newtons is acting? I think you people know how we can say that with confidence. How do we know the mass of this is 5 kilograms? We know how you got that from an earlier experiment. Now, we know how the numbers come in. The algebra is, of course, very trivial here. Then, the next problem is a little more interesting. Here I got 3 kg and I got 2 kg and I’m pushing with 10 Newtons and I want to know what happens. One way is to just use your common sense and realize that if you push it this way, these two guys are going to move together. And know intuitively that if they move together, they will behave like an object of mass 5 and the acceleration will again be 2. But there’s another way to do this and I’m going to give you now the simplest example of the other way, which is to draw free-body diagrams. By the way, when I say there’s 10 Newtons acting this way, you might say, “What about gravity? What about the table?” Imagine that this is in outer space where there is no gravity for now. The motion is just along the The free-body diagram, it says you can pick any one body that you like and apply
Notice I’m using the same acceleration for both. I know that if the second mass moved faster than the first one, then the picture is completely wrong. If it moved slower than the first one, it means it’s rammed into this one. That also cannot happen, so they’re moving with the same acceleration. There’s only one unknown Now, here’s another variation. The variation looks like this. I got 3 kg and I have a rope. I got 2 kg and I pull this guy with 10 Newtons. What’s going to happen? Again, your common sense tells you, “Look, you are pulling something whose effective mass seems to be 5, the answer is 2.” Let’s get that systematically by using free-body diagrams. Now, there are really three bodies here. Block one and block two and the rope connecting them. In all these examples, this rope is assumed to be massless. We know there is no thing called a massless rope, but most ropes have a mass, but maybe negligible compared to the two blocks you are pulling, so we’ll take the idealized limit where the mass of the rope is 0. Here is the deal. 3 kg is being pulled by the rope on the right with a force that I’m going to call
Now, you can do ## Chapter 6. Weightlessness [01:03:15]Now, for the- whoa! I’m going to give the last class a problem which is pretty interesting, which is what happens to you when you have an elevator. Here is a weighing machine and that’s you standing on the elevator. We’re going to ask, “What’s the needle showing at different times?” First, take the case in the elevator is on the ground floor of some building and completely addressed. Then, let’s look at the spring. The spring is getting squashed because you are pushing down and the floor is pushing up. You are pushing down with the weight By the way, that is a subtle thing people may not have realized. Even in the case of this spring, when you pull it, if you pull it to the right by some force. Remember, the wall is pulling to the left with the same force. So springs are always pushed or pulled on either side with the same force. We focus on one because we are paying for it, but the wall is doing the opposite. We don’t pay any attention to that. You cannot have a spring pulled only at one side, because then it will then accelerate with infinite acceleration in that direction. This spring is getting squashed on either side, and it’ll squash by certain amount Now, what happens if the elevator is accelerating upwards with an acceleration Let me just briefly look at the ride on the way down. As you start on the top and go down, your acceleration is negative. Remember, you’ve got to keep track of the sign of acceleration, so if you’re picking up speed towards the ground, [end of transcript] Back to Top |
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