 Hi, I'm Zor. Welcome to a new Zor education. Today we will talk about Newton's three laws, which are basically the most fundamental laws of classical mechanics. This is the heart of the classical mechanics. I mean probably all the problems related to any kind of motion are based on these three laws. Now this lecture is part of the course called Physics 14. It's presented on Unizor.com. On the same website you have Math 14, which is actually a prerequisite for this course. Especially important are topics like vectors and calculus in Math 14. So I do suggest you to brush up your mathematics. This course heavily depends on it. Also, the website is completely free. There are no advertisements. So I do recommend you to watch this lecture from this website as well and any other lecture because there are very detailed comments to every lecture. All right, so let's talk about three laws. Well, first of all, these are the laws which are related to each other, the main concepts of mechanics. Motion, most likely, the first one, and obviously the cause of the motion, which we are calling the force. Now the connection between the force and acceleration and the mass was presented in the previous lecture. So these concepts were introduced. Now this lecture is about their quantitative relationship, especially the second law of Newton. Now a couple of things which should be probably said before everything else. These laws, they are axioms, so to speak, in physics. I mean, in physics, we don't really like to use the word axiom. It's a mathematical word, but we basically take them without any kind of a proof except experimental proof. So there are certain experiments which show that these laws are actually true and that's kind of sufficient within the framework of our experiments, within the precision of which we are making these, conducting these experiments. So basically, there are no proofs of these laws. However, there are certain considerations which I'm going into which kind of prompt that these laws do make sense. All right. The second very important thing is that all these laws are true in the inertial frame of reference. Now, there are, well, there are no absolutely inertial frames of reference, as we know, but we have mentioned before frames of reference, systems of coordinates, which are related to stars, which are so far away that seem to be immovable. So there is, for instance, a heliocentric frame of reference where the origin of coordinates is at sun and the axes are directed to stars. There is a geocentric system, which is basically rooted in our planet and again, the axes are related to stars and there is a much more, I would say, common frame of reference, which is related to the ground where we stand. And for certain purposes, we're using one or another system of coordinates in most of the very simple experiments, which we will be talking about, the ground-based system is sufficient to be called inertial. I mean, it's not inertial because the Earth is rotating, obviously, but within the framework, which we need within the precision, which we are talking about, it's sufficiently inertial system. So anyway, in theory, three laws are only true in the inertial frame of reference. Okay, another very important aspect of this. In most of the cases, when we are talking about objects, we are talking about point objects, which means the object is infinitely small. So whenever we are talking about trajectory, we are talking about trajectory of the point, of the point object. And sometimes we will omit the word point, we'll just talk about objects, but we really have in mind that the object is infinitely small. So its coordinates is basically a coordinate of the point in three-dimensional world. All right. Now, the first law is basically something which we are already familiar with. It's the law of inertia. So the law of inertia is the first Newton's law. And as we know, the law of inertia is that an object in the inertial system of reference continues its state of rest or uniform motion with constant velocity vector, unless some unbalanced forces are applied to this object. Let's not talk a lot about what is balanced forces, non-balanced forces. You just consider it this way. If the object has certain forces applied to it and they're not balanced, then the object will change the velocity. And if the object does not change the velocity, it means there are no unbalanced forces. All right. We'll talk about balancing of the forces a little bit further down. That would be when certain forces are aggregate together as vectors. But that's a separate story. Okay. So that's the first law. I mean, there is nothing to talk about. We have already discussed it many times. That's the law of inertia. And we all are familiar with this. Now, the second law is basically the law of mechanics. It quantitatively connects three very important characteristics related to motion. The force, the mass and acceleration. Well, first of all, let me just add one thing. These are vectors. That's number one. Acceleration is, as we know, a second derivative from the position function. Position function in the Cartesian system of co-ordinate is basically three functions. So this is a vector which depends on time. And this is the position vector. Now, the first derivative would be the velocity vector when you have derivative here, here and here. And the second derivative when you have double prime here, double prime and double prime here, that would be my vector of acceleration. All right. So let's talk about this particular thing. Now, to make my life a little bit easier, I will not talk about vectors, and I will talk about movement within a straight line. That's just easier whenever we are talking about real three-dimensional movement. It's exactly the same, but instead of just an x-coordinate, you will have x, y and z. So at the very end, I will probably do some kind of a talking about this. But basically right now, let's consider that we are talking about the straight line movement and if it's a straight line, then acceleration is basically the second derivative of x-coordinate and the force is directed again along this straight line. Now, to make some kind of a quantitative equation, we need to know how to measure. How to measure acceleration? And that we do know because acceleration is the second derivative of the coordinate function. Coordinate function is a length. It's a distance, right? So distance is measured in meters per time. That's the first derivative is meter per time per second and the second derivative is meter per second per second, right? So this is a known unit. It's one meter per second square, which means that during one second the velocity will change by one meter per second. Let's say if it was 40 meters, not minutes, 40 meters per second. After one second, it will be 41 meters per second. That's what the change is, which dictates the acceleration of one meter per second square. But, talking about mass and force. Now, in the previous lecture, I have introduced the unit of mass, which is one kilogram, and unit of force, which is one Newton. Now, what we were talking about, and let me just repeat from the previous lecture, one kilogram was chosen basically arbitrarily. It's some kind of a cylinder made of platinum and iridium alloy, which approximately weighs as much as the cubicle decimeter of water, and that's what we just call a kilogram. Now, fine, that's okay. Now, what is one Newton? Well, one Newton is the force, which, if applied to one kilogram, will give one meter per second square acceleration. Now, if there are no force, then acceleration is zero, right? That's inertial system, and there is a law of inertia. But if there is a force, then it's supposed to be basically measured in these units. So, basically, what I'm saying is that Newton is equal to kilogram meter per second square. That's what I'm actually talking about. It's one kilogram, one meter per second square. And that's how we basically defined the units. So, in case we are talking about the force of one Newton, the mass, the measure of inertia, so to speak, one kilogram, and the acceleration is one meter per second square because that's exactly how we defined our unit called Newton. Then this particular equation, basically, well, it's true, right? One is equal to one times one. There is no problem with this. Now, okay, let's go a little bit further. I also suggested methodology how we can measure any mass, any object. Well, I said that if my force of one Newton applied to mass kilogram, it's supposed to be, it's supposed to give acceleration one over m meters per second square. So, that's how I defined mass of m kilogram. Now, I know what one Newton is. So, I know this force. This is the force which I have once established. It's one kilogram mass being pushed with the acceleration one meter per second. Now, if exactly the same force gives me this type of acceleration, then my mass is m. Or, if you wish, we can do it slightly different. If it gives me acceleration a, then the mass is one over a, which is exactly the same thing. Now, why is inverse proportional? Because mass is the measure of inertia. Inertia is a resistance of the movement, which means if the movement is greater, it means my resistance should be less. And that's why a is in the denominator. So, that's how I defined, again, this is a definition. So, if I want to know how much, what is the mass of a particular object, I will apply the force of one Newton, which I know already how to get it. I will apply the force of one Newton. I will measure my acceleration and I will take the one over that value and say that this is the mass. And, again, this is kind of a definition of the measure of the mass. So, it's definition. And, again, the same equation is true. One is equal to one over eight times one times times eight. We have exactly chosen this particular measurement style to preserve this equation. So, we know how to measure the mass of any object, right? We just measure it against the one Newton force and check the acceleration and reverse the value. Now, what if I want to measure the force? Well, very easily, I will take one kilogram and measure acceleration. And I said that this same value A would be the measure in Newtons of my force. And, again, it's chosen exactly to preserve this particular equation. So, this thing basically follows from definition, at least for these simple cases. Now, what if it's just a general case? Any mass and any force and any acceleration, will this be actually held? Well, let's just go back to experiments. Well, first of all, experiments do show that this is right. Not only in these simple cases, but in any case. Because, since now we can measure the mass and we can measure the force, we can always combine a new mass and know what exactly the value of this mass by doing this, applying the force of one Newton. We can then take any kind of a force and we can measure the force by applying it to one kilogram and see what kind of an acceleration we have. So, that's how we measure the force. What if I apply some force to some mass? What will be the acceleration? And it's supposed to be this way. It's also based on the property of additivity. Now, my mass is additive, which means that if you have one particular object of mass M1 and another object of mass M2, then combined together into one object, it will be a new object of mass M1 plus M2. So, M1 and M2 are measures which are additive to each other. It's the same thing as, let's say, an area. If you have an area of this figure and an area of this figure, then the area of a combined figure will be this plus this. Or if you have some other characteristics, lengths or whatever else. Now, from this additiveness follows that increasing the mass by certain numerical factor without changing the force will decrease my acceleration. That's what it basically means, and increasing the force without changing the mass will increase the acceleration. So, basically from this proportionality of the force and acceleration. So, I can just write it down this way. So, the force is proportional to acceleration. The mass is inverse proportional to acceleration, right? So, what I will do is, in this particular equation, I will increase the mass by factor of M. What will I have? Well, my acceleration kilogram. My acceleration would diminish, right? Meters per second. Because if my mass is increase, my inertia increase, and that's where my acceleration should decrease. So, I will still have this Newton. Okay. Now, what if I will increase the force now by the factor of M? I will, again, increasing the force will increase the acceleration. So, M newtons here, M kilograms here, and M times greater than this would be one meter per second. Okay. Let me now increase the force again by the factor of A. That will increase acceleration by factor of A. So, what happens? If I will take an object of the mass M, and I know that this object travels with acceleration A, the force must be the multiplication of M times A, which is basically this. So, it's not really a proof, like strictly speaking. It's not a rigorous proof of this formula. There is no proof. Basically, we are taking it as given as an axiom because there are experiments, et cetera. But this is kind of a natural explanation of how actually we came to this formula. It comes from the consideration of proportionality between force and acceleration, and inverse proportionality between mass and acceleration. And these proportionalities we actually borrowed from definition. Because in the very first place, we have defined mass, like this one, defined the mass of M as something which gives acceleration of 1 over M. So, we axiomatized the inverse proportionality between mass and acceleration. And similarly, we axiomatized the direct proportionality between force and acceleration. And from there, we derived this particular equation. Of course, if mass is inverse proportional to A and F is proportional to A, we must have the formula like this. Of course, right? So, I have spent all this time just to explain that the formula really makes sense. Now, I would like to go back to my vectors. So, now it's three dimensional case. And I think you should just understand that this is true because A is actually x of t, y of t, and z of t. Now, F also has three components. It's a vector, right? So, it's fx of t, fy of t, ffz of t. These are three components of the vector. And what I'm saying is times M. So, these three components and these three components of acceleration and the force correspondingly, they are related, which is actually a system of three differential equations of the second order. Because what it means is f of x of t is equal to M times d2 x of t dt square. And same thing for y and same thing for z. So, each one is a differential equation of the second order. So, I mean, you might not actually think about this formula F is equal to MA in these terms, but it is. That's what exactly it is. In the most general sense. And the only thing which I would like to have more than that is about two things. Number one is a very simple one. The first law actually can be derived from the second Newton's law. Why? If F is equal to M times A, if F is equal to zero, no force or forces are balanced together. Then A must be equal to zero. Now, A is the second derivative, which is derivative from derivative. Derivative of position is velocity. So, A is actually velocity. This is vector. This is also vector. So, A is a derivative of the velocity, vector derivative, all three components. So, this is something which has a derivative equal to zero. So, this is constant. So, velocity is constant. If derivative of V is equal to zero, then velocity is constant. And that's exactly what the law of inertia says. But in the absence of forces, velocity is constant. Maybe zero, in which case it's at rest. And the third law. The third law is not really quantitative. It's more qualitative. It says that if there are two objects and if one object acts against another, then that other acts against the first one, always. And not only just that, the forces are equal in magnitude and inversely directed into opposite direction. So, if I have a table and I have some kind of an object on this table, object pushes down, acting on the table. But table pushes back up towards the object. Now, here you have to really be very careful. Just talking like I was just talking, you can have an impression that these two forces are balancing each other and that's why the object is not moving on the table. That's not true. Because this force is applied by the object to the table. So, the table is a recipient of the force, so to speak. This force is how table reacts to the object, in which case object is recipient of the force. So, the application of these two forces is different. One force is applied against the table, another force is applied against the object on the table. And they cannot be combined. Only forces which are applied to the same point can be combined together into some kind of a resultant force. Not the forces which are applied to two different objects. They cannot balance each other by definition. So, why is the object then not moving? Well, because it's not these two forces only which are involved. There is also in this particular case force which is called gravity. So, the gravity is how earth pulls down this object. Now, this force, the gravity force, and this force which is reaction of the table, these two forces are actually applied against the object. Table is pushing the object up, gravity pushes the object down. And they're equal in magnitude and oppositely directed. And that's why object does not move, because these two forces balance each other. Now, what about the table? I'm pushing down onto the table. Object weighs something. So, it pushes down because it has weight. This gravity takes the object down and the object pushes the table with the same force, right? And table is not moving. Why? Well, because there is a floor. So, what's going down onto the floor? The weight of the table, which is gravity, and the weight of the object, which is also gravity, but the floor reacts back with the same force. So, these forces are balancing each other. So, they always have something which is balancing another if the object doesn't really move. So, if this particular marker is not moving, it means that the weight its gravity force, which pulls down, is equivalent and oppositely directed to my support, which I am basically pushing forward, pushing upwards this particular marker. So, that's very important to know which exactly is the point of application of the force. And don't mix together the action and reaction because they are directed and they are applied to different points. But the gravity, for instance, and reaction are applied to the same thing. And that's why they can be combined together and they are equal in magnitude and opposite in direction. So, that's basically the third law of Newton in details. And that's it. That's all I wanted to talk about today. I suggest you to go to the website and read the notes for this lecture. They are maybe more detailed in some cases than whatever I have suggested right now. Okay. Thanks very much and good luck.