 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about inertial frame references. Now, this is a kind of philosophical concept in physics, and it requires lots of explanation, which I will try to offer you as much as I can. Now, this particular lecture is part of the course called Physics for Teen on Unisor.com. And if you go to this website and navigate to this particular lecture, you will see very detailed explanation of whatever I'm talking right now, maybe even better than whatever I'm talking. Well, and obviously all other lectures have very detailed notes, so I do suggest you to watch this lecture from this website. Site is free, by the way, and it has no advertising, so all open for all. Now, so let's talk about inertial frames of reference. First of all, what is a frame of reference? It's just a system of coordinates. In most of the cases, I will use Cartesian system of coordinates, and that's why it's Cartesian frame of reference, and I will interchangeably use the term frame or frame of reference and system of coordinates. The shortest one is a frame, right? So we will try to use the short one. So, now let's talk about position of any object as it moves. Position we usually describe by position functions, right? So we have three position functions. It's coordinates in some system of coordinates, in some frame of reference, as functions of the time t. So whenever we are talking about some numerical characteristic of the position, we are implying that there is certain system of coordinates, certain frame of reference relative to which the position is basically defined. Now, sometimes we don't really explicitly specify this. We implicitly kind of understand what this actually is. Just for example, if the car moves along a straight road and we are saying, okay, the speed of the car is, let's say, 70 kilometers per hour. Now, what does it mean? Well, it means that we are considering certain system of coordinates where the origin is probably on that road. The, let's say, x-axis goes along this road, y and z are perpendicular to each other and they are always equal to zero since this is a straight road. And we are measuring the position x-coordinate in this system as a function of time t. And the first derivative of this x of t function, which is the speed, is actually 70 kilometers per hour. So we are implying that there is such a coordinate system. It doesn't mean that our car moves at speeds 70 kilometers per hour in some absolute sense. There is no absolute sense because the car is on the Earth. Earth is rotating around the Sun. Sun is rotating around some center of the galaxy, et cetera, et cetera. So everything is moving. So it's very important to specify what exactly the coordinate system we are talking about when we are talking about numerical characteristic of the position. Now, just as an example, if you have certain coordinate system and you have functions of position equals to this. So at time t is equal to zero, we are in the beginning in the origin. And then as the time increase, we are moving along the x-axis with increasing, quadratically increasing of time speed. Y and z are equal to zero. Okay, fine. Now let's have another system of coordinates. The system of coordinates which I suggest is three, zero, and minus five. So it's somewhere minus five. It's here. So zero and minus five, so how can I, it's somewhere here. All right? Now, and from this point, I also have a coordinate system. With axes parallel to my original one. So let's call it UVW. Now, if the law of the motion is as described here in this particular system, and I consider a different system with origin at point three, zero, minus five. Now, what would be the functions which describe the position in this system? Well, obviously that would be U of t, which is equal to, now, it used to be t-square, but now we are shifted by three along the x-axis. So it should be t-square minus three, right? So the point is here, and this is my x-coordinate. Now, this is my U-coordinate, which is by three less, right? Now, y-coordinate did not change because I have zero here, right? So it would be zero as well. And z-coordinate, since I moved my whole system of coordinate down by five, so it used to be z-zero, but now if my system is moved down by five, z would be equal to five, always. So we have a completely different system of positional functions because we have changed the frame of reference. Change of coordinate system causes change of the equation of motion, right? So it's very important, I just wanted to show again how important it is to always specify the system of coordinate we're talking about. Again, sometimes it's implied, like in the case of the car moving along the straight line, but sometimes you really have to explicitly say in which frame of reference your equations are actually true. So it would be nice if we had one and only absolute frame of reference, which is absolutely immovable. It doesn't change with the time, it just stands still in time and all our motions, all our coordinates will be relative to this absolute coordinate system, absolute frame of reference. Well, once there is no such absolute frame of reference because everything is somehow moving somewhere relative to something with some kind of a direction, speed, etc., etc. As I was saying before, we are on the surface of the Earth, Earth moves around its own axis, around Sun, Sun is moving around the galaxy center, etc., etc. So everything is moving, there is no such thing as an absolute frame of reference. However, well, for our modest needs, we can assume actually that there is a system which is related to stars, let's say we just position a couple of stars in some way. You see, stars relative to Earth are so far away that they basically seem like immovable, right? They are standing still. We all know they are not standing still, we all know that Earth is moving, etc. But within the limitation of our precision, whatever we require for our experiments, we can always assume that the stars are really not moving anywhere and we can tie our coordinate system, our frame of reference, with immovable stars and consider this as the, well, almost absolute frame of reference. Now, we are talking about inertia. First of all, what is inertia? Inertia is the property of the object to continue, well, in philosophical sense, to continue doing whatever it's doing without changes. In physical sense, inertia is actually the continuation of movement in exactly the same fashion as it was before, which means if the object was at rest, it will be at rest. If it moves in certain direction with a certain speed, it will always move in the same direction with the same speed. That's what basically inertia is. Now, by the way, we know that movement in the same direction with the same speed, which means the velocity vector is the same, it's called uniform motion. So, inertia is very much related to uniform motion. And even the state of rest can be also called a uniform motion with the velocity vector equals to zero, right? All right, so, we are talking about uniform motion. Now, let's recall our almost absolute system of coordinate related to star, what's called star-based frame of reference. And let's consider a comet which flies somewhere in the space far from the solar system, far from other solar systems in vacuum. So, the gravity is very, very low. Within our precision, we consider that there is no gravity. There are no other objects which actually force the comet to change its course. So, what happens with this comet? Well, within this frame of reference related to star, our comet would probably move in the uniform motion. And our experiments actually suggest that that's exactly the case. So, our star-based frame of reference has this very important property, that the object, let's say, a comet or anything, if it does not experience any kind of external forces or would be more precisely to say if the forces are non-existent or maybe they are balanced to each other. Well, if you have two forces against each other, it's like they're nullifying each other. So, if there is no unbalanced force, then our object will be moving in a uniform motion fashion. So, this is basically the law of inertia. That the object at rest will stay at rest. The object in uniform motion will continue the same uniform motion in the absence of unbalanced forces. Now, we didn't talk about any kind of system of coordinates here, right? Well, it's implied actually that we consider that there is some absolute system and obviously we're talking about something like a star-based system which is almost absolute in our sense. So, relative to this system, if the object was in the state of rest, it will be at rest, if it was in the state of uniform motion, it will continue this uniform motion, the velocity will be exactly the same. So, we have at least one star-based, in this case, frame of reference and we have this law of inertia, which we basically, while considering our experiments, we postulate it. It's our axiom. So, there is an axiom which is called law of inertia. The body will not change its velocity unless you have some unbalanced forces relative to this star-based system. And because we have postulated this particular law of inertia, we call this star-based frame of reference, we call it inertial frame of reference. Well, inertial frame of reference is the frame of reference where the law of inertia is true, right? Or we have basically postulated. Now, we found one, that particular system, that's star-based, and we found this, we postulated this law of inertia, that's good. However, to work with this particular system is very difficult, obviously. I mean, just consider, you have this very abstract kind of a system of reference, system of coordinates and it's very difficult to basically numerically explain the movement of something in terms of these functions if it's related to some stars. I mean, the numbers would be crazy and etc. So, it's not convenient and what do we do? And here is what we do. Now, I'm going to prove a theorem actually, that if you have one particular inertial frame of reference, then there are infinite number of others and I will show you how to construct these other frames of reference, which are also inertial in the sense that the law of inertia is true. If it's true in one system of coordinates, then I will construct the infinite number of other systems of reference, systems of coordinates, which are much more convenient to work with, which are also inertial where the law of inertia is true. Now, how do I do it? Here it is. So, let's assume that we have an XYZ frame and it's inertial. Well, for instance, it's our star-based system. We know it's inertial because it's an axiom. Now, let's consider another frame, UVW frame. Now, this frame is moving relative to XYZ frame uniformly. What does it mean? Well, it means that if this is one system and this is another system, then this is, let's say, a vector q0. That's vector from the origin to origin. So, this is XYZ. This is UVW. Now, what we are talking about moving uniformly, it means that from this initial position this coordinate system moves in certain direction, let's say this is the vector, and this is the velocity of moving of this particular point, the origin of the new system. It's moving with the velocity vector omega in some direction, and the omega is constant. So, in this particular origin of the new system from its initial position, which is characterized by vector q0 in the old system, it moves along certain vector omega which characterizes the direction and the speed, basically, in that direction. Then I'm actually stating that this new system of coordinates will also be inertial if XYZ is inertial, and it's actually very easy to prove. Now, how the movement of any particular point, let's say this point, is described in this XYZ system, I will put a PXYZ of t. That's the vector which is, now this is in this particular frame, this is uniform motion. So, I assume that this is a uniform motion, which means it's t times velocity plus some kind of a beginning position. So, if my object, initially, at point t0 is at this position and moves with constant velocity v in the XYZ system, then this would be co-ordinate at point t in this system. Now, what would be the position of this same object, this same position expressed in this other, the new system of co-ordinate? Well, this is the vector and you know the rules for addition of the vector. So, if you have a vector and then you have these vectors, then some of these vectors is this one, right? Some of these two is this and some of this and this is this. That's the rules for vector addition. So, I can say that the vector from here to here is equal to vector from here to here. That's the initial position of our origin, of new origin. Then, during the time t, my origin moved to point to this one, right? My new co-ordinate system is moving. So, during the time t, it moves along this vector omega from initial position, which is q0, to which position? Well, origin will move here, which is t times omega, right? It's same thing basically as here because we are talking about uniform motion of this point of origin. So, it's basically the same kind of a formula, but the vector is omega. That's the vector how my new system of co-ordinate is moving. Now, what is the co-ordinate of the same point in this new position of the new uvw frame of reference? Well, that would be from here to here, and this has certain co-ordinates, which are puvw of t. So, that's my new co-ordinate of the same point in the new system of co-ordinate. So, if the old system of reference gave me this formula, the new gives me this formula. So, basically, they are equal because my position, because I object in the same point, this is co-ordinate of this point in this co-ordinate system, which is shifted by this from the origin, and origin itself was shifted by q0. And this is the same position in my old system. Fine. This is basically the conversion from old to new or from u to old, whatever it is, omega and q0 are constants. This is initial position of my uvw frame of reference. This is a constant, omega is a constant velocity my uvw frame of reference is moving. Within the old XYZ system. And this is the new co-ordinates of the same point from the new position of the uvw frame of reference. Okay, that's good. Now, let's differentiate it because these are positions, right? So, the derivative of position is velocity. Here we are talking about uniform motion. We really have to watch the velocity. It's supposed to be constant. Well, let's do it. If we differentiate, this is a constant. This differentiation gives me omega, differentiation by t. This would be from the position I go to velocity, uvw of t. And this would be velocity of XYZ frame. So, what does it mean? If this is constant, this is constant because this is constant. So, whenever we are talking about one frame of reference moving uniformly relative to inertial system, the movement would be uniform in another system of co-ordinates as well. So, now, XYZ, I can always say this is my old-fashioned star-based system. So, if my system of co-ordinates, this one, is moving uniformly relative to my stars, then since the stars are inertial frame of reference, my system also is inertial frame of reference. Now, for example, if you have a system of co-ordinate which is fixed on Earth, and Earth is moving, well, obviously it's not moving along a straight line. So, I can't really say it's really uniform movement. But within certain practical precision, it is relatively uniform movement. And we don't really feel that there is something like acceleration towards something, etc. So, within certain degree of precision, we can assume that the movement within the Earth is basically the movement within some kind of inertial frame of reference. The point was that you can always invent some kind of inertial frame of reference, which is convenient for your particular problem based on this property. So, that's why we will assume that we always deal in cases of uniform motion, whenever we are talking about uniform motion, we are actually talking about uniform motion either relative to our star-based system or relative to some maybe implied, maybe explicitly specified system of co-ordinates, which is also inertial. Why do we need it? Well, because we need the law of inertia. Law of inertia guides basically the behavior of the objects which are uniformly moving. So, we expect that if I have a uniform movement, then it will continue to be uniform movement indefinitely. I kind of assume that the law of inertia is true. And that's how I can make some kind of calculations. So, that's why I'm saying that V is constant. That's why I know that it exists this particular kind of motion. The motion with a constant velocity and we call it uniform motion. So, that's what's very important. And now, as you saw, there are lots of explanation and there are a lot of approximation. We don't really have ideal absolute inertial system, inertial system of reference. But, within certain precision, we can always say that the star-based co-ordinate system is inertial and any system which is uniformly moves relative to the stars, at least within certain narrow, maybe, time period. I mean, yes, Earth, for instance, is rotating around its own axis, right? But if our experiment takes, let's say, a couple of seconds, then this movement is so insignificant that we can say that basically the Earth, relative to these immovable stars, also at rest. And that's why we can use this particular system of reference for our own experiments, whatever our experiments are. That's basically the bottom line of this inertial system. But again, I can definitely tell you there is no absolutely 100% inertial systems. Everything is moving somewhere. But within certain degree of precision, we can always assume that we are choosing the reasonably inertial frames of reference. And most of the problems, actually, which we will be dealing with, most of these problems will also be related to certain properties of the object moving within inertial frames of reference, maybe implying, not necessarily explicitly specifying, okay? This is my system of co-ordinate frame of reference. No, sometimes it's just implicitly specified, which is fine as well. All right, now I do urge you to read the notes for this lecture on Unisor.com, because maybe something is explained a little bit in more details, because I was actually trying to put the notes almost like basically as a textbook for this particular lecture. So good luck. Thank you very much. That's it.