 Hi, I'm Zor, welcome to a new Zor education. Today I would like to talk about specific kind of equilibrium, equilibrium of a point object. So we have some kind of a point object, which is a point with certain mass. And we're talking about equilibrium, conditions on equilibrium and kinds of equilibrium of this particular case. Now, this lecture is part of this course, Physics 14, presented on Unizor.com. I recommend you to go to the website to watch this lecture, because it has parallel text notes for every lecture, including this one. Also, there is a prerequisite course on the same website, which is called Mass 14s. You do have to know mass to succeed in physics on the level which we are talking about right now. And the site is completely free, there are no advertisements. So I do suggest you to go to the Unizor.com. Plus, there are certain exams which you can take if you want. It's completely anonymous if you want to. Obviously, you can sign on and basically your history of exams will be preserved. Alright, so equilibrium of the point. Now, point is characterized by its three coordinates in our three-dimensional world. Obviously, we will call them x-coordinate, y-coordinate and z-coordinate. Now, as we know from the previous lecture about equilibrium, that equilibrium means basically that all the forces which are acting on the particular object are balanced. Which basically means very simply that some of all the forces as vectors, let's say we have n different forces which are acting on the same point object, and their vector sum is supposed to be a null vector. Now, if that's the case, then we are talking about equilibrium. Well, now, this is some kind of an equation basically. It's a condition on the forces to be satisfied for the object to be in equilibrium. Well, when talking about equation, we are usually talking about algebraic equations. Now, this is a vector equation. So how can we translate it into algebraic form so we can basically solve the equation? Well, very simply. Every force has, again, three coordinates because any vector can be expressed as the sum of its three projections. So if you have x, y and z coordinates and there is some kind of a vector, we can always project this vector onto x, y. So this is y coordinate, this is x coordinate, and on the z, this is z coordinates. So this is f, y, this is f, x, and this is f, z. So the vector f is equal to fx plus fy plus fz. Well, actually I put it on the upper index, sorry. We need the bottom for different purposes. Now, in this particular case, we can always express this condition as condition on each component, namely, f, i, x plus f, i, y plus f, i, z. This is my f, i, right? Equals to zero, where i index i goes for every vector. Now, I can make a slight correction of this, well, not correction, but transformation rather. I will summarize not these vectors in these orders, which means first for i is equal to one. I summarize f first, x component plus f first, y component plus f1, z component, getting basically the overall f1. Instead, I will do it separately. I will do all x's and all y's and all z's. So basically, it's like three different kind of things. It's sum of fx, no, f, i, x plus sum of f, i, y plus sum of f, i, z equals to zero vector. These are all vectors so far, right? Now, but let's think about it. All fx are collinear. They're all within the x-axis. They're all projections of f, i onto the x-axis, right? So they're all collinear, and their sum is also some kind of a vector along the x-axis, right? Again, if these are my coordinates, so all f-axis are here of different lengths, obviously, and maybe different direction along the x-axis, but they're all collinear. They're all within the same line. They're all projections of f, i wherever it is onto the x-axis. So if they are all collinear, their sum is basically an algebraic sum. So I can have this whole, I can have a sum of these magnitudes with proper signs, plus or minus depending on the direction, but I can add as magnitudes. So this would be f, i without the vector sign, and this will be my x-component of 0 of null vector, and the x-component of the null vector is obviously null. So this is an x-component of this vector. This is y-component, because all these vectors are along the y-axis. And this is, again, a z-component of this, which means all of these and all of these must be, if I will summarize algebraically, all the magnitudes with the proper sign. They must be equal to 0. So this is not a vector equation anymore. This equals to 0 is a plane algebraic equation where f, i, x. Sorry, where f, i, x is x-component of the f, i vector. And same thing obviously for y, sum of f, i, y is equal to 0, and same thing for z, sum of f, i, z is equal to 0. So what do we have as a result? As a result of this equation and representation of each vector as sum of its component, we have three algebraic equations, namely sum of f, i, x equals to 0, sum of f, i, y equals to 0, and sum of f, i, z equals to 0, where f, i, x, f, i, y and f, i, z are correspondingly x, y and z components of the vector f, i. Now these are three algebraic equations. And this is a condition, algebraic, if you wish, condition for a point to be in equilibrium, to basically be in the state of rest, so to speak, right? Now there are only three equations. What does it mean? It means that if we would like to find the forces which are acting on a particular subject, on a particular object which result in this object to be in equilibrium, we should not have more than unknown variables, right? So forces are, if they are completely independent and there are a lot of them, then we will not be able to resolve this system of equation in certain unique way. There are no unique solutions in this case. We will have more variables, unknown variables than the equations. So basically that's what it is. If you have a system which contains a point object and certain number of forces which act on it, if we will be able to find no more than three unknowns from which these forces depend upon, and I will give you an example, then we can uniquely identify these forces. Otherwise there are infinite number of solutions most likely, and the system would not have a unique solution. Now let me just make a very simple example of this. Let's consider you have a chandelier which is hanging on three threads. So you have three threads from the ceiling. So this is the ceiling, and these are three points. Now these are three points from which we have hanging on a thread from underneath, right? So this is the ceiling. Okay, and this is our chandelier. Now let's assume for simplicity that we know the weight of this chandelier. Let's also assume that these three points where it's hanging from are forming an equilateral triangle, and these threads are of equal lengths, which is obviously the typical situation. Now my purpose is to find out the force of tension on each thread, so I will be able to use a proper material like a chain or a thread or a rope or something else. So I have to calculate based on the weight, I have to calculate what's the tension on each thread. Now all I need right now is basically to know certain things about geometry of this. So what I probably need as the smallest amount of information and only one particular parameter from which everything else depends I should know what kind of an angle with vertical each thread makes. So if you have some kind of vertical it goes from the center of this triangle down to the chandelier and I know the angle between these threads with the vertical. Let's call it phi. So if I know w and I know phi, now I can actually determine the tension forces on each thread very easily. Why? Well because of the consideration of symmetry, obviously the tension should be exactly the same because this is the equilateral triangle with the vertical lengths of the threads, so it's supposed to be evenly distributed. So there is a force of tension here, here and here. There are three forces of tension and they must be equal to each other. So if this is tension t then in the vertical direction from this particular force t times cosine phi, right? So if my vector t can be represented as the sum of two vectors, the vertical and horizontal. Now horizontal doesn't really do anything in the vertical direction, right? And all the horizontal components are acting in three different directions basically at the top. If I will look at this from the top I will have these three at 120 degree each and they will nullify each other so that's why my chandelier doesn't move horizontally. But as far as the vertical is concerned we should really neutralize the weight, right? So three times of these, because all of these projections from this force, from this force and from this force on each thread, tension on each thread, projection on the vertical, they're all collinear and that's why their sum would be just algebraic sum of the magnitudes. And that must be equal to w, from which t is equal to w divided by 3 cosine phi. So what does it mean? Well, first of all it means that if my angle is close to 0 then I will have this cosine of phi close to 1. Now as my angle is increasing so I'm separating the point from where I'm hanging further and further from each other so making my threads instead of this I will make it this, they're further from the vertical. That actually makes cosine smaller and smaller and since it's a denominator, the tension will be greater and greater. Alright, now imagine as our points of hanging is separating further and further and my thread from the vertical becomes more and more closer to the 90 degree because the point of hanging is really going further and further. Well, the cosine of 90 degrees is 0 so the whole thing is going to infinity. So the further we are from the vertical in our threads the tougher will be to hang really the chandelier and if it's 90 degrees as I was saying it's really infinity will be which means that if you have these threads hanging not from the ceiling but let's say from the walls so if you have a room, this is the room, I'll just have one particular and instead of a ceiling you are trying to hang your chandelier from the walls and you don't want this chandelier to go really deep. You would like it to be almost in the same height where your attachment points are. That's impossible. So whenever in case your chandelier is exactly at the same level as these points of hanging this will be 90 degrees and that's why the cosine will be 0 and the tension force will be infinite. So it must hang down. You cannot horizontally hang the chandelier. So that's all about chandeliers. Now we have three different types of equilibrium. This is simple and quick. Let me just draw the picture and you will know what I'm talking about. What happens if I have let's say a cup, a semi-spherical cup and you have a small bowl at the bottom. Well, at the bottom it just sits there, right? And the weight is equal to the reaction force of the cup and it actually stays in place in equilibrium. Now what happens if I will move it off the equilibrium point? Let's say I'm moving my bowl into this position. Now my weight still goes here. My reaction is always perpendicular to a surface, right? So the perpendicular to a surface would be something like this. This is the reaction force. This is my weight. So what would be the resultant of these two vectors? It would be this force which forces our bowl back to the original position of equilibrium. So if we will move it off the position of equilibrium, it will return back. So this is basically called a stable equilibrium. Now let's talk about a little bit more rigorous definition of the stable. How far can I move this particular object, in this case the bowl, from the position of stable equilibrium so it will still return back? Just think about, my condition can be something like this. What if I have something like this? If I will move my bowl within this interval from here to here, it will return back. But if I will move it a little bit further, it will go away, right? So the stable equilibrium is an equilibrium when there is some kind of a neighborhood within which I can really move this object from the equilibrium point. So around the equilibrium point must be some kind of a neighborhood when the object would return back into the equilibrium point. Now how big is supposed to be this neighborhood? Well, it actually can be any as long as it's not zero. So basically the relatively rigorous definition of stable equilibrium is it's such an equilibrium where exists some kind of an epsilon neighborhood and from the mathematics you know what I mean epsilon neighborhood of the point of the equilibrium where if I will move it within this neighborhood, epsilon neighborhood, it will return back and obviously it's supposed to be strictly greater than zero. That's what definition of a stable equilibrium is. Well, if you again remember the mathematics there is a concept of a continuity of the function and there is also like epsilon and delta neighborhoods are involved. Argument is supposed to be no larger than something if you would like that the function value to be deviated from another no more than something. Epsilon delta in this case is just one particular epsilon neighborhood where epsilon is greater than zero must exist. However small, so it can be one millimeter or it can be one micron but as long as there is such a neighborhood of the point of equilibrium this particular point of equilibrium is called a stable equilibrium point. Okay, now what is unstable? Well, look at the second picture. At the second picture obviously in this case we do have some kind of an equilibrium but as soon as I will move it by any value of the equilibrium point it will immediately go out from the equilibrium. So basically the definition of unstable is that there is no such epsilon greater than zero. There is no such epsilon neighborhood around the point of equilibrium where my return will be guaranteed. No matter how small epsilon is there is always some point within that epsilon neighborhood where I move my object it will immediately go out from the equilibrium point. This is unstable. And finally the neutral, well on a horizontal plane obviously this particular object, this ball can have any basically point as a point of equilibrium. Now if I will move it from the equilibrium point it will not return back but it will not go away, it will just stay where it is. So this is the neutral. So this picture is I think very good illustration of the concepts of stable, unstable and neutral equilibrium. And this basically completes my discussion of the equilibrium for point objects. Now my next lecture will be about solids. And with solids we have a little bit more complicated story. Why? Well because solids can rotate. Now the point object can move in three different directions. So it has three different degrees of freedom as we are saying. Along x, along y and along z-axis. Now whatever the combination of these movements along these three axes is basically tells about what kind of a freedom of movement my object can have. So it can move freely within any direction on the x-axis, any direction on y-axis and any direction on z-axis. Okay, what about the solid? If you have a solid then it can also rotate around x-axis, rotate around y-axis and rotate around z-axis. So these are additional three degrees of freedom. So the solid has six degrees of freedom. Well unless there are some restrictions. I mean we can think about construction restrictions when the solid can move only around one particular axis. But in any case, in theory if everything is free then these rotations around each of the three axes also gives certain degree of freedom. And if we are talking about equilibrium we should really think about the rotation. So we basically know how to deal with movement along the translational movement along the coordinate axes. And now we will talk, in the next lecture we will talk about rotation. And basically what I'm actually saying is that rotation around any axis can be really represented as rotation around x-axis, around y-axis and around z-axis. So any kind of rotation, whatever the rotation this particular object has, the solid object has, it can always be represented as rotation around three different coordinate axes. And as much as any movement within the space can be represented as movement along the x, y and z-axis. And that would be what we will discuss in the next lecture. Thank you very much, that's it for today.