 Hi, I'm Zor. Welcome to Unizor Education. Today we will talk about one particular mechanical device, Pentaloom, which everybody saw many times, I'm sure. But we will investigate how this particular mechanical device is working, what are the laws which govern its behavior, its parameters, whatever. And we will approach it strictly, mathematically, physically correct as far as we can do it. Now, this lecture is part of the course called Physics for Teens, presented on Unizor.com website, together with some other courses. For instance, Maths for Teens, and US Law for Teens, and some others. So, all these courses are completely free, there are no advertisements. So, I do recommend you to watch this lecture, and any other actual lecture of this and other courses from the website, because for every lecture there is a very detailed notes, and the lectures are arranged in certain logical sequence to make it a course. So, whatever we are doing right now, Physics for Teens, that's a relatively complete course of Physics for high school level students, which starts with mechanics and ends with whatever, pictures of atoms or whatever. Anyway, so, let's talk about pendulum. Today we'll talk about pendulum. Very simple device, everybody knows what it is, so we will define it as follows. We have certain vertical, and when I'm saying vertical, it means we are on the planet, like this planet Earth, for instance, and the vertical is obviously the direction towards the center of the Earth, because this is the source of gravity, and the force of gravity is the main force participating in the oscillation of the pendulum. Now, at some point, we fixed one end of the thread, and at another end we put some kind of a weight. This is a point object, which means it has a zero dimension, and certain mass, let's say mass is m. Now, the thread has certain lengths, obviously, we will call it L, and the way how we will operate this particular pendulum, which sometimes is called mathematical pendulum, because it's really an abstraction. This is not really a point, obviously, object. This is some real object, and whenever we are talking about real parameters, everything is changing, but right now we're talking about ideal mathematical pendulum, which is this is the point object of certain mass, and what we do is we will tilt it by angle phi from the vertical, and let it go. Our question is, what will happen? Well, I mean, we all know what will happen in the qualitative sense of this world. It will start oscillating back and forth, back and forth, and considering this is a mathematical pendulum, which means we ignore completely air resistance, some kind of a friction in this thread, etc., thread is considered to be non-stretchable, obviously. So, it will indefinitely make these oscillating movements. So, now the question is how we can really say what kind of quantitative parameters of this movement. Again, intuitively, it's probably obvious it will go from the angle phi to the right, to angle phi to the left, and then back and forth, back and forth. We are probably mostly interested in the period of these oscillations, and actually what's probably the most important characteristic of this movement is this function phi of t. Now, this angle probably should be symbolized as phi zero, which means it's an initial angle of tilting this particular thing. But as the angle, this angle is changing with the time, this basically defines the movement of this pendulum. So, that's our purpose to find out this function. Okay. Now, we should actually impose certain initial conditions, because as you know, lots of mechanical motions can be expressed using whatever the laws of physics are, like, for instance, Newton's second law, can be expressed in some kind of differential equations, and to solve differential equations, we need initial conditions. So, what kind of initial conditions we can impose on this particular function, phi of t? Well, first of all, obviously, phi of zero is equal to initial angle, right? So, that's at the moment t is equal to zero, because we have tilted and then let it go. Now, the fact that we let it go, it means we didn't push it in any direction. Now, pushing is something like initial speed or whatever. I mean, then it goes by itself, but initially we can't push it, but we didn't. So, we have to somehow specify that we didn't really push it. Now, what does it mean that we didn't really push it? It means that linear, this is trajectory, right? So, the linear speed of this point object at the end of the thread initially is zero. That's basically what it means we don't really push it. So, there is an initial speed along this orbit and it's zero. Now, how can I express linear speed along the trajectory on a circle? Now, this is obviously a circle, right? Since this is a thread. So, how can I express in terms of function phi of t that the linear speed along this trajectory initially is equal to zero? So, how speed along the circular trajectory can be expressed in terms of the angle? Well, very simply, as we know the length of the arc is always equal to radius times angle, right? So, if you have an arc, this is angle alpha in regions, obviously, this is radius r. So, this length is r times alpha. Okay, so, we know that which means our linear displacement is equal to l times phi of t, linear displacement from the beginning to position at moment t. Now, the derivative of this is my linear speed. l is a constant, so it's a derivative of phi of t. So, that's my linear speed in terms of angular displacement and the radius l of this thread, which means that I can say that my at moment zero is equal to zero. That's what it means. And now, we don't really need l because it's a constant. So, this is the second initial condition. So, the function phi of t has two initial conditions, one on its value at point t equals zero, another value of its derivative at point t equals zero. So, the initial tilt is equal to phi zero, initial speed is equal to zero. Okay, now let's talk about what kind of equation we can impose on the movement of this particular object. Well, what kind of forces are we are dealing with right now? Well, obviously, the first and most important force is the weight, right? This is mass of this object m times acceleration of the free falling. Probably, we're talking about planet Earth, so g is 9.8 meters per second square. Mass is whatever the mass is. That's the most important point, most important force, which is acting. And as we are considering, let's say, initial condition like this, what actually forces this particular point object to still be at length l from this point and at the same time to move along the circular orbit, circular trajectory? Well, very simply. What we have to do is we have to represent my weight as a superposition of two forces, one going along the along this thread and another going perpendicularly to the thread. So, this would be r and this would be f. So, basically, my weight is equal to a combination of function f plus r. Now, this point is always at the length l, which means that this component, the r, it's not stretching because this thread is not stretchable, which means that there is a reaction force r0 or whatever one we call it, which is exactly equal to the component r but directed in the opposite direction. And they are balancing each other. This is the reason why this point object is always on a constant length l from the point where the thread is fixed. And this component is the main component which actually forces our object to move along the circular trajectory. Now, obviously, this is angle phi. Now, this is also, now we can actually talk about phi, not phi0. So, it's some kind of a position in between. So, this is also angle phi. So, my function, my component f is equal to pi times, this is phi, this is phi sine. So, this is mg times sine of phi of t at any moment t. So, it means that this particular force is changing in both direction because it's always tangential to a circle and in magnitude because phi of t is changing in magnitude. Now, this component r, it's correspondingly, is equal to mg times cosine of phi of t, also changing in direction because we are moving. So, the r is directing this way or this way or this way and it's also changing in magnitude depending on phi. And we can actually completely ignore this force r, we don't really need it. We need really only one force which is the one which basically forces our pendulum to go along this circular object. And since we know the force, we can put together a second law, Newton's second law because acceleration along this particular orbit, we can definitely express in terms of function phi of t. So, we were already talking that the velocity is the first derivative from the L times phi of t. L times phi of t is basically the distance traveled until the angle becomes phi of t during the time t. So, this is the first derivative. So, this is basically absolute value of the velocity and the second derivative will be this one again because L is a constant. So, the second derivative is acceleration along the circular trajectory. So, at any moment of time, this force course is the cause of this acceleration of this particular object. So, let's just express it as a second law, Newton's second law. This is my force. Now, this is acceleration. So, we have to multiply mass times acceleration to get the force. So, it's mass times acceleration L times phi second derivative of t. They must be equal to each other. However, I would like to make one very important note here. As the angle is diminishing, force is always acting towards diminishing of this angle. So, whatever the position phi of t at any moment, then the force goes this way and it's trying to diminish this angle. So, the force must be with a minus sign. Otherwise this thing will not be a true equation. So, that's very important. The force is towards decreasing the function phi of t. So, that's why we're always talking about diminishing its speed and diminishing its acceleration. So, that's why acceleration is always negative in this particular case. It's always towards diminishing of the angle. And what's another very important thing is this m, m, which means what? Which means that our equation, and correspondingly the function phi of t, is independent of mass. So, it doesn't really matter how big, how heavy this particular object is. The heavy pendulum will have exactly the same type of oscillation as the light one. It will do in sync. So, if we will just take two pendulums like this, one with a heavy point object, another with a light one, and just tilt it at the same angle and let it go at the same time, they will do synchronously oscillations. So, the function phi of t, it's a solution to this differential equation, and m mass does not participate. So, all we have to do is to solve this particular differential equation. So, let me just write it here. I will do something like this. Okay, this is our differential equation. And solution to this equation is our function phi of t, which we are trying to determine. Now, obviously, it looks like it depends only on how the planet attracts objects on its surface, the freefall acceleration g, that's the property of the planet, right? On Earth, it's 9.8, and it depends on the length of the thread. Nothing more, everything else is dictated by this differential equation. Now, the good news is that we have a second order differential equation, right? Second order, because it's a second derivative here. And we have two initial conditions, the initial condition for the function and its derivative, and from the theory of differential equations that should be sufficient to find a solution. Well, that's the good news. Another good news was, by the way, that we have cancelled out the mass. So, the motion is independent on the mass of the point object at the end of the pendulum. Now, let me give you the bad news. The bad news is that we cannot solve this differential equation in simple algebraic functions. You cannot say that some algebraic function phi of t is equal to something square, something square root, something multiply, something divided. You cannot express in any kind of a traditional mathematical functions, trigonometric function, exponential function, whatever else you can do. You cannot express your solution in these explicit terms of nice algebraic functions, which we get used to. Well, that's quite unfortunate, I would say. So, if we want to know exact solutions to this thing, there is only one way. Use the computers and use some numerical analysis to basically plot the function as it goes as a function of t. So, for every specific time t, we can find the value of this function, but we cannot express it on the computer, but we cannot express it in any kind of a nice algebraic formula. Well, you might say that you are surprised that there is something which we cannot really express in normal mathematical terms. Well, don't be surprised. I would say that majority of the things which are really happening in the nature are much beyond in their complexity to abilities of mass to nicely express it. Well, say thanks that we have a differential equation and we have certain numerical procedures using computers to basically calculate the function phi of t for any specific time. But as far as expressing it in some nice way, probably most of the really occurring in nature things are not that simple. The law of gravitation is approximation. The second law of neutron is approximate. Everything is approximation to a certain degree. And even such a simple mechanical device as ideal pendulum, and I'm talking about ideal pendulum which the thread is not stretchable, you know, this type of thing. Our object is a point object, so it doesn't have any dimensions. Even in this ideal case, so-called mathematical pendulum, even here we have an equation which cannot be solved. However, what we can do, we can approximate the solution of this differential equation with another differential equation which is, well, kind of close to this one and we can solve it. And that's actually what I'm going to do right now. Here is my point. Now, if you went through a course of mathematics, especially if the one which presented on unisor.com mass proteins, you must be familiar with this particular limit. So, if you have graphics, this is sine and this is y is equal to x. So, here they are very, very close to each other. So, if my angle is in relative proximity to zero, if it's a small angle, so to speak. For a small angle, the difference between sine x and x is very, very small. It's nonexistent. I mean, it exists, but it's so small that we can really kind of ignore it. And instead of using the sine of something, of an angle, we can use the angle itself. Now, obvious question is what exactly are the boundaries of closeness which we, which we can really feel comfortable about if we replace sine of something with that something itself. Well, traditionally we are saying something like from minus 15 degree to plus 15 degree in this interval, it's really allowed to have sine of the angle to be replaced with an angle itself without much, without much difference actually. Which means that if our initial phi zero, if our initial tilt is really not a big one, it's not like 45 degrees, but something like 15 degrees. If our, so this is 45 degrees approximately, right? Now, 15 should be one-third of it, something like this. So, if we just a little bit tilt the pendulum in the very beginning and then let it go, then probably this particular type of motion can be approximated with replacing this differential equation with this one. So, I replace sine of phi of t with phi of t itself. Now, this is much simpler differential equation. This is the differential equation we can actually solve without any problems. And again, during the course of mass for teens, I did consider something like this as an example of differential, linear differential equation of the second order which can be solved. So, right now I'm not going into the details of how to solve it. I'll just put down a solution. Solution is phi of t is equal to C1 cosine square root of g over l t plus C2 sine of square root of g over l t. This is a general solution of this equation. Let's just verify that this is true. The first derivative of this would be C1. Derivative of cosine is a sine of square root of g over l t times the inner function which is just a multiplication by a constant should be differentiated and the derivative is square root of g over l plus C2. Derivative of sine is a cosine of this times derivative of inner function which is again square root of g over l. That's my first derivative. Now, my second derivative, so this is first. My second derivative is, well, this is a constant. This is a constant. So, we have C1 times square root of g over l is a constant. I think I made a mistake. Now, this should be with a minus sine. Sorry, minus C1 times derivative of cosine minus sine, not sine. Derivative of sine cosine, that's okay. So, we still have a minus here. Let's see if something is wrong. Times derivative of sine which is a cosine of square root of g over l t times derivative of the inner function which is also square root of g over l and we have already square root of g over l. So, if I will multiply, I will just get rid of the square root. Then this one, derivative of cosine minus sine. So, it's a C2, this constant, square root of g over l, cosine would be minus sine. So, minus here, sine here, square root of g over l t and inner function, it's square root of g over l and I will get rid of this. Now, what we have, the second derivative is equal to this and how this compared with this, that's exactly the same thing. You see minus g over l minus g over l minus g over l multiplied by C1 cosine plus C2 sine which is exactly our function 5 t. So, this is a solution. Great. And again, I did not want to get inside of how I obtained this solution. If you are interested, I suggest you to go to any textbook on differential equation or to the course, Mass 14's differential equation chapter where I discuss this. So, this is a solution, proved it. Now, what are C1 and C2? Well, we have initial conditions which will help us. So, if I will substitute 0 instead of t, I should have phi 0. Now, I substitute 0 here, that would be sine of 0 which is 0. And now, if I will substitute this 0, it will be cosine of 0 which is 1 times C1. So, it's C1 is equal to phi 0. Now, the first derivative, I wiped it out, but I can just write it again. C1 is equal to phi 0. Now, my first derivative is equal to minus C1 square root of g over l sine of square root of g over l t plus C2 cosine square root of g over l cosine square root of g over l t. That's my first derivative, right? And if I substitute t is equal to 0 into first derivative, this will be sine of 0 which is 0. This will be cosine which is 1. So, I will have C2 times this square root is equal to 0. So, C2 is equal to 0. So, we found a particular solution, not just a general solution, which is this one, but a particular solution which satisfies our initial condition. C1 which is phi 0 cosine square root of g over l times t and C2 is equal to 0. So, this is a solution. This is how our pendulum would actually oscillate. This is the dependency of the angle of tilting from the vertical as a function of time. But again, don't forget that this is only within certain very small angles. So, our initial phi 0, our initial tilting, should not be really very big. As I said, no more than like 10-15 degrees, which is relatively small. Then this is true. On a bigger angle, it's not working. So, whenever you're presented with the problem about pendulum and they're saying use whatever the formula is for pendulum oscillation like this one, you have to understand that, well, it's working more or less well within only certain restrictions on the initial angle not to be too big. Now, our problem can be formulated differently. For instance, instead of tilting at some initial angle phi 0 and then let it go, and then it goes by itself, we can impose different initial conditions. For instance, the pendulum is hanging vertically and we just push it, give them some initial speed along the trajectory. So, this would be 0 and this would be some constant. And that would result in different C1 and C2. They will be different law. But it will still be some trigonometric style. If this is equal to 0, it means that C1 should be equal to 0, right? And if this is equal to some constant, then C2 would be equal to something. So, it will be some coefficient times sine of this particular expression. So, just different variations of this general solution to cater to the particular initial conditions of our pendulum. And finally, I would like to basically say a couple of words this. Now, this is the cosine, right? It's a very kind of well-established trigonometric function we know all about it, etc. Cosine is from minus 1 to 1, but if multiplied by phi 0, it will be from minus phi 0 to plus phi 0, right? So, phi 0 is an amplitude. Now, how about the period? Now, period is very important for a pendulum because the period is something which we always talk about. What is the period of oscillation of this pendulum because the pendulum is used, for instance, in the clock, right? Old clocks, grandfather's clocks, right? There was a pendulum. Now, it's very easy. Now, we know from the theory of trigonometric functions that if you will multiply argument by some multiplier, then the period will diminish. So, from 2 pi, it will be 2 pi divided by this particular coefficient or multiplied by its reciprocal. So, this would be a period. This is obvious from the function from the analysis of the function cosine. So, again, if you multiply periodic function like cosine by some constant, then obviously the period will be diminished by this constant or I will multiply by its reverse. And this is the formula which is given to students in many textbooks on websites, etc. Well, just know that this formula is very approximate. It's based on the solution, on a simplified solution of our initial differential equation. So, it's true only for very small oscillations around the vertical. Well, that's basically it. That's all I wanted to talk about. And again, I would like to point out that the nature, even in such simple devices, this pendulum, is very complicated. And it's so complicated that even all the contemporary mathematical functions which we are typically familiar with cannot cover such a simple motion as oscillation of the pendulum. So, just have an appreciation to nature in this case and realize that our knowledge is really kind of restricted. Let's put it this way. Although we do have certain ways to analyze something which is not really algebraically well expressible, we have this power of computers and we can probably solve this particular differential equation numerically, which means this function will be numerically evaluated for every moment of time t. We can find its value again to a certain degree of approximation. Okay, that's it. Thank you very much. And I do suggest you to go to the website and just read the notes to this lecture. I think it's very helpful. Thanks and good luck.