 Hi, I'm Zor. Welcome to a new Zor Education. Today we will talk about mechanical power. This is by the way its definition, but we will talk about this. This lecture continues a couple of lectures which precede this about mechanical work. Now, the whole set of these lectures is part of the course called Physics 14 presented on Unisor.com. It's a completely free website where all the lectures about math for teens, mathematics for teenagers are presented as a pre-course for this particular course of physics. So, you really have to be relatively familiar with simple concepts of mathematics needed for the physics course, which is primarily vectors, calculus on a relatively simple level. So, if you are not familiar with these subjects, I do suggest you to take the course Math for teens on the same website. Now, if you have found this lecture somewhere on YouTube or any other website except Unisor, then you will definitely have the benefits of listening to this particular lecture. However, again, this lecture is part of the course and the course is presented in its logical sequence on the website Unisor.com. So, that's where I suggest you to go. The site is free. There are no advertisements, no financial strings attached. You don't really have to even sign in if you don't want to. Okay, now, back to power. Let's imagine that you are traveling along some trajectory. Now, as you are traveling, you are covering certain distance. And obviously, there is a function, how much you have covered based on the time. So, from some initial point, you travel during the time t, certain distance, and the function distance as a function of time basically describes how you travel along the road, along the trajectory. Now, I would like to make a similarity between this, between the traveling along the trajectory and the force which is acting upon object, forcing it to move and doing some work and the work which we have already discussed before. So, you remember that the work basically is the force times the distance. Well, obviously, there is a little bit more complicated case when the force is variable and the distance is also some kind of a function. But anyway, if there is a constant force which is applied during certain distance and forcing basically an object to move this distance, then that's the work which this force performs. So, again, the object is traveling and the distance is a function of time. Now, the force is acting on the object and the work which is performed by this force is also a function of time. As the motion continues, you have both this and this functions. The distance covered and amount of work which the force which initiates this distance and causing this distance, amount of work performed, both are monotonously increasing functions. Now, we usually attempt to characterize motion not only by the distance which is covered but also by the speed which is the first derivative of the distance by time. This is the rate at which we are covering the distance. We are making some kind of an infinitesimal time frame from t to t plus dt. Measure the distance and the distance covered is differential of the function s of t and then we divide basically these two things. And whenever we are making our time interval shorter and shorter, we basically come up with the derivative, obviously. So, this is the rate at which we are moving. Now, the power is actually the rate at which this force performs the work. So, basically the power as a function of time is the first derivative of the work by time. So, as we progress along the road, this is the instantaneous speed at which we are moving. As we are performing, as the force performs certain work, this is an instantaneous rate at which the work is performed. Sometimes we have to produce more work during the unit of time, sometimes less. Just as an example, if you consider, for instance, the car, it starts from the state of rest, it first accelerates and for which you definitely need more power because your force is stronger, right, to make an acceleration. At some point, you have achieved your maximum speed and you go along the straight road, force is very, very minimal. Basically, the force, if the car is moving with a constant speed, is actually to work against the friction of the wheels and the air resistance and then both are relatively minimal. So, at that particular moment, the power which your engine is supposed to produce in the car should be less. And if you notice, you have, for instance, the, how is it called, RPM, revolutions per minute. And this is the glitch on your dashboard in the car. And usually, in the beginning, it goes all the way up and then as the car actually moves along the same road in a straight line motion, the revolutions per minute are going down, which means the power which is basically the engine exhaust, then basically it's much less. So, the power can be more or less. It's the rate at which the work of the engine or force, whatever, it's the rate at which this work is being done. So, this is the definition of the power. Okay. Now, let's consider a relatively simple case. You have the car which has achieved its maximum speed and goes along the straight line against the forces of friction and air resistance. Now, with a certain speed, these are, well, relatively known variables. I mean, we can measure it somehow. But anyway, let's assume we know that there is a force which engine is supposed to develop to force the car to go in a straight line with a constant speed against the friction and against the air resistance. All right? Now, let's say it covers certain distance s. Then obviously, the work which is performed is this. Now, what is s? If our speed is constant, then the distance is equal to the speed times the time, right? And now, if we are talking about, now, and this is not just w, it's basically w as a function of time, right? So, at the very beginning, t is equal to zero. We start from this particular point. And then, as the time goes, the work developed from that moment forward is basically calculated using this formula. f is a constant because there is a constant air resistance and constant friction of the wheels. d is a constant. Again, we have basically postulated this v is constant. This is the function which represents the work which is being done by the engine. So, what is the power in this case? Well, as I said, this is the first derivative of the work by time, which is the first derivative of this function, which is f times v. These are constants, both. So, the power is constant. We can actually drop as a function of t and say that, okay, if the car moves along a straight line with a constant speed, the engine is supposed to produce a constant power, which means a constant amount of work is being performed during every unit of time, during the first second, during the second second, etc., etc. So, this is a very nice formula, and as I was saying, this formula is actually applicable to the case whenever you are a constant force and constant speed and then the constant power. Well, let's make this a little bit more complicated. What if the speed is not constant and the force is not constant, etc., and again, the perfect example is the car, which starts from the state of rest, accelerates, and moves forward with different speeds, obviously, etc. So, again, we will start with the function s of t, whatever that s of t is. I don't know the way how it's calculated, but whatever the way it is, it's some kind of function. Now, obviously, speed would be a derivative of this function by time, okay? Now, as I was saying, in the beginning, car is accelerating, so there is even acceleration. Now, what is acceleration? Again, it's a function of time, generally speaking, and if you remember, acceleration is the first derivative of the speed by time, right? So, this is the first derivative, and this is the derivative from the derivative, which is actually the second derivative of the function s of t, right? So, this is how the distance is measured. Okay, fine. Now, I don't know really the value of the force. However, I do have the Newton's second law. At any moment of time, if this is my acceleration, now this is my force, the mass, I suppose, is constant. So, well, to tell you the truth, mass is not exactly constant if you are talking about the car because the engine is working and there are some exhausts, et cetera, et cetera. But let's assume it's minimal and let's assume that the mass is constant. Now, in a similar case of the rocket, it's even more obvious that the mass is not really constant because there is a huge amount of gases which are going out and the mass is basically diminishing. But again, this is the case when we have a constant mass and some kind of a force actually forces this particular object, car or whatever else, to move forward. Okay. Now, if this is the case, then what we can say about the amount of work which is performed, we are talking about power, right? So, the power is amount of work during certain infinitesimal interval of time divided by this interval of time, right? Dt is differential of time. So, that's what we would like to somehow come up with. Now, let's think about what is the differential of the work. The amount of work which this force, f of t, performs during infinitesimal interval where our object is moving by the distance, this differential of s of t. So, the increment of the distance, since we are talking about infinitesimal interval of time, we consider function f to be constant, obviously, during this infinitesimal and equal to f of t from the moment from t to t plus dt. So, f of t is considered to be constant. Now, ds of t is basically, what is ds of t? It's f of v of t times dt, right? Now, as we know, f of t is mass times af t times v of t times dt. So, this is a differential of the work at times t. So, if this is a differential during the time increment, basically, infinitesimal increment during the infinitesimal increment of dt. So, the first derivative, which is this, equals to m times v of t times af t. Mass is constant, we assumed, right? So, if my mass is moving forward and v of t is at speed and af t is its acceleration, not necessarily constant, acceleration also might change. Again, for instance, when you are starting the car, first acceleration is relatively high and then, as you have achieved the maximum speed, acceleration will be zero. So, acceleration is variable and speed, obviously, is variable as well. The mass is the only which is a constant. And this is the power. So, the power, in case of a non-uniform motion of the object of mass m, where v of t and af t are, basically, it's, well, geometrical, if you wish, characteristics of the motion. All of, both of them are basically related to this function. So, the first derivative of this is the second derivative from the distance as a function. So, if we know the distance as a function of time, we can actually calculate what power required to achieve this particular movement of this particular object of mass m along the distance if this is the function, distance of time. Because these are just two derivatives of this function. So, this is the formula for a non-uniform motion. And I specifically put as a function of t in both cases, because sometimes you can find this, which is correct. However, this doesn't really tell you that all of these except mass are actually functions of time, which means as the speed is changing or acceleration is changing, power consumed by this particular movement is also changing. Okay, that's basically it. Now, let's talk a little bit about how we measure the power. But, again, let's start from the work from which actually the power is derived. The work is measured as force times distance. It's measured, obviously, in newtons and the distance in meters. So, this is the unit of work which work is measured. And this is called JOL. So, one JOL is the work performed by the force of one Newton which is performing the motion of one meter. Okay? Now, since power is basically a rate of performing work per unit of time, and the unit of time is second, so the power which is actually differential of this by this. So, this is measured in JOLs. So, it would be JOLs divided by seconds. And JOL divided by seconds is called WAT. Now, WAT is a unit of measurement of the power in this international system of measurements, standard physics, standard of measurements, C. Now, obviously, there are some derivatives like kilobat, which is 1000 Watt and megawatt, which is 1 million Watt. And the game is all named in honor of James Watt, the Scottish physicist of 18th century, which did a lot of experiments with what kind of measurements of the power we can actually perform at that time. Now, one of the very interesting things is that he was measuring the power of the horse, and basically suggested another unit of measurement, the horse power. Now, the horse power, well, unfortunately, there are more than one definition for a horse power, but at least I know two, for instance. There is a metric horse power and there is a mechanical horse power. Now, metric horse power is the power which is needed to lift one kilogram of force, which is actually 75 kg of mass times 9.8 m2, right? This is M and this is A. So, this is 75 kg of force with a speed of 1 m per second, which means this is about 735.5 Watt. So, this is a horse power, which is called metric horse power. Now, mechanical horse power probably was the one which James Watt was talking about, because it has older units of measurement. It's actually 33,000 pound feet per minute. So, time is measured in minutes, not in seconds. Force is measured in pounds and the length is measured in feet. So, if you will translate this into, again, the units of C, which is in watts, it would be a little bit different. It would be 745.7 Watt. I think this one is actually used in many cases. Whenever you're translating horse power into, I think whenever people are saying horse power, they probably meant mechanical horse power, but I'm not really sure, frankly. It doesn't really matter right now. So, in any case, these are two old historical measurements of the power and the cars, whenever the cars engines are measured, they're measured in mechanical horse power. So, whenever you're talking about the car which has a power of 200 horsepower, that means 200 times 745.7 Watt. Okay. Now, we finished with units of measurements. What is the one? And now we'll talk about rotational movement. Now, rotational movement is slightly different, obviously. But if you remember the kinematics and dynamics of rotational movement, there is a lot of similarity between the uniform motion along a straight line and uniform rotation. So, I'm talking right now about uniform rotation only. Now, let's consider you have a well, let's say a well. And this is your thing and this is your bucket. So, there is something which turns this wheel. Let's say the wheel has a radius r and the constant angular speed omega. Now, what does it mean? It means that the speed we are lifting the bucket is v equals to r times omega. So, this is the linear speed on the surface of this cylinder and this is basically the linear speed of the bucket as it's being lifted from the well. Now, let's assume that we are lifting this bucket with a constant speed. So, this speed is omega is constant. Now, that means that there is no acceleration, which means that all the forces must balance each other. Now, the force down is m times g. That's the weight. So, the force up which is developed by rotating this wheel is equal to mg and it's constant. So, this is constant, this is constant. Okay, now, power means f times v as we were saying in the previous part of this lecture and this is also constant and it's equal to mg r omega. Okay, now let's talk about different things. You remember that whenever we are talking about rotation, we usually assume that angular speed would be very similar to linear speed in the uniform motion along the straight line and the torque would be similar in some way to the force which is developed during the motion along the straight line. So, now, let's think about this. mg is the force, right? This is the force and r is the radius. So, what is mg times r? That's the torque. That's the definition of torque. So, I would like you to actually look at this formula and compare it with a similar formula for linear motion. The linear motion we were talking is equal to force times speed and the same power exactly can be measured from the rotation of the wheel as the torque times angular speed. You see? Instead of force, we can use torque. Instead of speed, we can use omega. We don't really know these two things. Well, the force actually, we don't know. But the speed we don't really measure, we measure everything in terms of angular speed, right? We don't really measure the speed of the surface of the wheel. So, the angular speed and torque are characteristics of the power needed to do whatever we were doing which means uniformly moving upwards this bucket of water. Now, let's assume that there is some kind of an engine here and this is the shaft of this engine. Now, usually if there is an engine which is manufactured, let's say it's an electric engine, an electric motor, then usually there are characteristics to this motor, the power and the torque. Same thing if you remember in the car engine. The car engine has two characteristics, the power, the power and the torque, right? So, this is how they are related. The power and the torque are related like this. Now, I was comparing how this formula actually colorates with real numbers produced by car engine manufacturers. So, in the text which is accompanying this lecture on Unizor.com, I actually put a graph which is just taken from, I don't remember which car engine I took, where the power and the torque are actually specified for different angular speeds, different numbers of revolutions per minute. And I do the calculation and, lo and behold, this is approximately correct. Well, approximately because the car engine is not really like ideal engine. It has certain limitations, etc. But during, you know, certain point in this graph, this is a true formula to the degree of precision which basically I consider acceptable. So, this is the correlation, this is basically dependence between these two very important parameters of any engine which basically does the rotation like electric motor or car engine, etc. The power and the torque are related to angular speed. Well, I encourage you to take a look at this text which is accompanying the lecture on the Unizor.com. It basically contains whatever I was just talking about. But it's in writing, so basically when you're reading this, it's like a textbook. And then, well, you can start solving certain problems which I'm going to put into this course. So, the next lecture will be dedicated to the problems with power. And as usually, I consider solving problems a very important part of this course. That's it. That's it for today. Thank you very much and good luck.