 So the topic of this video is a review of the basic principles of work and energy. And really positioning where we are to get to this topic is basically you start with, we're starting with statics and dynamics, which builds into mechanics of materials, which is a strong foundation or a main motivating foundation for what we talk about in machine component design. And so I'm really going to do a review of these principles of work and energy because they they talk about or influence a lot of what we'll do later as we get into more topics in machine component design. So I'm going to go ahead and get started on that and kind of talk about what we're what we're looking at here. So principles of energy, basically energy and work there, I mean work is a form of energy. So we have work, which is, if I get my mouse in the right place here, work is equal to the integration of a force over a distance. And basically what that means is if we apply a force to a mass or something and move it over a distance, we're doing work on that something. So if I come in and I say, okay, I have some arbitrary mass and it has a weight, I'm not going to worry about that too much. But I'm going to apply a force to it F. And that's going to result in it moving some distance D. Then I can say that I'm changing its position, right? I'm changing position from, let's say R one. And we're moving it and it goes to some new position, R two. So then back here in my integration, I'm doing work over F. So as F is applied over this distance or change in distance D, R. So I'm applying the integration from R one to R two. And if I can assume that my force is constant in that it's not changing, well, what happens when we have constants in integration? If we have a constant, then this force F can go outside the integration. We integrate from R one to R two over DR integration of DR is just R apply from R one to R two. I think you get the idea. And basically what we would then have is, okay, well, if we do all of that, F is constant and blah, blah, blah, we can resolve for this or excuse me. So we have work then is equal to F times R. Sorry, my pen is being difficult. I'll figure this out as we go equal to R two minus R one. So that's just the basic integration, which as I've defined R one and R two over here, that's just D. So this is equal to F times D. So force times distance. So that's pretty straightforward, I think. Now, of course, then we can see that units of work are force and distance. So Newton meters, foot pounds, depending on whether you're in SI or engineering, American engineering units. And that I think makes sense. Of course, we can do the same thing if we're talking about rotation. So if I have a rotating object or rotating system, I'll come over here and do this. I have a circle. And it's rotating about some center. And that circle has a radius R. There's a torque applied to it. Or equivalently, I don't even need to necessarily say there's a torque, I can say I'm applying a force F to that circle. And my torque then slowly I'll figure out how to make this work nicely. The equivalent system would have a torque. And that torque, of course, is equal to F times R, because it's just a force applied at a distance. Now, if my point that I apply this force, say it's at the top here, it is rotating around the circle. So I'm just saying that point is moving. That's something we call arc length, which is the same thing as saying r theta. So arc length or arc position S is the same as a rotation around a circle by some angle theta, if that circle has a radius R. Therefore, I can come back and try to formulate work for this, right? So work, which is a force moving over a distance, what's that distance? Well, that distance is S, right, as I move around the circle. That's how much where my position that I apply the force to is changing. So force is distance, which means that if I solve my equation T equals FR for force, I can write this as T over R S. I write as R theta. And we can see that the R's cancel. So then I'm left with work equals T theta. And that kind of makes sense, right? If we think about rotational translation of the linear version, the linear version work equals FD, right? Force times distance. The rotational version is torque times theta. So load applied over distance that it changes. So we can go ahead and see that that should work for us. Now, we're not always exclusively interested in work. So I'm going to come back here and just remember that this is work. And we're not exclusively interested in work, right? Because many times we want to know how long it's going to take to do something. And how long it takes to do something requires some information about power. Power is work per unit time. So if I go ahead and come up with a formulation for power, typically I'm going to write power as W dot. The dot here indicates that it's the first derivative with respect to time. That's just a mathematical notation. So we'll say W dot. And if I take the derivative of work, force again being a constant, distance, derivative of distance with respect to time is velocity, right? So I can write power as W v. Now, the reason we might be concerned with power is that, well, it can lead to heat buildup, right? The amount of time that I take to do work causes heat that's energy into the system. So we want that heat to be able to dissipate, which means we need to have an understanding of how much time it takes to do that work. So power then I can write as F v, which is Newton meters per second, Newton meters per second. That's the same thing as Joules per second. Newton or Joules are just another name for the Newton meters. And of course, that would also be Watts. Now, Watts are the primary unit for power in SI. If we're talking about engineering American engineering units, we'd be talking about horsepower. You know, as we've probably encountered many times before, horsepower is usually a little, or American engineering units are a little funnier to work with. So in this case, we're talking about 550 foot pounds per second. So they're not as nice, you know, it's not powers of 10 and all that. When we're talking about power, we might also then be interested in efficiency, which I'll use Ada and efficiency generally speaking is power out over power in that we can, we can have a good, pretty good understanding of. All right. So we've got work, we've got power. Next, we want to talk about conservation of energy. So conservation of energy is one of those key things that we can use then to understand what's going on in our system. Sorry about my handwriting. It's not going to be great on these notes. So you'll have to do your best with interpreting that. So conservation of energy, generally, we have two forms of energy, right? We have kinetic and we have potential. So for broadly, talking about these things, we have kinetic and potential energy. Now kinetic energy, we'll often write as Ke, potential energy Pe, and kinetic energy is the kinetic of, or is the energy of motion. Things move, they have energy. So the general basic equation for kinetic energy, one half MV squared. And for potential energy, we can write it as MGH, or one form of it as MGH. And MGH in this case is the energy of something being at a height, right? So if I have an object up here, and it's going to move down to here, that's a change in potential energy because I'm changing H. G is gravity, M is mass. So as that thing moves, it changes energy. However, we can also have potential energy in stored in a spring or stored in something that from elastic deformation. So in which case, the equation for potential energy would be one half K delta squared. Now in this case, K is the spring constant. It's how much force it takes to compress the spring or whatever it is by a certain amount. And then delta is how much that thing is compressed or distorted. It's the deflection. And so we can put all that together. Now with these forms of energy, I can write a conservation of energy equation, which is going to look like delta Ke. So that's changing kinetic energy plus delta Pe plus delta U. And delta U here is internal energy. So energy internal to the system equals Q plus W. So Q in this is heat transfer. Generally speaking, we're not worrying about that in machine component design. But as in terms of a system, there's of course heat transfer. And then W is work, work done on the system. Either we extract work from a system or we put work into a system depending on what we're talking about. So that kind of gives us a broad understanding of what we're talking about when we say conservation of energy. We have all these things happening in a system, and we need to put them together into one equation that we can then use to balance and solve for other things. So I'm going to go ahead and stop the video there and pick it up in a minute.