https://youtu.be/Mu9pwGN9XzY
PHYS 1101: Lecture Sixteen, Part One
Welcome to lecture 16. This lecture, we’re going to modify our main starting equation for doing energy conservation problems. Let me remind you that when you read a problem, if you see the words, or it occurs to you that you’re connecting, or you’re comparing speed, so not velocity, but just how fast it’s going and position, then energy conservation is a great tool to consider. In the end, it’ll turn out to be the quickest way to solve the problem. Often, the problem can be solved using Newton’s Second and Third Law as well, but energy conservation will be very convenient for you. Here’s the situation.
Last lecture. I introduced this equation as our main starting equation that defines these constraints of energy conservation. This equation physically meant that we could consider forces that are acting on this object, that are working to speed it up or to slow it down, as adding or removing energy from that object. What kind of energy? We call that kinetic energy. It’s an energy of motion, if you will. The sheer fact that an object has some speed and mass represents energy, a form of energy. That energy that’s added or removed is, then, equal to this change in kinetic energy, or this energy of motion.
All we’re going to do today is realize that we can break this main equation down into a more useful form. At first, it doesn’t appear more useful, but it turns out it’s very convenient to use, once you get the hang of it. So, today’s lecture is about learning how to use this equation and learning what each of these terms means, how to translate this mathematics, go back and forth between the problem and the math.
The left side of the equation now, it’s no longer all of the work that gets done on the object, but it’s the work done by all forces except gravity, the force due to gravity. We have to do the same exercises of identifying all forces on this object, calculating the work that each one does. The only thing we’re going to leave out now is the potential work the force of gravity does. This work that’s due to this NC here, means non-conservative forces.
For our class, that means all forces, basically except gravity. That work has to be equal to the change in kinetic energy. This is the same thing we had before, but now we have a new set of terms on the right here. You see. Those terms are called the potential energy terms. It’s the change in potential, the potential energy at the final instant. Then, I subtract the potential energy at the initial instant. This last combination, it captures the impact of the work that gravity does from start to finish.
In the end… Okay. Here are the warm-up questions for this quiz, for this lecture. These have to do with circular motion and seeing if you’re developing this physical connection between these physics, quantities, vectors, these mathematical tools, and then what is happening physically to that object. So, I’ve got a comparison here. a versus b, of the same object with the same mass, that’s undergoing uniform circular motion. So, I’m showing you a snapshot where that object is at this location. Based on what you see in these two drawings and how they differ, which of these two objects is moving faster, has the higher speed?
Question 2: Focus now on a ball on a rope that’s being swung around vertically with a rope. So, this is you’re standing up, watching this. This is the side view. This is the view you would see then. The ball’s going down, up. The force of gravity is straight down, just to orient you. Question 2: When the ball is at the highest point of its circular motion, which of these two sketches has to represent going the fastest, the largest speed at that instant? For the same scenario, the ball going around in a circle, which of these two scenarios represents the ball going the fastest again? Now though, you notice the snapshot to consider is when the ball’s at its lowest point, as it goes around this circular path. Does a or b, given the way I’ve drawn the schematics, represent the larger speed?
Here’s the overview of last lecture. We began our discussion of using energy conservation to solve physics problems, to determine aspects of the motion from a problem, and here was the concepts behind it. We know now, in this class, what we mean by forces. Some of the forces on the object are there to provide an equilibrium condition. For example, if it’s an object sliding along a surface, the vertical forces do nothing more than balance to provide equilibrium along that direction.
Some forces though, or some parts of the forces, some of their components, do directly contribute to speeding up or slowing down an object. This means that these forces have some component that’s in the direction of motion. The name of the game with energy conservation is to focus on only those parts of the forces or those forces that are in the direction of motion. We can calculate what’s called the work that they do on the object.
So, every single force, as I draw here my free body diagram as an example, I may have a collection of forces. For every one, I need to consider the work done. That means I need to have a clear picture of the direction of motion. The clear way to indicate that is with this displacement vector.
Then, to calculate it, for one particular force, f, the work that it does is going to be equal to the magnitude of f, the hypotenuse, if you will. I then multiply by the cosine of the angle, and then times the distance that the object moves. This is from the initial snapshot to the final snapshot, from start to finish. What angle is that? Well, drag your force over to where you’ve got your displacement vector drawn, and place it there like hands on a clock. The two tails are together. Then, draw the arc starting from s, and swing around until you hit the force vector. That arc you’ve swept out is the angle, Theta. Turns out, it doesn’t matter if you want to sweep in the counterclockwise direction, or if you swept around this way. When you take the cosine of either this smaller angle or the cosine of the bigger angle, you get the same number.
This scenario not only gets you the right value for the energy, but the sine, it will also be correct. It will tell you if this force is adding energy to the object, meaning this number will be positive. Or is it taking energy away from the object? In which case, this number is going to be negative. That sine is properly represented or taken into account by using this cosine of the angle as our mathematical tool. It gets us not only the right part of the force, but if the force… And also then, the right direction of the force compared to the displacement. If the force is, in general, in the direction with the motion, it’ll end up adding energy. If the force, like friction, is opposing the motion, it’ll end up taking the energy away.
Okay. Then, that energy. What’s going to show up on the right side of our basic energy conservation equation is these kinetic energy terms, because that work that gets done is equated to or is represented by the change in kinetic. At any instant, I can calculate the kinetic energy of an object as being one half times the mass of that object times its speed squared.
So, to emphasize, this is the speed at an instant. What we’re going to care about is how the kinetic energy compares from the instant that defines the start of our problem to the kinetic energy at the instant that defines the last instant of our problem or the end of our problem. So, the initial and the final are these two snapshots that sandwich the scope of this motion that’s occurring in the problem.
That balance of the energy added or taken away, having a net effect in terms of changing the energy of the object, is represented by this basic equation. This, therefore, is the fundamental starting equation and the equation we want to always work with as we solve the problem. So, it’s all about starting with this equation and customizing it to our particular problem to solve for it. We need to stick with this equation because this is what enforces this physics constraint that all of the energy added or taken away equates to this change in the kinetic energy.
So, what we’re going to learn today is a more useful form of this starting equation. And now, the problems that we did last lecture, for example, can be solved with this equation, as we saw last lecture, or in fact more easily, they can be solved with this new equation, the slight modified version I’m going to show you today.