https://youtu.be/TWDRCc4kLbo
PHYS 1101: Lecture Fifteen, Part Two
Okay, let me start our new material now with putting it in context, giving you a big picture now of where this new tool, this energy conservation tool, fits in. Here’s our schematic of the tree again, where we’re trying to identify the basic physics that’s at the heart of a group of problems and then developing problem-solving skills that allow us to derive our own solution to a particular problem. That means we can start with the basic physics equation and we can customize that logically, appropriately, to solve a particular problem.
We’ve done this for what we’ve called kinematics, Chapters 2 and 3. We spent a lot of time with another tool, Newton’s laws, that’s Chapter 4. A special case of Newton’s laws was circular motion, that was Chapter 5. Chapter 6 is a whole new tool. I’m going to schematically indicate that with this circle that is around a set of problems that’s separate from these types of problems.
So our starting equation is going to be different, but again, I’m going to give you problem-solving steps that help us go from our starting equation, the basic physics behind all of these problems, that lead us to the specific solution we’re after. So here at the base of this tree, our starting equation is this net work, the energy added or taken away from this object, is equivalent to the change in its kinetic energy. And let me just call out that this whole collection of leaves, this whole category or type of problems, I’m going to label as “energy considerations”.
And the whole point to this lecture is breaking this down and showing you what this means and how to use it, and so how to solve for these particular problems.
Let me say some general things about trying to recognize when you need to use this tool, or when it’s going to be helpful to you to apply energy conservation to solve a problem. Here’s the rule of thumb: you’re going to see that the energy considerations I’m going to describe for you are in essence capturing the connection between forces causing acceleration, and in particular, a force causes an object to speed up or to slow down. When we did Newton’s laws in general, remember in some directions the forces balanced and I didn’t have acceleration, but yet, perhaps, in a perpendicular direction, say, along the slope, I had an acceleration.
We’re going to find with these energy considerations that we’re going to end up focusing only on the parts of the force or the forces that directly either speed up or slow the object down. What we’re going to find is that if the question asks us in any way to compare or about the speed and position at some initial instant of time compared to a speed and position at a later instant of time, that energy considerations is probably going to be our best tool to use to go after it.
Here’s a collection of some scenarios for which this is true. We have a ball that rolls down a hill, and then after rounding this corner, it’s going to fly up into the air at a certain height. We have a roller coaster at this initial position, initially at rest, that rolls down the hill, picks up speed. We have a person with an initial speed slowing down, changing their position, their height.
In all three of these scenarios, if you imagine drawing the forces on the object — let’s take this ball, for example — gravity is a force on the ball and a normal force. Notice that the ball speeds up going down the hill. It’s only part of the gravitational force, wx, if you’d like to call it, it’s only this part of the force that’s in this direction of motion at all. Therefore this is the only part of the force that is directly contributing to this acceleration, the ball speeding up. The normal force is perpendicular to that. It’s not impacting the speed, slowing the ball down or speeding it up. If gravity is the only force that’s contributing to the slowing down or speeding up, energy considerations is going to be a great tool for you to use.
So let’s motivate that with a simple picture. Let’s start out with a very generic or simple picture where I’ve got a single force on an object. I know that causes acceleration. Therefore the object speeds up. The speed is going to increase. So let’s assume it starts out at rest. This force is applied. Or let’s start it out, I guess, heading to the right, some small velocity. Because of this force, I know from Newton’s Second Law that I have acceleration to the right, meaning I have a horizontal change in my velocity. So this horizontal velocity then just directly gets bigger. From the initial time after the ball has moved some distance, it’s now going faster because of that applied force.
And that’s what the energy consideration directly connects. It’s the connection between the force acting on this object over a distance and the implications of that in terms of how much it has to speed up. Okay?
So if you go to your textbook, they’ll show you a detailed derivation of this equation that we’re going to talk about. I’ll save you the details of the derivation, but let me just say something in general, that this equation is nothing more than the logical consequence of Newton’s Second Law, which tells us the connection between forces and acceleration, and then some kinematic information, which tells us the connection between a and the displacement, the x minus x0.
You remember that third kinematic equation that we used on occasion, which was v2 equals v02 plus 2a times the displacement? Let me draw that up here. That was equation 3. That’s the connection that we’re combining with Newton’s Second Law in order to come to this conclusion. The force causes an object to accelerate. That acceleration, because it happens over some distance, is related to how the speed changes. To match the notation in this chapter, I would still label this my initial velocity, but this I would put as “f” for final. The book’s switching the notation a little bit on you.
Okay, so the book derives that for you. Let me just show you what the result is. So, Newton’s Second Law, our kinematic relationship logically tell us that this has to be the case. If you multiply the size of the force times the distance that the object moves while this force is applied, that is equal to what’s called the change in kinetic energy. The math derivation shows that each of these terms is a combination of one-half multiplied by the mass of this object times the speed squared. So it’s the speed at the final instant, at this final snapshot, minus one-half mv02, where this is the speed initially for this object.
So here’s the words I put to that. On the left, that term is called the work done by force F on the object from the initial to the final position. This reflects the total energy that’s either added or taken away from this object as it moves from the initial snapshot to the final snapshot. That energy is the cause of, is equivalent to, the change in kinetic energy. So let me highlight. Each of these is called the kinetic energy. This first one is the kinetic energy at the final instant. That’s because it’s the speed at that last instant, the final instant, that is used to calculate this. And then this last term is the kinetic energy at the initial instant.
Let me say a few more things about these terms. The left side is called the net work. We’ll label it wnet. We’re going to see that in general the left side has to include the work that’s done by all forces acting on that object. That’s the only way that we will fairly represent all of the energy that’s either added to or taken away from this object. And for shorthand, remember that the essence behind the work that a force does is force times the distance, so I’m going to put it in quotes here, “force times distance”.
Now, you’ll see that this is going to need some fine-tuning as we look as specific examples, but in essence that’s what you’re going to be after. The units of work are force, which is in Newtons, times distance, in meters. A Newton times a meter is called a “joule”. Well, we used the symbol “J” to represent the units of work. So if you go on to further physics classes, all forms of energy have the units of joules. If you go through these kinetic energy terms, you’ll discover that they have the units of Newton times meter, as well. Each of these has units of energy, a joule. So I’ve just written that bullet comment there for you.
So in our class, we’re focused on only one simple kind of energy, which is called the mechanical energy of an object. And in this first, simple introduction of the idea, we’re looking at specifically what’s called the kinetic energy of an object. It’s somewhat of an abstract idea, but I suspect you have some physical intuition for an idea of kinetic energy. You have some intuition, I’m sure, that if an object is moving along with some speed, it has the ability to exert a force and accelerate something else if it runs into it. The faster it’s going, the more potential it has for accelerating something else, the more kinetic energy it has.
So let me write that equation in this shorthand form, which is what people end up writing most of the time, just to write it down more quickly. On the left side, we’re going to write that the net work, the overall effect of all of the forces times distance, is equal to kinetic energy final minus kinetic energy initial. So you have to remember or know that the 0 and the F corresponds to those two instances of time during which or between which this work gets done, or these forces contribute to speeding up or slowing down the object.
On the right side, we’ll sometimes use this notation of Δ, which, again, is a Greek symbol that in physics we use to mean change, so the work is equal to the change in kinetic energy. And then you have to remember to replace this or substitute in, expand it, if you will, that this represents the final minus the initial kinetic energy, and then where each of these terms, you can expand it, substitute in, one-half times the mass of the object, times the speed at that instant squared.
Let me make some important points about this equation and how it’s different than what we’ve been working with. This equation, which says the work that gets done is equal to the change in kinetic energy, it’s not a vector equation. It’s what’s called a scalar equation. These are just numbers, these collections of terms.
That means, then, something very important in terms of the sign of these quantities. Only in a vector equation does the sign represent direction. In an equation like this, a sign, you’re going to see, is going to have the physical meaning of positive energy or negative energy, or adding energy versus taking energy away. These quantities themselves are always going to be a positive number. The v‘s that show up here, that’s not the vector velocity. These are just the speeds. It’s the speed and then it’s squared. Mass is always a positive number. So these terms intrinsically are always positive.
However, it could be that this term ends up being larger. Perhaps my initial speed is larger. So the combination here could end up being negative. If the right side is negative, the left side is negative. Negative work would then have the physical meaning that energy is taken away from this object. It’s going to slow down. The final speed is going to be smaller than the initial speed.
Of course, force and the displacement are vector quantities. But when we calculate the left-hand side here, you’re going to see in the process of preparing the numbers or the equations that get plugged into this, that we’re going to take the direction into account, and the direction of F compared to the direction of s. So when it gets plugged into this equation, it’s not going to be a vector anymore. The sign is just going to represent positive work or negative work.
All right. What is the more general, useful form of this equation? This simple version is only for that picture I drew with a single force acting on the object as it accelerated in a straight line over some distance, s. What do we do if we have multiple forces, and how do we handle this direction content or information about the forces compared to the displacement? How do we handle that?