https://youtu.be/cdBlIveUyDc
PHYS 1101: Lecture Sixteen, Part Three
Okay. One more definition for you and then we’re ready to move on to more quiz questions and examples. First definition, or a main new definition, scientists refer to the mechanical energy of an object as being equal to the sum of the kinetic plus potential energy.
So this mechanical energy, it’s a unique value at any instant in time. At that instant, I need to say what’s the kinetic energy, what’s the speed of the object, it’s mass, calculate kinetic energy at that instant, and then I want to add to it the mgh, the potential energy at that instant. This combination, it turns out it’s useful to group these together because you’ll see for some cases that you can talk about the mechanical energy being conserved or not changing for an object. So here are my two examples. Say at the initial instant and the final instances which sandwich this motion we’re looking at. At those two snapshots I can calculate and think of the total mechanical energy, the kinetic plus the potential at the initial. Kinetic plus potential at the final instant.
With these definitions, let’s go and just look again at our main starting equation. And here I’ve got it copied it again for you. Work non-conservative is equal to the change in kinetic plus the change in potential, and this term here on the right, I just want to rearrange it. Let’s group together the kinetic and the potential at the final instant.
So I’m pulling out these two terms and just writing them first with parentheses around them. Then let’s group together the kinetic and the potential at the initial instant and just move those around so they’re grouped together. If I then kind of look at this equation with fresh eyes I see that another way to physically translate this math is to say that the right-hand side represents the total mechanical energy at the end minus the mechanical energy at the beginning. This is the change in our mechanical energy.
So if I make those substitutions then my starting equation, another way to think about it, is that I have the change in mechanical energy on the right, so this includes both kinetic and potential energies. Any change in the mechanical energy has to result from work that forces do other than gravity.
Okay. So the logical conclusion then, or the consequence, is if you’re working with a problem and only gravity is doing work, then the left side of this equation, there would be nothing to calculate. It would be 0. There would be no other forces doing work. Nothing else that’s adding energy to this object. So if the left side went to 0 that would say that the mechanical energy can’t change. As long as you always add the kinetic and the potential at any instant, it’s not going to change. It should always give you the same number. That’s a very convenient and very powerful physical tool to use or physical consequence to infer to help you learn about the motion.
So this is what we call that “the mechanical energy is conserved.” Let me write here, e.g., mechanical energy, which is the sum of the kinetic energy and the potential, at any instant will be the same number.
Okay. How often is this true? When would this be useful to you in solving a problem? Well, more often than you’d think. Let’s go through some quiz questions, and I’ll try to help you develop your own intuition and appreciation for the usefulness of this conclusion for many different problems.
Okay. First question. Consider question five. So question five is Baby Max. Here he is in his Halloween costume on the slide and he slides down this slide. And let’s assume it’s perfectly slick, it’s a very fun slide, so friction isn’t going to do any work. There’s no frictional force on baby Max as he slides down the slide. How much work does the normal force do in that process?
So imagine, let me draw a little cartoon here off to the side. Let me just picture kind of a side view of part of that motion. Here is baby Max as he slides down the slide. He’s always, of course, following the surface. The normal force we know is always perpendicular to the surface. What other forces are on this object? Baby Max as he slides down the slide. Okay. And then for question five, in particular, if you focus just on this one force, how much work does it do?
Question six. For this problem of sliding down a slide, is mechanical energy conserved? Okay. One way to answer that is to say if I could calculate the kinetic plus potential at the top of the slide and the bottom of the slide I could see if those numbers are the same. But another way to answer that question is to look at the left side of the equation. You know that the right side will be 0 if the left side of this equation is 0. When could this be 0? This will only be 0 if gravity is the only force doing work. So is that true when you slide down the slide?
Okay. Next quiz question for you. We’re thinking this through some more. I’ve got different slide combinations: A, B, C, D. Different geometries for the slide. So now when a child slides down each of these it’s the same small child. In all cases I can ignore friction. You need to rank in order from largest to smallest the speeds at the bottom. So the child starts at rest at the top in all cases. I have so much potential energy when I’m at the top of the slide. That’s going to be the same potential, notice, for all the slides because I’m at the same altitude. That’s all that matters for potential energy: my altitude is the same. How do the speeds compare at the end?
Okay. For any of these questions you’re probably inclined to just think that you could mull it over and come to your conclusion or your answer right away. If you’re not sure, and this is a tricky question so you shouldn’t be sure, I want you to work with this equation and see what the mathematics tells you. Work with the equation that the work, our basic equation, the work that all forces do except for gravity would have to be equal to the kinetic energy final minus initial plus the potential energy final minus initial.
Write down what these terms are. Where does that variable show up that you’re after here in the problem to try to compare. Where is that a variable? You just plug in for these values and then translate the equation. See what the math tells you when you solve it for what you’re asked to focus on, the speed at the bottom, that’s V final. When you rearrange this and solve it for that see what the math tells you about how these speeds are going to compare.
And, perhaps, to help you out here the variables would be, here is my h final, my lowest altitude called 0. Here is my h initial and it’s the same for all of them. You could call it just h, or make up a number if you feel more comfortable. Make this a 10-meter high slide in all cases. Draw the free body diagram. Start working with this equation. See what it tells you. Translate the math. Which is the right choice? A through D. See what the logical consequences are of the physics, which is captured by that equation.
For your next quiz question I encourage you to do the same thing. The stakes are high on this problem. I’ll give you ten points if you get it right. Test out via discussion board. I’ll look at it. I’ll help you out. But it’s another example where I encourage you to go to your starting equation that governs how the speeds in these positions have to compare to answer the question. See what this equation, what the mathematics of the equation tells you.
You’re comparing, I have three different balls. They’re fired with the same initial speed. It’s all about comparing speeds at the beginning and position so when they hit the ground they’re going to be at the same height. When they started out they were at the same height. So this is the words that should flag to you that if you apply this equation to this problem it will tell you what this relationship has to be. So do that. For this scenario work out what this equation tells you that the speeds have to be or how they have to compare when all three of those balls hit the ground. The only thing that’s different is the direction that they’re fired.
So the velocity vectors are different initially, but the magnitude of those vectors is the same. Their speeds are all initially the same. Okay. And they all have equal mass too. Don’t forget and miss that point.
Okay. Here’s a series of clicker questions of concepts having to do with this idea of kinetic energy and potential energy just to see if you, or give you an opportunity to really think it through and have the ideas settle. You’re going to be considering a car on a roller coaster as it goes over this geometry of a roller coaster. So this is not a motorized car at all. There’s no driving force that speeds the car up or slows it down, so it’s literally just rolling along and consider it to be frictionless, a frictionless track as it rolls over this terrain.
Location one and two are meant to represent the same altitude, the same height above the ground. And what you’re asked in the following questions is how do these various quantities compare at these different locations. So the first one, as an example, asks you to consider how the kinetic energy of the car is going to change, how it’s going to compare when the car is at location one, compared to rolling over the hill and reaching location two, speeding up, going down the hill, and passing location three, etc.
Okay. First one. Question 9 is about the kinetic energy. You have to rank them. How does the kinetic energy of the car compare at locations one, two, three, and four?
Question 10, you then are ranking the potential energy of the car, those locations. Question 11, now you need to rank the mechanical energy.
Question 12, you have a physical picture by question 12 of what’s going on. Does that picture mean, or can you conclude that the mechanical energy is conserved for this roller coaster for the car in this roller coaster, the ride?
Question 13, is the left side of our starting equation: Wnc equals, etc. Is this left side equal to 0 for the car of the roller coaster?
Question 14, during a time interval on the same roller coaster all you know is that the car moves uphill during this interval. What can you conclude about the kinetic energy comparing the end of the time interval to the start?
Question 15 says during that same time interval when the car has moved up a hill, how has the potential energy changed?
Question 16, during, again, the same time interval as the car rolls up the hill, how has the sum of the potential energy plus the kinetic energy changed?
Question 17 is our last question about the roller coaster and it’s going to take a little bit of thought to answer it correctly and to motivate you to take that time and to think it through carefully. I’m going to give you more points. You’ll get eight points for this if you get it right. Again, take advantage of the discussion board. I ask you in question 17 if the speed of the car at location four depends on the mass of the car.
Let’s give you a clear starting point. Let’s assume that the car starts at rest at the top of the hill, at the top of that crest that you saw in that picture. So that’s between locations one and two. The car starts there. It’s at rest. And it then was just the slightest push but still consider it has an initial velocity of 0. It rolls down the hill. When it reaches location four, the speed that you have there, so Vf is the speed at spot four.
Ask yourself if that speed depends on the mass of the car. Get there by doing kind of a careful analysis or thinking logically through starting from our basic equation. For that scenario where my initial is at the top of the hill at rest, my final snapshot is when the car passes spot four. What would these values be for that car on this roller coaster? Substitute in your one-half MVf squared, your one-half MV0 squared, your MGHf, your MGH0. Substitute these in, simplify that equation, do some algebra, rearrange it so that you ultimately solve for Vf because that’s what you’re interested in. How does Vf, I’m sorry. V4, which is Vf, how does it depend on the other variables that are left in that equation?