Physics: Conservation of Mechanical Energy

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We've defined kinetic energy, energy of motion, K, and potential energy. We've called it the symbol U. Potential energy refers to the energy associated with where you are. For instance, gravitational potential energy, mgy, tells you your capacity to do work just by virtue of where you're sitting when you're in a gravitational field. I could add energies and define mechanical energy of an object. The mechanical energy is just the total energy, kinetic energy plus potential energy. And it's a nice definition. We've spent a lot of effort coming up with formulas. The formula for kinetic energy is fairly simple, . And the formula for potential energy requires doing an integral. But once you have specified a particular conservative force, we know what the potential energy is as a function of position. So this is a quantity. It's a property of an object, which is fairly easy to compute. There are no vectors in here. It's just numbers you pretty much just plug into a calculator.
So it's easy to compute, but what does it teach us? Why have we been worrying so much about computing this mechanical energy? The reason is a little fact. In fact, it's a big fact. It's one of the most important ideas in physics. If you have a system which is complete, that is to say if all of the energy of the system can be expressed in terms of kinetic energy and potential energies--So I'm not going to worry for the moment about frictional forces that don't have potential energies associated with them. If you have conservative forces with all the energy in the form of potential energy, then mechanical energy is a constant. What's the constant? Well, that depends on what system you're looking at. I can't tell you that in advance. But once you give me a system, I can measure its kinetic and potential energies at one time. And then I'll know what this constant is for my system, and it will never change. It's a constant. We say mechanical energy is conserved. And it's a wonderful idea. It tells us that a very simple easy to calculate property of the system is always the same. Motion can be complicated. You can have kinetic energy at one time, like a ball thrown up in the air. And that might convert into potential energy at a later time, but when you add those two numbers, it's always the same.
In terms of practicality, this equation is wonderfully useful in a large number of problems. Not every problem, but there are a lot of problems where, for instance, I give you what's going on at the beginning of the story. I tell you essentially what are the kinetic energy and the potential energy at the beginning so you know what this constant is. And then I ask a question later like how fast is it going when it's here? Now according to this formula, that's very easy to compute. If you want to know how fast it's going, that's the v in one half mv^2. You simply compute the potential energy which is a given, simple formula. The potential energy is just mgh or mgy for gravitational potential energy, and so on. There'll just be a little formula.
Let me just give you an example to think about. Suppose you have a funky roller coaster, and you started off up here at rest at some initial height h. And then you go through a loop to loop, and you go up over some hills, and you end up over here. And what are you concerned about? One of the things that's fun on a roller coaster is the wind whipping in your hair, how fast you're going. And the roller coaster designers are very interested in how fast you're going too. They don't want you going too fast, so that things start flying apart. So a lot of physicists and people who are interested in the roller coaster problem, what they would like to know is how fast are you going? Now I've just shown you that with the principle of conservation of mechanical energy, assuming as I said that there's no friction forces to worry about, we'll get to that later, if mechanical energy is conserved, it's easy to compute our initial mechanical energy. If we started at rest, for instance, it's just purely potential energy, mgh[i]. So I know how much energy we've got. And then at the end, I know how much potential energy we've got, mgh[f]. And it's easy to solve for the final speed. It's really easy. It's just numbers, and a simple algebraic calculator problem.
Imagine what we would have had to do without the principle of conservation of energy. Look at this problem and think Newton's second law. Up until now, Newton's second law was the central guiding idea, the main important formula in physics. So you start off here. There's a gravitational force down. And there's the normal force of the track. And off you go. And at every moment in time, there are these two arrows. So even at an instant in time, it's a slightly difficult problem. You have to know the angle of the track. You've got to know the size of the arrows. And you've got to figure out the sum of the two forces, gravitational and normal. We're neglecting friction. And then you know the total force. So Newton's second law says you know the acceleration. It's not constant. You can't use our lovely constant acceleration kinematics equations. Throw them out the window. You have to integrate on a computer if you know acceleration as a function of time, yes, you can figure out velocity. It's a nightmare problem. And trying to figure out the velocity at the end, I wouldn't even really want to think about having to do that. So it's a problem, which, on the one hand, looks like an enormously complicated problem. And yet there's a fundamental, underlying simplicity here. And that simplicity is seen when you just look at this equation. Kinetic energy plus potential energy is a constant.
Let me derive this formula for you, because it's so important. It's really the central idea in modern physics. Like I said, Newton's second law, , isn't even completely correct. If you go down into the quantum mechanical world, you have to fix it up. If you go into the world of ultra high speeds, Einstein's special theory of relativity deals with speeds near the speed of light, and again you have to fix up Newton's second law. But the idea, the principle of conservation of energy is one, which will continue to hold, even to the extremes of physics that we know about today.
So how would you derive conservation of mechanical energy? Let me remind you of the two pieces that go into energy. Kinetic is one half mv^2, and remember the change in potential energy is what we have defined. The change in potential energy is the negative integral from where you started to where you ended of F[x]dx, where F is the force associated with this potential energy. So if this is gravitational potential energy, this would be the gravitational force. This was the one-dimensional formula. In two or three dimensions, it's just F dot dr, a line integral.
So that's kinetic and potential, and what's the work-energy theorem say? Remember change in kinetic energy is equal to the net work done by all forces. I'm focusing my attention on an object. I'm thinking roller coaster in the back of my mind. And there are a bunch of forces acting on it. It could be gravity. It could be springs. Let me stick to the situation right now, where all of the forces acting in our problem are conservative forces. Gravity and springs are two really nice examples. So in that case the net work would just be pure 100% conservative forces doing work. And there might be a bunch of them, so this would be the sum of all the forces' contribution to the work. In principle there might be other forces, I could call them W other, which generate another work. In other words, work is the integral of force dot displacement. And I'm breaking that integral up into the integral of all the conservative forces dot displacement plus the integral of all the other forces, the ones that aren't conservative, like friction, or some external force that isn't taken into account on my conservative forces. In many problems we can just neglect this W other. Mostly it's friction, and we're going to often be just not worrying about friction, because it's going to be small.
Let's focus our attention on the work done by conservative forces. Each one does a certain amount of work. How much work does a conservative force do? The formula for work is . Right here, that's the formula for work done by a force. So if delta U is the negative, the work done by the conservative force is the negative of delta U. Just stare at that formula and remember what work means. And then there's work other. Add delta U to both sides of this equation, and you get delta K plus delta U is equal to W other, . This is very close to what we started off with. If there are no other forces, nonconservative forces in the problem, then delta K plus delta U is equal to zero. And this is the story that I was really focusing on in this tutorial. If delta K plus delta U is equal to zero, what does that tell me? It says K plus U is not changing. Delta is the change in a quantity. So if the total change is zero, it means that K plus U is equal to a constant. The proof is kind of mathematical. And really the details of the proof aren't so important. As long as you can understand and use this formula, these are the same formulas. Change in mechanical energy is equal to zero says mechanical energy is conserved. It's a constant.
In life, when you've got a situation involving only conservative forces, it can look very complicated. I'm jumping on a pogo stick, and I'm bouncing around, and I'm hopping around, and it seems like, boy, what a difficult problem to describe and to understand using Newton's law. And it is a difficult problem to understand from that perspective. From the perspective of mechanical energy, it's kind of much simpler. What's going on when I'm hopping around on a pogo stick? I'm simply converting energy from kinetic energy into potential energy, back and forth. I jump onto the pogo stick and I compress the spring. So all of the energy of the system is potential energy of a compressed spring. This U here, remember, can be the sum of all the potential energies of all the conservative forces in the story. In the case of a pogo stick, we've got two, the potential energy of the spring plus the gravitational potential energy. So this U here really means U spring plus U gravity. Then what happens? The spring expands, I'm flying up into the air. All of my energy is in the form of kinetic energy. Kinetic energy is everything. Potential energy is zero. But the sum is still the same as it was in the beginning. So if I knew how much the spring had been compressed at the start, I can immediately compute how fast I'm going when I'm flying up into the air. And then I go up, and I am momentarily at rest up in the air. I'm at my high point. Still my total energy is that same old constant. It's no longer kinetic. I'm at the top. It's no longer potential energy of the spring. It's all gravitational potential energy.
So conservation of energy has this sort of new way of thinking about the world. Instead of thinking about pushes and pulls, you can think about energy flowing back and forth between kinetic energy, motion, and potential energy due to some conservative forces. It's kind of a neat way of thinking about how all the activity in the world around us is occurring. Energy is just flowing from one part of the system into another part of the system and back again. And that's what explains everything that's going on.
Conservation of energy, it's a very basic and really important principle of physics. It's also very practical and useful in solving a whole bunch of problems. And we'll be working some examples to just get some experience, to see how to plug in numbers and use it.
Energy
Conservation of Energy
Understanding Conservation of Mechanical Energy Page [3 of 3]