Physics: Energy in Orbital Motion

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We've discussed the kinematics and the dynamics of orbital motion, especially circular orbital motion. We've used Newton's Second Law to understand and explain the behavior of some satellite, for instance, in circular orbit around the earth. That's fine. Newton's Second Law does explain motion. But sometimes it's nice to look at problems using the perspective of energy and energy conservation. It often teaches you something new and, if you ask the right questions, energy conservation can sometimes answer questions more easily than using Newton's Laws directly.
So, let's think about energy and energy conservation for orbital motion. If you want to do that, you really have to start with the potential energy function, which we already know. The potential energy of the gravitational potential energy, between two objects is . So, we're talking about say, some little mass m in orbit around a big mass, capital M (like the earth) divided by r, and there's minus sign, a very important minus sign in that formula. And here's the graph. -. And you look at this graph and you say, what do I do with this? Potential energy curved teaches you a lot of physics just staring at it and thinking about conservation of energy.
Conservation of energy is valid for gravity. It's a conservative force. Total energy is kinetic plus potential. So, if I've got a graph of potential, this immediately relates total energy, energy of motion, and energy of position. So, let me just give you a specific example. Supposing that I have some fixed amount of total energy to start with. You'll always have some certain amount of energy to start with and then no matter where you are in your motion, as you cruise around and it doesn't have to be in orbit. You could be talking about up in the air or in some funny elliptical orbit. It's guaranteed that kinetic energy plus potential energy will always equal to that same number.
So, for instance, you might be just sitting still out here at this large radius r, and have no kinetic energy. And so you look down, and you've met the curve right here. All of you energy, E[tot],is equal to potential energy at that moment. And what happens next? Well, this is a ball at rest in the gravitational field, it's going to head toward the center of the earth, if it's a ball held above the earth. So, you would imagine that this object would follow the potential energy curve down toward the center. It will to a smaller and smaller radius. And at any given height, you can immediately use this graph to deduce how fast it's going. You just say, let's say, when we've gotten to this radius, the difference between the potential energy and the total energy is the kinetic energy. So, that immediately tells you how fast it's going.
And you can work it upside down as well. If you toss the ball up in the air, and you know its initial potential energy, that means where it started, and its initial kinetic energy, that you can deduce, for instance, how high it's going to get, the turning point on this graph.
It's not true that if you know the total energy, you know exactly what the motion is going to be. There's lots of different motions with the same total energy. For instance, this rock being thrown up, or a rock falling down, or for that matter a rock going out around in a circular orbit, a satellite; they can all have the same total energy. And, if you look at this total energy, it's negative. That makes sense. If you're throwing a ball up in the air, it's not going off to infinity. It's stopping somewhere. If you're energy total is negative, that's exactly what that means. Our potential energy function has been defined to be zero off in infinity. So, you would need to add energy to that system in order to get it free, away from the earth and sometimes people call the amount of energy that you need to add to a system to break it free, the binding energy. Chemists use that term quite frequently. It's the amount of energy more that you would need to add, so it makes sense that the total energy is negative in this case. And because of that word, we say this is bound object. It's stuck to the earth because its total energy is negative.
If you have a circular orbit, what radius will that orbit have? If you look at this graph, you might instinctively think that this would be the radius. But that's wrong. This is the radius where the object is at rest. And object at orbit is not at rest. So how do we go from this graph to understanding the relationship between energy and kinetic energy and potential energy for an object in the circular orbit. Well, we just have to work out a few equations and they all amount to conservation of energy. Total energy is kinetic, energy plus potential energy. And let me just write down the formulas. Kinetic energy is always mv^2. Potential energy for an object in a gravitational potential, well, is always negative -that minus sign is important. So these are the formulas for total energy.
Now, I'm going to specialize to a circular orbit. This is true for any old object, any old orbit. If you have a circular orbit, r is constant. So that's nice. And speed is going to therefore be constant. And what are those constants? Well, let me use Newton's Second Law in the form F - Ma. The force on a circularly moving satellite is just the force of gravity. It's . Notice that force gets the r^2 and potential has the , and force equal mass times acceleration. It's moving in a circular orbit by my assumption and so the acceleration is , and look: I've got this formula here, it involves both the velocity and the radius. I want to know either one individually. So, let me get rid of velocity in favor of radius. Let me write this equation totally, just in terms of radius. I'll use this one to do so. Multiply both sides by r and divide by 2 and you get mv^2. In fact, you get mv^2 equals . One of the r's canceled and it's positive, which is good. It's a kinetic energy. And let's just plug that right in here and we get minus and that's just a subtraction problem and you get minus . So, there we go, if you know the total energy of an object in the circular orbit, you can immediately predict the radius of the orbit. And vice versa.
It's negative and that makes sense. When you're in orbit, you're still stuck to the earth, you still have a total negative energy. you're not running away off to infinity, you're just in this circular orbit. You're still bound.
It also true, if you look at the derivation, we have the formula for kinetic energy was plus , so E[tot] is minus kinetic energy. That's a slightly weird looking formula, but it's correct; and it makes sense because energy is negative and kinetic energy will be positive. So the minus signs are weird to look at, but they're quite correct. And since kinetic energy is mv^2, this is immediate the formula that connects total energy to speed. So I can either connect energy to radius or energy to speed; and of course I can also connect speed to radius. And that's neat. If you know the radius of an orbit, you can use energy principles instead of solving Newton's Second Law, actually either one, it's just about as easy, but this gives you velocity as a function of radius.
And finally, if you look back at our formula for potential energy which was just -, you see that potential energy is minus twice kinetic. So you stare at these formulas and their fine, you can use them when you're solving problems and trying to understand what's going on in an orbital motion problem. These minus signs are a little bit weird and it's nice to just look at a graph which summarizes these results.
So, let me graph total energy, kinetic energy and potential energy versus r. So, I'm just using these equations. And here's what the graph looks like. Kinetic energy of course, is always positive. And the total energy was just negative K. So this red curve is just sort of a mirror image of that kinetic energy curve. This is total energy, remember these are circular orbits that I'm graphing. It's a very special case. So, when you have a circular orbit, the energy is always negative. And the potential energy was minus twice the kinetic energy.
So you look at this graph and you imagine that you've got some object sitting out here at some radius r, and this immediately how fast is it going, what's it's total energy. The difference here would be how much energy you would have to add if you wanted to boost this thing away from the planet earth. There's lots of information coming straight from those formulas.
Imagine that this is a satellite in orbit, a circular orbit. And imagine there's a little bit of friction. So, the total energy of that system will no longer be conserved. Slowly, but surely the energy of that system will get smaller and smaller. Now what does that mean smaller? Doesn't mean smaller in absolute value. It means smaller, it means that you have to go to a more negative energy: that's what smaller means. So, if you're going to lower and lower energy, than the particle must be moving to smaller and smaller radius.
Well, that makes some sense to me if you've got friction. I expect that the satellite will sort of, head in towards the earth. We say it's spirally in, although that doesn't mean it's doing this, it means it's going in a circular orbit with a steadily smaller and smaller radius. And eventually it will burn up and crash into the earth. Makes sense, although there's one really weird aspect, which you would never have guessed, probably. But conservation of energy insists as it heads, get closer and closer to the earth-I'm assuming that it stays in a nice circular orbit, which is an approximation-its kinetic energy is going up. So, there's friction and it's speeding up because of the friction. And that's correct. It's going faster, and faster, and faster, and that's what satellites do as they head in and then finally burn up; and it's just because of conservation of total energy. Their kinetic energy is going up, but their potential energy is going down even more, cause of that factor of two. And in the end, it's true that their total energy really is going down.
So, understanding orbital motion in principle, you can just use F = Ma; that's all you ever need in principle to understand any physics. But in practice, for instance in this situation, with a gravitational force and a frictional force in the story, it might be difficult from a practical point of view to try to figure out what's happening. You'd have to add the forces and then solve Newton's Law with this force that's pointing in a weird direction. I wouldn't really know to go about doing that, but conservation of energy principles-or I should just say energy principles-really allow me to understand the physics in a much direct way. And even solve for observables, like how fast it's going as a function of radius, in a quick and easy and reasonably intuitive way.
Force of Gravity
Orbital Motion
Energy in Orbital Motion Page [3 of 3]