As I threatened last week, I want to talk some about central forces. They’re forces in which one particle attracts another with a force that depends only on how far apart the two are. Last week’s essay described some of the assumptions behind the model.
Mostly, we can study two particles interacting as if it were one particle hovering around the origin. The origin is some central reference point. If we’re looking for a circular orbit then we only have to worry about one variable. This would be ‘r’, the radius of the orbit: how far the planet is from the sun it orbits.
Now, central forces can follow any rule you like. Not in reality, of course. In reality there’s two central forces you ever see. One is gravity (electric attraction or repulsion follows the same rule) and the other is springs. But we can imagine there being others. At the end of this string of essays I hope to show why there’s special things about these gravity and spring-type forces. And by imagining there’s others we can learn something about why we only actually see these.
So now I’m going to stop talking about forces. I’ll talk about potential energy instead. There’s several reasons for this, but they all come back to this one: energy is easier to deal with. Energy is a scalar, a single number. A force is a vector, which for this kind of physics-based problem is an array of numbers. We have less to deal with if we stick to energy. If we need forces later on we can get them from the energy. We’ll need calculus to do that, but it won’t be the hard parts of calculus.
The potential energy will be some function. As a central force it’ll depend only on the distance, r, that a particle is from the origin. It’s convenient to have a name for this. So I will use a common name: V(r). V is a common symbol to use for potential energy. U is another. The (r) emphasizes that this is some function which depends on r. V(r) doesn’t commit us to any particular function, not at this point.
You might ask: why is the potential energy represented with V, or with U? And I don’t really know. Sometimes we’ll use PE to mean potential energy, which is as clear a shorthand name as we could hope for. But a name that’s two letters like that tends to be viewed with suspicion when we have to do calculus work on it. The label looks like the product of P and E, and derivatives of products get tricky. So it’s a less popular label if you know you’re going take the derivative of the potential energy anytime soon. EP can also get used, and the subscript means it doesn’t look like the product of any two things. Still, at least in my experience, U and V are most often used.
As I say, I don’t know just why it should be them. It might just be that the letters were available when someone wrote a really good physics textbook. If we want to assume there must be some reason behind this letter choice I have seen a plausible guess. Potential energy is used to produce work. Work is W. So potential energy should be a letter close to W. That suggests U and V, both letters that are part of the letter W. (Listen to the name of ‘W’, and remember that until fairly late in the game U and V weren’t clearly distinguished as letters.) But I do not know of manuscript evidence suggesting that’s what anyone every thought. It is at best a maybe useful mnemonic.
Here’s an advantage that using potential energy will give us: we can postpone using calculus a little. Not for quantitative results. Not for ones that describe exactly where something should orbit. But it’s good for qualitative results. We can answer questions like “is there a circular orbit” and “are there maybe several plausible orbits” just by looking at a picture.
That picture is a plot of the values of V(r) against r. And that can be anything. I mean it. Take your preferred drawing medium and draw any wiggly curve you like. It can’t loop back or cross itself or something like that, but it can be as smooth or as squiggly as you like. That’s your central-force potential energy V(r).
Are there any circular orbits for this potential? Calculus gives us the answer, but we don’t need that. For a potential like our V(r), that depend on one variable, we can just look. (We could also do this for a potential that depends on two variables.) Take your V(r). Imagine it’s the sides of a perfectly smooth bowl or track or something. Now imagine dropping a marble or a ball bearing or something nice and smooth on it. Does the marble come to a rest anywhere? That’s your equilibrium. That’s where a circular orbit can happen.
We’re using some real-world intuition to skip doing analysis. That’s all right in this case. Newtonian mechanics say that a particle’s momentum changes in the direction of a force felt. If a particle doesn’t change its mass, then that means it accelerates where the force, uh, forces it. And this sort of imaginary bowl or track matches up the potential energy we want to study with a constrained gravitational potential energy.
My generic V(r) was a ridiculous function. This sort of thing doesn’t happen in the real world. But they might have. Wiggly functions like that were explored in the 19th century by physicists trying to explain chemistry. They hoped complicated potentials would explain why gases expanded when they warmed and contracted when they cooled. The project failed. Atoms follow quantum-mechanics laws that match only vaguely match Newtonian mechanics like this. But just because functions like these don’t happen doesn’t mean we can’t learn something from them.
We can’t study every possible V(r). Not at once. Not without more advanced mathematics than I want to use right now. What I’d like to do instead is look at one family of V(r) functions. There will be infinitely many different functions here, but they’ll all resemble each other in important ways. If you’ll allow me to introduce two new numbers we can describe them all with a single equation. The new numbers I’ll name C and n. They’re both constants, at least for this problem. They’re some numbers and maybe at some point I’ll care which ones they are, but it doesn’t matter. If you want to pretend that C is another way to write “eight”, go ahead. n … well, you can pretend that’s just another way to write some promising number like “two” for now. I’ll say when I want to be more specific about it.
The potential energy I want to look at has a form we call a power law, because it’s all about raising a variable to a power. And we only have the one variable, r. So the potential energy looks like this:
There are some values of n that it will turn out are meaningful. If n is equal to 2, then this is the potential energy for two particles connected by a spring. You might complain there are very few things in the world connected to other things by springs. True enough, but a lot of things act as if they were springs. This includes most anything that’s near but pushed away from a stable equilibrium. It’s a potential worth studying.
If n is equal to -1, then this is the potential energy for two particles attracting each other by gravity or by electric charges. And here there’s an important little point. If the force is attractive, like gravity or like two particles having opposite electric charges, then we need C to be a negative number. If the force is repulsive, like two particles having the same electric charge, then we need C to be a positive number.
Although n equalling two, and n equalling negative one, are special cases they aren’t the only ones we can imagine. n may be any number, positive or negative. It could be zero, too, but in that case the potential is a flat line and there’s nothing happening there. That’s known as a “free particle”. It’s just something that moves around with no impetus to speed up or slow down or change direction or anything.
So let me sketch the potentials for positive n, first for a positive C and second for a negative C. Don’t worry about the numbers on either the x- or the y-axes here; they don’t matter. The shape is all we care about right now.
Now let me sketch the potentials for a negative n, first for a positive C and second for a negative C.
And now we can look for equilibriums, for circular orbits. If we have a positive n and a positive C, then — well, do the marble-in-a-bowl test. Start from anywhere; the marble rolls down to the origin where it smashes and stops. The only circular orbit is at a radius r of zero.
With a positive n and a negative C, start from anywhere except a radius r of exactly zero and the marble rolls off to the right, without ever stopping. The only circular orbit is at a radius r of zero.
With a negative n and a positive C, the marble slides down a hill that gets more shallow but that never levels out. It rolls off getting ever farther from the origin. There’s no circular orbits.
With a negative n and a negative C, start from anywhere and the marble rolls off to the left. The marble will plummet down that ever-steeper hill. The only circular orbit is at a radius r of zero.
So for all these cases, with a potential V(r) = C rn, the only possible “orbits” have both particles zero distance apart. Otherwise the orbiting particle smashes right down into the center or races away never to be seen again. Clearly something has gone wrong with this little project.
If you’ve spotted what’s gone wrong please don’t say what it is right away. I’d like people to ponder it a little before coming back to this next week. That will come, I expect, shortly after the first Theorem Thursday post. If you have any requests for that project, please get them in, the sooner the better.