00:01
Here i'll be looking at how you can use the potential energy function, which is a function of position, how you may use that to determine some things qualitatively about the motion of a particle.
00:15
So i've drawn a potential energy function down below, which may just look kind of abstract, but does mimic what's called the effective potential of a particle undergoing gravity, where you get down to, a single variable r, the distance of the particle from the force center.
00:39
So the idea is that force is simply negative the derivative view with respect to that variable r.
00:54
And so we can use a little bit of calculus.
00:56
We see that where the function has a minimum in this case, the force is zero because the slope of the potential energy is zero.
01:18
Okay, so that is an equilibrium position.
01:26
Furthermore, if we take the second derivative of view with respect to r, we may use a curvature test.
01:40
So if that is positive, the potential energy is curved up, and there is a minimum, which is a stable equilibrium.
01:54
And likewise, if it is curved down, that is an unstable equilibrium, like a marble sitting on the top of a hill.
02:07
And of course, if it's zero, it's a very flattened equilibrium that exists between kind of an inflection -type region.
02:19
But those are rare.
02:20
So here in the sketch i've drawn, we have a minimum and we have a stable equilibrium.
02:29
Furthermore, the second derivative of you with respect to that variable can be thought of as an effective spring constant.
02:45
So if you are near that equilibrium point, let's call that r0.
02:52
Here are zero, you will, not you, but the particle, the particle will oscillate like a simple harmonic oscillator.
03:15
So i'll abbreviate that s -h -m for simple harmonic motion.
03:23
And you may find the frequency of that oscillation as the square root of the effective spring constant over the mass.
03:35
So these are all things that help to determine some qualitative things about the motion of the particle.
03:43
Furthermore, the energy level, where that comes in on the graph, will determine possible qualitative motions.
04:08
So in the diagram above, there are two sort of possible regions, if we want to think about it.
04:14
But there's three possible levels.
04:18
We'll call them e1, e2, and e3.
04:24
And we're assuming, yes, that this is conserved energy, no rockets or things coming in.
04:32
So e1, for example, e1, which is less than zero, crosses the potential at two points called turning points because those are the maximum energy allowed.
05:01
And that's where all the potential is wrapped up into the energy.
05:09
You at the turning point is equal to e1.
05:15
Okay.
05:15
So for those energies that are similar to e1, the particle will execute some sort of oscillatory motion, not necessarily simple harmonic.
05:33
So particle goes back and forth between those two turning points.
05:56
E2 is equal to zero, and there the particle, what does it do? that's not even that special.
06:09
E2 is equal to zero.
06:12
It's similar.
06:13
Sometimes e equals zero is a special point, but it is similar to e3.
06:19
There is just one turning point.
06:34
So a particle starts out heading to smaller r, going to the left in this case.
06:52
It will turn around and never come back.
07:05
In other words, it'll head off to r equals infinity, not looking backwards.
07:11
Okay, we are going to apply these ideas and even more.
07:15
There's one last thing that's a little complicated to get into without.
07:21
Doing sort of a more specific example because it does involve an integral, but we'll get there and involves conservation of energy.
07:30
And that's the last point that i want to make.
07:33
Conservation of energy means the energy level, i can't spell conservation.
07:44
Conservation of energy.
07:51
Simply means that you have a constant energy level, and it's a trade -off between your kinetic energy of the particle plus you.
08:05
Okay, so as an example, we're going to be looking at a particular force function.
08:11
I'll start with the u as a function of x given up there.
08:16
U of x is 18 over x squared minus 9 over x.
08:22
That came from a force field.
08:24
I can write that force field down is 36 over x cubed minus 9 over x squared.
08:40
So it's a conservative force field.
08:43
But we can see off this graph that the u has a minimum, a place where its slope is equal to zero.
08:57
We may find that position either from the graph or we can take its derivative and see.
09:02
Set it equal to 0.
09:09
And we can also find the k -effective spring constant of our system.
09:19
So let's see.
09:20
We have 18 over x squared minus 9 over x is our potential function.
09:27
And we want to take its derivative and set it equal to zero.
09:34
And here, x is taking the place of r.
09:37
X is greater than zero.
09:40
So it's kind of like a distance.
09:45
X is kind of like a distance, but just looks like x to make it a little bit simpler.
09:50
So we have minus 36 over x cubed plus 18 over x squared, not 18, just plain 9.
10:09
Excuse me.
10:10
Getting a little carried away here, a sign over x squared.
10:15
So we can see that that is the negative of, the force.
10:22
The derivative of u is that negative derivative is just equal to the force field.
10:33
Okay, and that's a good check on the math that we did.
10:37
But the way we got it was u is equal to minus f of x d x with infinity as a reference point.
10:49
Okay, so we're going to set that equal to 0, and then we can multiply by x cubed minus 36 plus 9x is equal to 0.
11:11
And that means x cannot be 0.
11:16
And then we have x is equal to 36 over 9 is equal to 4.
11:25
And that matches nicely with the graph.
11:29
We can see that that's a minimum.
11:30
But if we can also try the second derivative test for curvature, b squared u by the x squared, is minus twice.
11:45
So that's three times 36 over x to the fourth minus 18 over x to the cubed...