00:01
The particle in orbit with an attractive force determining the orbit that looks like this, negative kr.
00:09
It's kind of like a spring force.
00:11
And we want to know a couple of things about this force.
00:14
So first off, we want to show that the potential energy is 1 .5kr squared.
00:19
So the potential energy is just going to be the negative line integral of the force.
00:26
And so hopefully this isn't too hard to see that this just comes up to 1 .5k.
00:31
R squared.
00:33
That should be pretty easy.
00:35
Part 2 is a little bit more tricky, so we have to undergo a bore quantization scheme and find an expression for the velocity and the radius of the trajectory of the particle.
00:46
So the bore quantization scheme says that the angular momentum of a particle is equal to n times h -bar, planks reduced constant.
00:55
And l, remember, is like mvr in terms of the magnitude anyway.
00:59
So the magnitude of the angular momentum is equal to this.
01:03
So what we have, there are a couple ways to get at this.
01:05
One, we could start with our original force expression and note that if we're in a circular orbit, we have our centripetal force being supplied by this sort of hooks law, a little force, spring force.
01:18
And if we do some manipulation, we get like mv squared equals kr squared.
01:26
And mv squared, so if you multiply both, sides by m and by r squared what we'll get is m sorry m squared v squared r squared equals m k r to the fourth the reason we want to do that let me clean this up a little bit the reason when we do that is because on the left hand side this is just our angular momentum squared and so here's where plank's constant can enter into this so we just set this equal to n h bar so i'm going to do that really quickly so this is n squared hbar squared and now we we can solve for the radius the allowed radii of this so if we do that r is going to equal i'm going to write it as the fourth root of n squared h bar squared over mk so there's our allowed radii and then we can go back into this formula and plug this in to find the allowed velocities so because of course the velocity is going to be you know in h bar over m r basically i set this equal to n h bar divide by mr so these are our velocities and what we get when we plug all that stuff in um should be another fourth root so n squared h squared i just want to make sure i'm writing everything correctly n squared h squared yes k over n cubed and this is to the one -fourth power so this should be our velocities let me just make sure that's right i believe so yes okay so now we have those so our radii and our velocities as a function of then now we want to find what are the allowed energies so to do this i'm going to clear this out a little bit one thing we so there are a couple ways to do this one is to use the virile theorem which in this context basically says that the you know the average energy the average kinetic energy is like negative one -half times the average potential energy in this context and what that ultimately means is if you know if we add these two together what we're going to get is the the total energies the sum of kinetic impotential energies the total energy is going to be like negative one -half times the potential energy sorry i should use a u instead of a v here so, if we plug this in, we get negative 1 4k r squared, and now we just use our loud values for r...