00:01
All right, so let's say a harmonic oscillator has an amplitude a and a frequency omega.
00:05
Then the total energy of the system is going to be like one half, and we'll call this m omega squared times a squared.
00:14
That's the total energy.
00:15
Presumably, you know, it has a mass as well.
00:18
So we want to know what's the magnitude of the displacement and velocity when the potential energy is equal to the kinetic energy.
00:24
So our potential energy, we can write as one -half m -omega -squared x squared.
00:29
Our kinetic energy, of course, we can write as one -half mv squared.
00:33
When these are equal, what we'll have is basically m -o -m -o -m -squared, x -squared.
00:41
That's our kinetic and potential energy added together.
00:44
It is going to equal 1 -5m -o -m -o -m -a -squared a -squared.
00:48
So you can see x is going to be a over the square root of 2.
00:52
And then the velocity, which is going to be, we can do kind of a similar kind of calculation.
00:59
One half, or sorry, we'll have mv squared, which is our kinetic energy at this particular instant is one half m omega squared a squared.
01:13
And so we can see that v is basically going to be like the square root of two times its maximum, or the maximum velocity divided by the square of two...