[Verse 1]
I want to talk today about things that shake
And I hope my words aren't too opaque
One degree of freedom moving to and fro
Just how it moves, we’d like to know
We can represent all kinds of things
By a single mass between ideal springs
Each spring's connected to a wall
So the outer ends don't move at all

[Verse 2]
Let the mass be m, spring constant k
But don’t let friction get in the way
Use Newton's Laws, and what do we got?
F equals m psi double dot
F is also minus k psi times two
So now we have a diff E Q
And we can write down the general solution
For the simple harmonic time evolution

[Verse 3]
Let omega be root two k over m
Here's the answer. Won't repeat again
Psi is A cosine omega
Plus a phase, call it phi
So it's all very simple. And you can see
For any initial psi and velocity
We can find the constants A and phi
And the equation's exact for all time t
[Verse 4]
Now look again at the diff E Q
It's homogeneous and linear, too
So if you add two solutions together
Their sum's a solution that’s even better
We call it the principle of superposition
You can use it to fit the boundary condition
In fact, there is no contradiction
If we use it in a system that does have friction

[Verse 5]
In a real system (Nothing’s perfect, of course)
We have to include the frictional force
Suppose it goes as the velocity
Write minus m gamma d psi d t
Now if the damping is not too strong
Our old solution is close but wrong
See, it starts out with some amplitude A
But after a while it just dies away

[Verse 6]
The amplitude decays exponentially
As you can see experimentally
As e to the minus half gamma t
Now, it's almost right, but you see
The frequency is lower as we can compute
Omega is now given by square root
Of the quantity k over m times two
Minus quarter gamma squared, and now we’re through
[Verse 7]
So now we have the complete solution
For an oscillator's time evolution
And when there's damping, as everyone knows
The amplitude decays, and the frequency slows
If we have two solutions, no matter how chose
You know we can always superpose
And since you all find physics such fun
Do problems twelve, eighteen, and twenty one

[Outro]
Class dismissed