DanielSank BernardoMeurer 23 August 2016

Bernard Meurer

21:04

Okay, I'm ready

DanielSank

So we have $x(t) = A \cos(\omega t) + B \sin(\omega t)$ where $\omega \equiv \sqrt{k/m}$.

Yes?

Bernard Meurer

Yep

DanielSank

Ok, this is the standard solution to the harmonic oscillator problem and it's one of the most important things in physics.

It comes up all the time, and for a good reason which you'll learn.

Now let's go back to the original differential equation this came from.

We had $\ddot{x}(t) = - (k/m) x(t)$.

Bernard Meurer

Yes, and that had come from $ma = -kx$ right?

And this all comes from a fucking box attached to a spring lol

DanielSank

Yes.

Actually, let's writ it $x(t) = A \cos(\omega_0 t) + B \sin(\omega_0 t)$.

$\omega \rightarrow \omega_0$.

Bernard Meurer

21:08

Who are $A$ and $B$ again?

DanielSank

Aha! We don't know yet. They're free parameters. A second order differential equation has two free parameters.

Bernard Meurer

Whats $\omega \rightarrow \omega_0$ ?

DanielSank

These would be determined by the initial conditions of the problem.

Bernard Meurer

@DanielSank So now we guess?

DanielSank

@BernardMeurer Just changing notation, nothing more.

Bernard Meurer

21:09

@DanielSank Got it

DanielSank

@BernardMeurer Guess what?

Bernard Meurer

@DanielSank A and B?

DanielSank

@BernardMeurer No no, they're indetermined.

Bernard Meurer

Ah, alright

DanielSank

They depend on the initial conditions of the problem.

Now please rewrite the original differential equation in terms of $\omega_0$. This is a simple plug-in task.

Bernard Meurer

21:11

$$\ddot{x}(t) = - (k/m) x(t)$$ this one?

DanielSank

Yes.

Bernard Meurer

Lemme get a paper

DanielSank

good

Bernard Meurer

$$\omega_0 = - \frac{\ddot{x}(t)}{x(t)}$$

I feel like I can simplify it further, but I'm not 100% on how

DanielSank

I just wanted you to write $\ddot{x} = - \omega_0^2 x(t)$.

You missed the $^2$.

Bernard Meurer

21:20

Oh, right, there was a square root!

DanielSank

You can't simplify it any more.

I just want you to burn this into your mind:

$\ddot{x}(t) = -\omega_0^2 x(t)$

That is the equation of motion of a simple harmonic oscillator.

Bernard Meurer

Dully noted

DanielSank

In the box/spring problem that $\omega_0$ came from the spring constant and the mass.

However! Suppose we have an $LC$ oscillator circuit.

Then if you solve for the time dependent voltage you get the same equation but with $\omega_0 = \sqrt{1/LC}$.

Bernard Meurer

$LC$?

DanielSank

$L$ is inductance and $C$ is capacitance.

Bernard Meurer

21:23

AH

DanielSank

LC circuit

An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency. LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal. They are key components in many electronic devices, particularly radio...

Bernard Meurer

Makes sense, makes sense

@ACuriousMind Howdy

DanielSank

Ok Bernardo, back to the main point.

Bernard Meurer

Yes

DanielSank

You solved the equation with Wolfram.

Now please check that it's correct. Plug the solution into equation of motion and see if it all checks out, ok?

Bernard Meurer

21:25

I feel like a plot twist is coming

DanielSank

Not quite yet.

So far I just would like you to get some mechanical familiarity with this business.

Bernard Meurer

Wait, what do you want me to check?

DanielSank

Check that our solution for $x(t)$ satisfies the differential equation.

Bernard Meurer

How do I check that of Wolfram?

DanielSank

Don't use Wolfram.

Use your pencil.

Bernard Meurer

21:27

Ah

The solution is $\ddot{x}(t) = -\omega_0^2 x(t)$

Where do you want me to plug it?

Or not the solution, that's the equation of motion of a simple harmonic oscillator, duh

DanielSank

You have an equation of motion and something Wolfram told you is the solution.

Your job is to make sure Wolfram is right.

Bernard Meurer

$$x(t) = c_2 \sin(t \sqrt{k/m})+c_1 \cos(t \sqrt{k/m})$$

Which was this, right?

DanielSank

Yes but let's clean it up and use this:

$x(t) = A \cos(\omega_0 t) + B \sin(\omega_0 t)$

$\ddot{x}(t) = - \omega_0^2 x(t)$

Your job is to make sure those equations are consistent.

Bernard Meurer

Thinking

Yeah I dunno how to do this, I'm sorry

DanielSank

That's ok.

Hint: compute $\ddot{x}$.

Do you know $(d/dt) \cos(\omega_0 t)$?

Bernard Meurer

21:33

I just don't remember how to do that

DanielSank

oh

Hmmm.

ok

Bernard Meurer

I told you I was weak :/

DanielSank

Do you know $\cos(x) = \sum_{n=0,2,4\dots} (-1)^n \frac{x^n}{n!}$

Bernard Meurer

That looks vaguely familiar

taking notes

DanielSank

I see. If you aren't comfortable with calculus we could go about this all a different way.

In fact, we could make this a lot simpler.

Bernard Meurer

21:35

I barely know calculus, I just know what I learned by myself for the SAT

DanielSank

Side note: Focus carefully on calculus and linear algebra in university, and do well. Those courses are your basic wings for the rest of your life in CS and physics.

I can invent a different problem to show you the power of linear algebra, or we can keep going with the calculus example, and just make sure to spend time developing comfort with the necessary calculus.

Up to you.

Bernard Meurer

Well, my grandfather just had a heart attack, so I'm going to go see him in about 20 minutes, but in general I feel like we should go with the calculus approach. I gotta get over this

DanielSank

@BernardMeurer Oh I'm very sorry to heat that.

Go see your grandpa.

Bernard Meurer

Alright, I'll be heading out then, I'll read on calculus tonight :)

DanielSank

ok see you

0celo7

22:37

lol

Transcript for

DanielSank BernardoMeurer 23 August 2016