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- [Instructor] Alright so
there's some terminology

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you gotta get used to

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when dealing with simple
harmonic oscillators

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because people and books
and teachers and professors

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are gonna throw these
terms around like crazy,

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and if you are not used to them,

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it all can sound like
mathematical witchcraft.

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So the first term you gotta know

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is that if you displace
a mass from equilibrium,

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and why wouldn't you do that.

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That's how you get the thing to oscillate,

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by displacing it from equilibrium.

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The maximum magnitude of displacement,

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so this amount right here,
whatever that distance is here,

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is called the amplitude.

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So we represent the
amplitude with a capital A,

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and it's called the amplitude,

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and it's defined to be the
maximum magnitude of displacement

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for that oscillator.

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So if this mass only ever
makes it this far away,

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so from here to here.

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And I'm drawing arrows,
but this is not a vector.

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It's the magnitude. Right?
Magnitude of the displacement.

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So it's the magnitude of the vector,

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so it's always positive.

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So we can draw that over here.

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If we want to, we can just
say this amplitude here.

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This would also be the amplitude,

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because we're just talking about

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the maximum magnitude of displacement.

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So it's gonna get displaced
equally on either side

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of the equilibrium position
and that maximum amount

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is called the amplitude.

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So in other words, if I pulled
this mass back 20 centimeters

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then that means 20 centimeters
would be the amplitude,

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or if you wanted it in meters
it would be point two meters,

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and then, that means,

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when it shoots through
the equilibrium position,

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it would also come over here
and compress this spring

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by 20 centimeters on this side.

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So it's always equal on both sides.

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Now there's another term
that you gotta get used to,

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and that's the period.

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So the period is represented
with a capital T.

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Why is the period
represented with a capital T

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when there's no T in the
word Period? Not sure.

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But capital T is kind of like time,

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so T might stand for time.

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Maybe they thought that was a good idea.

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Because what the period means,

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is the time required for an entire cycle.

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So what does this mean. An entire cycle?

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What we mean is that you
got oscillations going on.

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So this process is repeating itself.

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So in other words: if you
start with the mass over here,

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it's gonna eventually make it
over to this end over here,

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right? Goes over here,
compresses the spring,

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then it's gonna come
back. The time it takes,

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oh, that's a little hard to see, sorry.

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Let me draw that up here.

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So the time it takes for it to go to here,

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and then come all the way
back after this happened,

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the whole thing just repeats.

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Now it's back here,

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the spring is gonna pull
it back to the left,

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and go to the right.

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It's gonna pull it back to the left,

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push it back to the right.

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So this process is repeating itself.

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There's not something new happening.

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It's just the same process over and over.

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The time it takes to go
through one entire cycle;

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ie: The time it takes
to reset, essentially,

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once this entire system
resets to the same position,

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that's the period.

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And so it's gonna be the same.

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Wether I count this as from
this point back to that point,

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or if I imagine just
starting my clock here,

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from this point is gonna go over to here,

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then it's gonna come back here,

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that would also be the period,

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because it's the time it took to reset.

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So the time it takes for
this process to reset

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is what we call the period.

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It will be given in seconds,

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so for the sake to making
this a little less abstract,

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let's say for example,

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the period of this mass on
the spring was six seconds.

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What would that mean?

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It would mean that it took
six seconds for the mass

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to go from this point and then all the way

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back to that point resetting itself.

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Now, this is getting kind of messy.

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And honestly, for that
reason people often draw

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what the simple harmonic
oscillator looks like on a graph.

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It turns out to be
particularly elegant and useful

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to represent these ideas on a graph.

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Because, look it. If you
just drew what's happening,

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you'd be like: alright,
the mass goes here,

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and then there, and then
there and then there,

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you are drawing all over yourself.

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So that's kinda ugly looking.

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It's better to represent this on a graph.

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What would that look like?
So let me get rid of this.

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It would look like this.

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You would have a graph of
the horizontal position X.

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So what does that mean? That means this.

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So we are essentially
graphing what this is.

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This is X. The horizontal
position has a function of time.

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Now already you might be upset.

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You might be like: Wait a minute.

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Why did we stick the horizontal position

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on the vertical axis?

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Isn't that a dumb thing to do?

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Well, perhaps.

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But long ago physicists
decided: You know what?

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Time, if time is involved,

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we are sticking that bad
boy on the horizontal axis.

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This is just designated.

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This is just by default.

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It's gonna go on the horizontal axis.

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So if you have anything else
you wanna graph with it,

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that's gotta go on the vertical axis.

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And so unfortunately
we're gonna be graphing

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horizontal position on this vertical axis.

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What that means is that
this equilibrium position,

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remember this is the
point where the net force,

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the restoring force, that
net restoring force is zero.

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The only force on this mass, in this case,

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is the spring force which
is given by Hook's law

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and that means this equilibrium position

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is gonna be the point where X equals zero.

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If I want my force to be zero,

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I better have x equals zero.

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So this equilibrium position right here,

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this is the line right here,
let me give it a special color,

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this equilibrium position,

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is essentially just
this X equals zero line.

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Right? These two lines are
representing the same thing.

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They represent X equals zero.

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And if I go this way, if I
pull this mass to the right,

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I'm essentially going up on this graph.

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Because I am going towards
positive horizontal positions.

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And if I go to the left, if
I push this mass to the left,

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I'm essentially going down towards

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negative horizontal
positions on this graph.

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So hopefully that doesn't
freak you out too bad.

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Let me show you what this looks like.

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If we do displace this mass,

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Let's say we pull it to the right.

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So like we had over here, right?

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We have this mass, we
pull it to the right,

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and if we started 20 centimeters

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from the equilibrium position and let go.

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What's that gonna look like on this graph?

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Well it started to the right.

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If it starts to the right,
I'm gonna start way over here,

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at this point is my initial position.

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That means I'm gonna start up here.

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I'll start up here at X
equals 20 centimeters.

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If we put that in meters,
technically SI units,

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you should have meters
for the default units,

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so this would be point two.

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Zero point two meters, and
that's also the amplitude.

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So remember, this is the amplitude.

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So this distance here is the amplitude.

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Then what does the mass do?

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Well it shoots back toward equilibrium,

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that's X equals zero.
And then it oscillates.

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It goes through that point and comes back,

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so essentially what
you're gonna have on here

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goes toward equilibrium,
so it looks like this,

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goes toward equilibrium, BOOM!

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Hits equilibrium. And that's
when it's at X equals zero.

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Passing through this point right here.

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Then it's gonna come back down,

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so eventually is gonna
compress the spring and stop.

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That's when you're way over
here and you've then stopped.

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The mass has been stopped by this spring.

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And it's gonna come back up

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and this process is gonna repeat,

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it's gonna go back through
the equilibrium position

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and come back up.

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Which by up it means over here
back to this initial point.

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That's one whole cycle.

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Look it, that has gone
through a whole cycle.

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I kinda made this a little too high.

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Let me make that a little better.

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It should never go any
higher that it started here.

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So it's gonna look something like that.

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Come back down and this
whole process repeats

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over and over and over.

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And If I was drawing this perfectly,

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it'd be perfectly smooth,

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but hopefully you get the idea.

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And this is great! 'cause
now we can draw the variables

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we talked about earlier like amplitude,

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because amplitude is the maximum
magnitude of displacement

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from equilibrium.

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That would equal point two meters.

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That's what we represented
on this graph here.

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And we can also represent the period.

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Remember, the period was the time it takes

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to go through an entire cycle.

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So if our mass started here,

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to go through an entire cycle,

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it better get back to that
point and have reset completely,

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so that would be to here.

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So on this graph, this is the period.

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So the time it took to do
that is one whole period.

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That would be the period T,

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which if we recall what we said earlier,

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we said that the period was six seconds.

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So if it really is six seconds,

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we can say that this here would be,

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if we count this is time T equals zero,

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this would be six seconds,
this would be three seconds,

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that would be half of the
period, or half of the cycle.

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This would be nine seconds,
this would be 12 seconds,

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which would be two whole periods.

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And in a sense, it has gone
through two whole cycles

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once it gets back to that point.

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Now notice, you didn't
have to measure the period

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from peak to peak.

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You could have measured it from,

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sometimes people call
these troughs or valleys,

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so you can measure it trough to trough,

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or valley to valley,

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took three seconds to nine seconds.

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That's a time of six seconds.

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It took six seconds to
go from three seconds

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to nine seconds.

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That's still one whole period.

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Or you can go from this point here,

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I guess this would be like
seven point five seconds

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all the way to what is this, 13.5 seconds?

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That would also be one whole period.

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Just make sure you don't do this:

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Sometimes people are
like: oh, a period, eh?

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Repeat a whole cycle, eh?

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Alright I'm gonna go from
this equilibrium position

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back to that equilibrium position.

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That's not a whole cycle. Look at it.

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This point the mass is going that way,

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and this point the mass is going that way.

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So you can't start your clock

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when the mass is going that way.

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Stop it when the mass
is going the other way

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and see if you have gone
through a whole cycle.

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Because that hasn't fully reset.

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If you're gonna fully reset,

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you gotta go from mass heading to the left

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through equilibrium,

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all the way back to
mass heading to the left

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through equilibrium.

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00:09:21,636 --> 00:09:23,862
So you would have to go
from this equilibrium point,

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00:09:23,862 --> 00:09:25,907
all the way to that equilibrium point

258
00:09:25,907 --> 00:09:27,741
to have a full cycle.

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00:09:27,741 --> 00:09:31,590
A cycle would look like this
whole process right there.

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00:09:31,590 --> 00:09:34,500
So recapping: the amplitude of
a simple harmonic oscillator

261
00:09:34,500 --> 00:09:37,086
is the maximum magnitude of displacement

262
00:09:37,086 --> 00:09:39,167
from the equilibrium position.

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00:09:39,167 --> 00:09:41,331
You can measure it that way or
you can measure it this way,

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00:09:41,331 --> 00:09:42,428
you would get the same amount.

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00:09:42,428 --> 00:09:44,383
And the period is the time
it takes for an oscillator

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00:09:44,383 --> 00:09:46,649
to complete one entire cycle,

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00:09:46,649 --> 00:09:48,559
which you can find on a graph by measuring

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00:09:48,559 --> 00:09:51,512
the time it takes to go from peak to peak,

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00:09:51,512 --> 00:09:52,907
from valley to valley,

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00:09:52,907 --> 00:09:56,477
or from equilibrium position,
skip an equilibrium position,

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00:09:56,477 --> 00:00:00,000
and then get to the next
equilibrium position.