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- [Instructor] So say you had
a mass connected to a spring

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and that spring's tied to the ceiling

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and you give this mass a kick,

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well, it's gonna start oscillating,

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will oscillate down
and up and down and up,

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but if I were to try to draw this,

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all those drawings would overlap,

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it'd look like garbage.

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Forget that.

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Let's just get rid of all this.

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Let's say we just drew what
the mass' position look like

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every quarter of a second.

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So, this is where it started at,

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wait a quarter of a second, now it's here,

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quarter of a second, now it's here,

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so this is one mass

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and we just took pictures of it

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and we move those pictures
right next to each other.

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If I were to connect the dots,

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I get a graph

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which is basically the height of this mass

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as a function of time.

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It looks like this.

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So if I were to plot this legitimately

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on a y or vertical
position versus time graph,

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you'd get something like this.

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It starts in the middle

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and it goes up and then it comes back down

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and this process repeats

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and you know what this is.

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This looks like a sine graph.

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But here's my question.

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Let's say we didn't start
the mass in the middle

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and give it a kick upwards.

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Let's say at t equal zero,

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we start the mass all the way
at the top and we let it drop.

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In other words, we do this.

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We start the mass up here.

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We let it drop.

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The graph would look like this.

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This time, if we were
to plot this over here,

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we'd get a graph that looked
a little something like this.

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It would start at the
top then would drop down.

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It reached some lowest point.

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It would come back up.

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Get up here.

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This process would repeat
over and over and over again.

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My question is,

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are these graphs the same
or are they different?

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Well, they're obviously different

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but almost everything is the same.

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Their amplitude is the same.

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They have the same displacement
from the equilibrium,

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that's the amplitude, that's the same.

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Their period is the same.

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The period is the time
between oscillations,

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so the periods are both the same.

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The only thing that's different

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is that one graph is shifted
compared to the other.

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In fact, check this out.

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If I were to take this green
graph, the initial one,

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and just shift it left,

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we get the exact same graph.

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They're almost exactly the same.

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One is just shifted,

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and so the word physicists
use for this idea

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that two graphs can differ
by the amount one is shifted

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is the idea of the phase.

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We would say that these two
oscillators are out of phase.

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How out of phase?

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Well, one whole cycle would be from here

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all the way to there.

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They're not that out of phase.

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They're just shifted by this much

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and that it turns out,

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for this case I've drawn here
is only a quarter of a cycle.

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So you could say these are a
quarter of a cycle out of phase

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or if you think of the unit circle,

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we know one quarter of a cycle

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would correspond to 90 degrees,

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either 90 degrees or pi over two.

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That's what these are.

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These are out of phase by pi
over two radians or 90 degrees.

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So how do we describe this
idea of phase mathematically?

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Well, if we were to try
to write down an equation

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for this green oscillator,

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say t equal zero started there,

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I might write down that,
okay, y is a function of time,

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the height of this oscillator
is a function of time

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would be the amplitude times,

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since this is starting at zero,

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I'm gonna use sine because
I know sine starts at zero,

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when t equals zero,

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of two pi over the period

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times the actual time.

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This little t is a variable.

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This little t represents the
actual time at a given moment.

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So let's make sure that this
equation actually works.

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If I were to start off at t equal zero

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and plug in t equal zero in here,

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the sine of zero

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is just zero,

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so this whole thing becomes zero

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and that's what I should have.

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I should start at t equal zero

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at a y value of zero, so that's good.

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And then as t gets a little bit bigger,

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this inside amount gets
a little bit bigger,

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the sine of a tiny positive amount,

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be a tiny positive number,

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so that's why this graph
goes up from there.

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Eventually, it gets to the peak.

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Where will it reach its peak?

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It will reach its peak

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when it gets through a quarter of a cycle.

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Remember, a whole cycle
is this whole amount here.

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So a quarter of a cycle

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is just when it gets to
the amplitude from zero

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and that will be at a time,

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t equals the period over four

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and the reason is that's
a quarter of a cycle.

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Does this math actually give us that?

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It does 'cause watch,

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if I plug in little t, the time variable

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as the period over four,

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I'll get the sine of
two pi over the period

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times the period over four,

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'cause that was my time.

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The periods cancel out.

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I'd get sine

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of

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two pi over four is pi over two

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and pi over two,

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that's 90 degrees.

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Sine of 90 degrees or sine
of pi over two radians,

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that's just one.

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That's the biggest that sine can be.

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That one times the amplitude

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is gonna give me a value for
the height of the amplitude.

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This is gonna describe
my oscillator perfectly

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'cause it's gonna give what the height is

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at any given moment of time.

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So that wasn't too bad,

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but what do we have to change

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in order to describe
the purple oscillator?

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So now, I wanna describe
this purple oscillator.

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You might say, "Oh, well, easy.

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"Just make this sine a cosine"

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and yeah, for this case,
it turns out that works,

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but pretend like you
didn't know you can do that

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or if you wanna make it harder,

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say this was not shifted a
perfect quarter of a cycle,

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maybe it was only shifted
like a ninth of a cycle,

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then cosine's not gonna do it for you.

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You need a more general way

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to adjust how much this wave is shifted

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and that's gonna be some sort
of phase constant in here.

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Where do we put the constant?

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You might think, "All right,
if I take my green graph,

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"I wanna shift it to the left."

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You might think, "Well, should I subtract"

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"some, like, amount out here?"

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That's not gonna work.

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That's gonna take your whole graph

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and subtract from the value
you get for the height,

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a constant amount every single time.

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Because of that, that
will just take your graph

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and shift it downward.

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That's not gonna work.

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If we added a value of B up here,

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that's also not gonna work.

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That would just end up
shifting our graph upward.

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All right, this B value
is not gonna do it for us.

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So, you might want that
in certain situations.

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That's not gonna shift
the graph left to right.

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It turns out, what we have to
do to shift to left to right

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is add a constant within this
argument of the sine here.

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So, if I wanted to describe
my purple oscillator,

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I'd say that y as a function
of time is the same amplitude.

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Let's just, again, use sine.

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It would be two pi over the period

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times time

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and then here's where the
phase constant comes in.

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I have to actually add a phase constant.

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You might think subtract.

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You might think subtract
because we want it to go left.

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It turns out, adding a phase constant

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will shift the graph left.

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This was counterintuitive to me.

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That's always freaked me out.

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I'd always forget this.

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How come adding a phase constant

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shifts the graph to the left?

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Well watch.

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If we take this equation now

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and instead of putting phi in there,

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this is the symbol we use

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for the phase constant in general, phi,

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and in this case, we
know what it should be.

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It should be a quarter of a cycle

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and for a sine graph, a quarter
of a cycle is pi over two.

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So let's just see if this works.

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At t equal zero,

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we used to get zero,
which is what we wanted,

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but now for the purple graph,

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I need to start at a maximum value.

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So at t equal zero,

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this whole amount right here becomes zero

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and I'm just left with sine of pi over two

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and sine of pi over two is one.

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That's a maximum value,

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so times amplitude

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would give us the amplitude
which is what we want.

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We want a graph that
starts at the amplitude.

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And this is better than
just putting cosine

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or at least it's better to know about

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because now, even if this
phase shift was pi over four

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or pi over nine or pi over 27,

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you could shift by any amount
you want using sine or cosine.

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Now you know how to shift these things.

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Adding a phase constant
will shift it to the left.

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Subtracting will shift it to the right.

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And the larger the phase
constant, the more it's shifted.

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You don't ever really need to
shift it by more than two pi

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since after you shift by two pi,

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you just get the same shape back again.

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So this constant in here,

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it's pi over two in this case.

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In general, it would look like this.

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You'd have some oscillator.

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It's got some amplitude.

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You could do sine or cosine

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plus a phase constant

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and this phase constant will determine

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how much this oscillator
is shifted left or right.

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And I should say be careful.

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Physicists can be sloppy here

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and use the same word for multiple things.

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Sometimes the word phase

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is used just for this little part here,

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this little added constant part,

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but sometimes, by phase,

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people really mean this
whole thing inside of here,

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this whole term that you're taking sine of

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because this is what's determining

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where you're at in your actual cycle

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and these ideas don't just
apply to a mass on a spring.

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You could write down
the equation for a wave.

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You have an extra term in
here for space, not just time,

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and guess what?

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There'd be a little constant
at the end that you could add.

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That would be the phase constant.

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So this idea of phase gives you a way

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to describe how two oscillators
or two waves are shifted

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with respect to one another

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and it lets you account for
all kinds of properties,

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like we had for this mass on a spring.