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Let's see if we can use what we
know about springs now to

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get a little intuition
about how the

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spring moves over time.

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And hopefully we'll
learn a little bit

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about harmonic motion.

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We'll actually even step into
the world of differential

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equations a little bit.

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And don't get daunted
when we get there.

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Or just close your eyes
when it happens.

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Anyway, so I've drawn a spring,
like I've done in the

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00:00:18,340 --> 00:00:19,410
last couple of videos.

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And 0, this point in the x-axis,
that's where the

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spring's natural resting
state is.

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00:00:25,500 --> 00:00:28,890
And in this example I
have a mass, mass m,

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attached to the spring.

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And I've stretched the string.

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I've essentially pulled it.

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So the mass is now sitting
at point A.

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So what's going to
happen to this?

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Well, as we know, the force, the
restorative force of the

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spring, is equal to minus some

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constant, times the x position.

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The x position starting at A.

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So initially the spring
is going to pull

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back this way, right?

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The spring is going to
pull back this way.

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It's going to get faster and
faster and faster and faster.

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And we learned that at this
point, it has a lot of

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potential energy.

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At this point, when it kind of
gets back to its resting

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state, it'll have a lot of
velocity and a lot of kinetic

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energy, but very little
potential energy.

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But then its momentum is going
to keep it going, and it's

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going to compress the spring all
the way, until all of that

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kinetic energy is turned back
into potential energy.

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Then the process will
start over again.

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So let's see if we can just get
an intuition for what x

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will look like as a
function of time.

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So our goal is to figure out x
of t, x as a function of time.

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That's going to be our goal
on this video and

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00:01:31,900 --> 00:01:33,610
probably the next few.

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So let's just get an intuition
for what's happening here.

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So let me try to graph x
as a function of time.

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So time is the independent
variable.

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And I'll start at time
is equal to 0.

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So this is the time axis.

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Let me draw the x-axis.

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This might be a little unusual
for you, for me to draw the

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x-axis in the vertical, but
that's because x is the

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dependent variable in
this situation.

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So that's the x-axis,
very unusually.

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Or we could say x of t, just so
you know x is a function of

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time, x of t.

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And this state, that I've drawn
here, this is at time

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equals 0, right?

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So this is at 0.

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Let me switch colors.

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So at time equals 0, what is
the x position of the mass?

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Well the x position
is A, right?

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So if I draw this, this is A.

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Actually, let me draw
a line there.

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That might come in useful.

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This is A.

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And then this is going to be--
let me try to make it

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relatively-- that
is negative A.

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That's minus A.

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So at time t equals
0, where is it?

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Well it's at A.

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So this is where the
graph is, right?

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Actually, let's do something
interesting.

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Let's define the period.

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So the period I'll do
with a capital T.

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Let's say the period is how long
it takes for this mass to

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go from this position.

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It's going to accelerate,
accelerate, accelerate,

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accelerate.

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Be going really fast at this
point, all kinetic energy.

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And then start slowing down,
slowing down, slowing down,

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00:03:17,080 --> 00:03:17,800
slowing down.

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And then do that whole process
all the way back.

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Let's say T is the amount of
time it takes to do that whole

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00:03:22,780 --> 00:03:24,560
process, right?

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00:03:24,560 --> 00:03:31,850
So at time 0 today, and then we
also know that at time T--

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this is time T-- it'll
also be at A, right?

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I'm just trying to graph some
points that I know of this

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function and just see if I can
get some intuition of what

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00:03:43,050 --> 00:03:46,590
this function might
be analytically.

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So if it takes T seconds to go
there and back, it takes T

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over 2 seconds to
get here, right?

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00:03:54,090 --> 00:03:56,470
The same amount of time it takes
to get here was also the

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same amount of time it
takes to get back.

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So at T over 2 what's going
to be the x position?

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00:04:05,880 --> 00:04:08,780
Well at T over 2, the block
is going to be here.

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00:04:08,780 --> 00:04:10,690
It will have compressed
all the way over here.

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So at T over 2, it'll
have been here.

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00:04:12,550 --> 00:04:15,440


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And then at the points in
between, it will be at x

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equals 0, right?

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It'll be there and there.

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Hopefully that makes sense.

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So now we know these points.

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But let's think about what the
actual function looks like.

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Will it just be a straight line
down, then a straight

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line up, and then the straight
line down, and then a

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straight line up.

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That would imply-- think about
it-- if you have a straight

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line down that whole time, that
means that you would have

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a constant rate of change
of your x value.

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Or another way of thinking about
that is that you would

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have a constant velocity,
right?

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Well do we have a constant
velocity this entire time?

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Well, no.

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We know that at this point right
here you have a very

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00:04:55,200 --> 00:04:57,650
high velocity, right?

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You have a very high velocity.

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We know at this point you have
a very low velocity.

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So you're accelerating
this entire time.

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And you actually, the more you
think about it, you're

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actually accelerating at
a decreasing rate.

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But you're accelerating
the entire time.

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And then you're accelerating and
then you're decelerating

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this entire time.

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So your actual rate of change
of x is not constant, so you

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wouldn't have a zigzag
pattern, right?

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And it'll keep going here and
then you'll have a point here.

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So what's happening?

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When you start off, you're
going very slow.

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Your change of x is very slow.

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And then you start
accelerating.

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And then, once you get to this
point, right here, you start

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decelerating.

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Until at this point, your
velocity is exactly 0.

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So your rate of change, or your
slope, is going to be 0.

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And then you're going to start
accelerating back.

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Your velocity is going to get
faster, faster, faster.

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It's going to be really
fast at this point.

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And then you'll start
decelerating at that point.

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So at this point, what does
this point correspond to?

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You're back at A.

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So at this point your velocity
is now 0 again.

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So the rate of change
of x is 0.

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00:06:06,350 --> 00:06:08,650
And now you're going to
start accelerating.

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Your slope increases, increases,
increases.

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This is the point of highest
kinetic energy right here.

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Then your velocity starts
slowing down.

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And notice here, your slope
at these points is 0.

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So that means you
have no kinetic

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energy at those points.

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And it just keeps on going.

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On and on and on
and on and on.

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So what does this look like?

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Well, I haven't proven it to
you, but out of all the

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functions that I have in my
repertoire, this looks an

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awful lot like a trigonometric
function.

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And if I had to pick one,
I would pick cosine.

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Well why?

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Because when cosine is 0--
I'll write it down here--

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cosine of 0 is equal
to 1, right?

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00:06:47,360 --> 00:06:50,610
So when t equals 0, this
function is equal to A.

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00:06:50,610 --> 00:06:59,880
So this function probably looks
something like A cosine

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00:06:59,880 --> 00:07:05,730
of-- and I'll just use the
variable omega t-- it probably

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00:07:05,730 --> 00:07:08,650
looks something like that,
this function.

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And we'll learn in a second
that it looks

167
00:07:10,570 --> 00:07:11,110
exactly like that.

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But I want to prove it
to you, so don't just

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take my word for it.

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So let's just figure out how we
can figure out what w is.

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00:07:17,490 --> 00:07:21,100
And it's probably a function of
the mass of this object and

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also probably a function
of the spring

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constant, but I'm not sure.

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So let's see what we
can figure out.

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Well now I'm about to embark
into a little bit of calculus.

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00:07:31,000 --> 00:07:32,380
Actually, a decent
bit of calculus.

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00:07:32,380 --> 00:07:34,470
And we'll actually even touch
on differential equations.

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This might be the first
differential equation you see

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00:07:36,790 --> 00:07:39,620
in your life, so it's a
momentous occasion.

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00:07:39,620 --> 00:07:41,120
But let's just move forward.

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00:07:41,120 --> 00:07:42,670
Close your eyes if you don't
want to be confused, or go

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00:07:42,670 --> 00:07:46,290
watch the calculus videos at
least so you know what a

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00:07:46,290 --> 00:07:47,510
derivative is.

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00:07:47,510 --> 00:07:52,190
So let's write this seemingly
simple equation, or let's

185
00:07:52,190 --> 00:07:54,570
rewrite it in ways
that we know.

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00:07:54,570 --> 00:07:57,620
So what's the definition
of force?

187
00:07:57,620 --> 00:08:00,030
Force is mass times
acceleration, right?

188
00:08:00,030 --> 00:08:05,820
So we can rewrite Hooke's law
as-- let me switch colors--

189
00:08:05,820 --> 00:08:11,020
mass times acceleration is
equal to minus the spring

190
00:08:11,020 --> 00:08:15,600
constant, times the
position, right?

191
00:08:15,600 --> 00:08:17,840
And I'll actually write the
position as a function of t,

192
00:08:17,840 --> 00:08:19,010
just so you remember.

193
00:08:19,010 --> 00:08:22,230
We're so used to x being the
independent variable, that if

194
00:08:22,230 --> 00:08:24,470
I didn't write that function of
t, it might get confusing.

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00:08:24,470 --> 00:08:26,770
You're like, oh I thought x is
the independent variable.

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00:08:26,770 --> 00:08:28,350
No.

197
00:08:28,350 --> 00:08:31,150
Because in this function that we
want to figure out, we want

198
00:08:31,150 --> 00:08:33,270
to know what happens as
a function of time?

199
00:08:33,270 --> 00:08:35,280
So actually this is also
maybe a good review

200
00:08:35,280 --> 00:08:38,080
of parametric equations.

201
00:08:38,080 --> 00:08:39,630
This is where we get
into calculus.

202
00:08:39,630 --> 00:08:40,880
What is acceleration?

203
00:08:40,880 --> 00:08:44,670


204
00:08:44,670 --> 00:08:51,650
If I call my position x, my
position is equal to x as a

205
00:08:51,650 --> 00:08:52,980
function of t, right?

206
00:08:52,980 --> 00:08:56,500
I put in some time, and it tells
me what my x value is.

207
00:08:56,500 --> 00:08:57,510
That's my position.

208
00:08:57,510 --> 00:08:58,890
What's my velocity?

209
00:08:58,890 --> 00:09:02,300
Well my velocity is the
derivative of this, right?

210
00:09:02,300 --> 00:09:06,320
My velocity, at any given point,
is going to be the

211
00:09:06,320 --> 00:09:07,890
derivative of this function.

212
00:09:07,890 --> 00:09:11,130
The rate of change of this
function with respect to t.

213
00:09:11,130 --> 00:09:13,340
So I would take the rate
of change with

214
00:09:13,340 --> 00:09:17,080
respect to t, x of t.

215
00:09:17,080 --> 00:09:23,030
And I could write
that as dx, dt.

216
00:09:23,030 --> 00:09:24,490
And then what's acceleration?

217
00:09:24,490 --> 00:09:26,410
Well acceleration is just
the rate of change

218
00:09:26,410 --> 00:09:28,340
of velocity, right?

219
00:09:28,340 --> 00:09:30,560
So it would be taking the
derivative of this.

220
00:09:30,560 --> 00:09:32,940
Or another way of doing it, it's
like taking the second

221
00:09:32,940 --> 00:09:36,460
derivative of the position
function, right?

222
00:09:36,460 --> 00:09:41,850
So in this situation,
acceleration is equal to, we

223
00:09:41,850 --> 00:09:44,930
could write it as-- I'm just
showing you all different

224
00:09:44,930 --> 00:09:49,650
notations-- x prime prime of t,
second derivative of x with

225
00:09:49,650 --> 00:09:50,480
respect to t.

226
00:09:50,480 --> 00:09:53,930
Or-- these are just notational--
d squared x over

227
00:09:53,930 --> 00:09:55,910
dt squared.

228
00:09:55,910 --> 00:09:57,000
So that's the second
derivative.

229
00:09:57,000 --> 00:09:58,280
Oh it looks like I'm running
out of time.

230
00:09:58,280 --> 00:09:59,480
So I'll see you in
the next video.

231
00:09:59,480 --> 00:10:01,510
Remember what I just
wrote. just wrote

232
00:10:01,510 --> 00:00:00,000