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- [Voiceover] A block
is initially at position

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x = zero,

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and in contact with an uncompressed spring

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of negligible mass.

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The block is pushed back
along a frictionless

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surface from position x = zero

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to x = -D, as shown above,

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compressing the spring by an amount

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delta x = D.

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So, the block starts here,
and it's just in contact

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with the spring, so it's initially,

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the spring is uncompressed.

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And it's just touching the block.

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And then we start to compress the spring

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by pushing the block to the left,

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and we compress it by an amount, D.

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They tell us that, right
there, delta x is = to D,

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so we compress,

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we move this block back
over to the left by D,

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that compresses the spring by D.

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The block is then released at x = zero,

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the block enters a rough part of the track

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and eventually comes to rest at position

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x = 3D.

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So when we compress the spring,

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we're actually doing to
work to compress the spring,

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so that work, that energy
from the, or the work

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we're doing, gets stored
as potential energy

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in the spring-block system.

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And then when we let go,
that potential energy is

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going to be converted to kinetic energy,

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and that block is going to be accelerated

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all the way until we get back to x = zero,

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then the spring is back to uncompressed,

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so it's not gonna keep
pushing on the block

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after that point.

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And then the block's going
to have this kinetic energy

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and if there was no friction
in this gray part here,

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it would just keep on going forever.

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And if there's no air
resistance, and we're assuming

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no air resistance for
this, for this problem,

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but since there is
friction, it's just going to

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decelerate it at a constant rate.

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You're going to have a
constant force of friction

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being applied to this block.

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So, let's see, they say, they tell us

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that it's going to come to rest at x = 3D,

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the coefficient of kinetic
friction between the block

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and the rough track is mu.

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Alright, on the axes
below, sketch and label

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graphs of the following two quantities

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as a function of the position of the block

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between x = negative D and x = 3D.

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You do not need to calculate
values for the vertical axis

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but the same vertical scale should be used

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for both quantities.

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So they have the kinetic
energy of the block

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and the potential energy
of the block-spring system.

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So let's first focus on
the potential energy, U,

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because when we start
the first part of this,

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when we're compressing the spring,

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that's when we're starting
to put potential energy

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into this spring-block system.

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So you have to think about
what is the potential energy

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of a compressed spring?

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Well, the potential energy

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the potential energy

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is equal to one-half
times the spring constant

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times how much you compress
the spring squared.

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So if we wanna say delta x is how much you

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compress the spring, that squared.

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Now, if what I just wrote is

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completely unfamiliar to you,

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I encourage you to watch
the videos on Khan Academy,

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the potential energy of
a compressed spring or

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the work necessary to compress a spring,

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cause the work necessary
to compress the spring

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that's going to be the potential energy

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that you're essentially
putting into that system.

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And so, for this, as we
compress the spring to D,

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you are, you're going to end up with

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a potential energy of one-half
times the spring constant

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x our change in x is D,

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our change in x is D.

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x D, x D squared.

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So let's plot that on
this right over here.

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So right,

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whoops,

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right when we are at x = zero

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there's no potential energy in our system,

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but then we start to compress it,

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and when we get to x = D,

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we're going to have a potential energy

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of one-half times the spring
constant times D squared.

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So let's just say this, right over here,

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let's say that over there,

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actually let me do a,

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let's see that one is,

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actually I'll do it over here so

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it'll be useful for me later on.

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So, let's say that this,
right over here, is one-half

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times our spring constant times D squared.

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So this is what our potential
energy's going to be like

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once we've compressed the spring by D.

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And it's not going to be a

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linear relationship,
remember the potential energy

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potential energy is
equal to one-half times

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the spring constant,

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times the spring constant,
times how much you've

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compressed the spring squared.

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So, the potential energy
increases as a sqaure

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of how much we've compressed the spring.

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So when we've compressed the spring

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half as much, we're going
to have one-fourth of

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the potential energy.

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So it's going to look
like this, it's gonna be

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you can view it as the
left side of a parabola.

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So it's going to, going to look something,

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something like this.

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So that's the potential energy.

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Now, when you're in this
point, when the thing is

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fully compressed, and then
you let go, what happens?

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Well that potential energy is
turned into kinetic energy,

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so as the spring, as the
spring accelerates the block,

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you're gonna go down this
potential energy curve,

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as you go to the right, but then, it gets

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converted to kinetic energy.

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So the potential energy
plus the kinetic energy

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needs to be constant, at
least over this period from

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x = negative D to x = zero.

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So the kinetic energy starts off at zero,

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it's stationary,

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but then, it starts, the block
starts getting accelerated.

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It starts getting accelerated.

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And the sum, the sum of these two things

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needs to be equal to one-half
times our spring constant

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times D squared.

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And so you can see if you,

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if you were to add these two curves

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at any position, you are going, their sum

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is going to sum up to this value.

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And so right when you
get back to x = zero,

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all of that potential energy
has been converted into

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kinetic energy.

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And then that kinetic
energy, we would stay at that

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high kinetic energy if
there was no friction

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or no air resistance.

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But we know that the
block comes to a rest at

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x is = to 3D.

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So all the kinetic energy
is gone at that point,

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and you might say well, what's
that getting converted into?

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Well, it's gonna get
converted into, into heat,

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due to the friction.

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So that's where, ya know, energy cannot be

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cannot be created out of thin
air or lost into thin air,

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it's converted from one form to another.

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And so the question is,

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what type of a curve is this?

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Do we just connect these with a line?

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Or is it some type of a curve?

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And the key realization is:

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is that you have a
constant force of friction

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the entire time that the
block is being slowed down,

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the coefficient of
friction doesn't change,

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so the force of friction,
and the mass of the block

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isn't changing, so the force of friction's

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going to be the same.

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And it's acting against
the motion of the block.

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So you can, you can view
the friction as essentially

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doing this negative work,

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and so it's sapping the energy away,

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if you think about it
relative to distance,

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in a given amount of
distance, it's sapping away

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the same amount of energy, it's doing that

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same amount of negative work.

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And so, this is going to
decrease at a linear rate.

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So, let me draw that.

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So it's gonna be a linear decrease,

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just like that.

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And the key thing to remind yourself is:

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is this is a plot of
energy versus position,

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not velocity versus position
or velocity versus time,

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or energy versus time.

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This is energy versus position,
and that's what gives us

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this linear relationship right over here.

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So, we have the kinetic
energy, k, of the block.

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That's what I did in magenta,

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so this is the kinetic energy.

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Kinetic,

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kinetic energy,

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and in blue, just to make
sure I label it right,

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this is the potential energy,

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potential,

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potential energy.