WEBVTT

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Let's consider an example
of motion, in which we want

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to use our energy concepts.

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So suppose we have a ramp
which is circular, radius r.

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And we have an object here.

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And now, here's a surface
a certain distance,

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d, that has friction, where
the coefficients of friction

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is non-uniform.

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So we'll write it mu-0 And
we'll write it as mu-1x.

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We'll take x equals zero.

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And then here, there's
a spring and a wall.

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And here there is no friction.

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Now as we drop, assuming we
have enough height here, h,

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so that this block will
slide across the friction,

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make it all the way across and
start compressing its spring.

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What I'd like to
find is how much

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the spring has been compressed.

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OK, now how do we analyze this?

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Well the key is to use
our energy principle,

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where we have w-external
equals the change

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in mechanical energy.

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And the key is to,
what we're going

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to do is the tool that
we're going to use

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is what we call energy
diagrams for the initial.

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So what does this mean?

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We want to choose, we
want to first identify

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the initial and final states
that we're referring to.

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So, in our picture, here
is our initial state.

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And I'll draw the final state
in when the spring has been

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compressed a distance, x-final.

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So I drew it on my diagram.

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So we have initial
and final states.

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And now what we
want to do is choose

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reference points,
zero-points, zero-point

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for each potential function.

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And show that on our diagram.

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So for the potential
energy of gravity, here,

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if we chose this to be
y, then this y equals 0.

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And u at y equals 0 is 0.

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And we'll call that the
zero point for gravity.

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And the zero point
for the spring

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is when this x, now I'm going
to call here x equals 0.

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So, let's call this a variable,
I can call it anything I want.

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I'll call it u-final.

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And this is where u is zero.

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So u-final just measures
the stretch of the spring.

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And so at y equals 0 is 0.

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And u spring little
u equal 0, 0.

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So now I can
identify my energies.

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So let's talk about
the initial energy

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is all gravitational potential.

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That's mgy.

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It's starting at rest.

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And e-final well, here,
this is the distance

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where it comes to rest, also.

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So there's no final
kinetic energy.

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There's no gravitational
potential energy,

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because we're on the
surface at y equals 0.

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But our spring has
been compressed

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by 1/2 k little-u-final square.

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Now, in terms of
our external work

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equals the change in
mechanical energy,

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we have now identified
the right-hand side

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using these tools of
the energy diagrams.

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And I can write my description
u-final squared, minus mgy.

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Now what I have to do is
think about the friction

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force, f kinetic friction,
as the object moves.

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This friction force
is non-uniform.

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If we were to draw n, and mg,
then our friction force here

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is equal to the integral minus
the integral from x equals 0,

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to x equals d.

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Equals 0 to x equal d,
of the friction force,

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which is equal to the
coefficient of friction, ukndx.

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Now that 0 to d, notice that
our coefficient of friction

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is varying.

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And I chose it intentionally
to show you that friction

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is really an integral.

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So we have mu-0 plus mu-1 xmgdx.

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Let's put our integration
variable in there.

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And these are just two
separate integrals.

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The first one is easy.

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It's minus mu-0 mgd.

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And the second one is -mu-1 mg.

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And we're just integrating
x-prime, dx prime

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between 0 and d.

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So that's simply
d-squared over 2.

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And that's equal to
ku-final squared minus mgy.

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Let's write that as y-initial.

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And we're starting it
at y-initial equals h.

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And so, even though
this is complicated,

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I can now solve for how much
this spring has been compressed

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with a little bit of algebra.

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And so I'm just going to
bring a bunch of terms

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over to the other side.

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And take the square root of 2,
divided by k of mgy-initial.

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Minus mu-0 mgd.

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Minus mu-1 mgd-squared over 2.

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And that's how much the
spring is compressed.

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Notice what I did not
do was divide this up

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into a bunch of
different motions.

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I picked an initial state.

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I picked a final state.

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I drew my energy diagrams
with my zero points.

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I described the key parameters
of initial and final states,

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y-i and u-final.

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I defined the initial
mechanical energy,

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the final mechanical energy.

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And then applied
the work energy,

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work mechanical
energy principle.

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I had to integrate the
friction force because it

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was non-trivial and solve
for how much the spring has

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been compressed.