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In problem solving,
when we're working

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with problems with potential
and kinetic energy,

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remember we have
our main concept

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that the non-conservative
work causes

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potential energy and
kinetic energy to change.

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K final minus K initial plus
U final minus U initial.

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Now, how are we going to
apply this in problem solving?

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Well, what we'd
like to introduce

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is a tool, which we're going
to refer to as our energy state

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diagrams.

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So the way this works is-- let's
have an example to think about.

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Suppose you have a dome.

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So here's some dome.

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We can think of this
as the MIT Dome.

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And you have an object
that is initially

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at the top of the dome.

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And then this object is
sliding down the dome

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and at a later time it's at a
point somewhere along the dome.

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Now eventually it's going
to fall off the dome,

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but that's a separate question.

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So how can we use
our energy ideas

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to analyze this situation?

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And the way we do
it is that we choose

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first initial and final
states that we're considering.

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Because our first
idea in the energy

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diagram is to compare the
change in kinetic energy

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between the initial and
change of potential energy

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between the initial
and the final states.

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So in our dome problem, we
would choose our initial state.

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Let's draw our
object at the top.

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And in our final
state, separately,

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let's draw the object over here.

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And then we want to
parametrize these states

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by some type of
coordinate system.

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So here we're going to have
some type of coordinate system

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that we use for both.

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Energy is a scalar, so we
don't have to worry about it.

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And what I'll do is I'll
define an angle theta final.

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Here from the vertical,
here theta initial is 0.

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And I've now parametrized
my initial and final states.

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Now, the important thing that
we'll do in the energy diagram

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is to choose our 0
potential energy.

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Where are we going
to choose this?

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So we could say either
a surface-- we'll

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call that potential energy.

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And in this diagram,
I can choose my 0

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for potential energy
anywhere I want.

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But I want to draw it on my
initial and my final states.

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So I'm going to
choose it right here,

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and I'll denote
this as u equals 0.

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And now I can now make a
list-- so I want to identify.

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So step three is to identify
the K and U for each state.

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So here we have K
initial, it's at rest 0.

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And U initial, well, I
didn't introduce a parameter

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R. U initialize is how high the
gravitational potential energy.

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So in this particular
case, Mg times R.

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Now over here, K final,
is 1/2 M V final squared.

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And U final, we can denote
it here if I plot this out--

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and that's R. And then
you can see that this

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is R cosine theta final.

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So that's MgR
cosine theta final.

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Now, the fourth step is to
identify W non-conservative.

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Is there any
non-conservative work?

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Now, here let's assume
the dome is frictionless.

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And that means that W
non-conservative is 0.

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And our last step 5 is
we can apply the energy

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principle, which is
W non-conservative

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equals delta K-- let's write
out everything explicit

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now that we've defined K
final-- minus K initial plus U

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final minus U initial.

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And you see the power of these
diagrams and this methodology

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is we've now defined very
explicitly every single term

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that appears in the
energy principle.

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And so we can write out
our result that 0 K final,

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1/2 M V final squared,
K initial, 0 plus U

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final, MgR cosine theta final,
minus U initial, MgR, and there

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we have applied the
energy principle

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using the tool of
energy diagrams.