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Now that we've calculated the
change in potential energy

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between some initial
and final heights

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for the gravitational problem,
mg of y final minus y initial.

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For this conservative
force of gravity,

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and we had our coordinate
system like that,

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we were able to calculate the
change in potential energy.

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And remember our theorem is that
non-conservative work equals

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Delta K plus Delta U.

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Now what we'd like
to do is establish

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the concept of a reference
point for potential energy.

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And we'll do that
as follows, see

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the change in potential
energy only depended

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on say, our initial state
and our final state.

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What we'd like to do is
introduce the concept

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of a reference point.

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So let's identify some point
Yp to be our reference point.

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And we'll define a potential Yp
to be the reference potential.

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And if we want to talk
about the potential energy

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in a final state,
we'll always refer it

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to the difference between
that and the reference point.

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So this is Delta U between the
final state and the reference

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point.

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Likewise, for the
initial state, we'll

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always refer that with respect
to our reference point.

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So any state that we
have, we can always

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refer the potential energy
difference between some state

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and some reference point.

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Why do we do this?

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Because notice that if we look
at U final minus U reference

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point, and we subtract from
that U initial minus U reference

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point, then the
difference-- the reference

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points here we have a
minus, here we have a plus--

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this is just equal
to U final minus U

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initial, which is Delta U.

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So the change in potential
energy between any two states

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is independent of how we
choose our reference potential.

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But it can make
calculations easier

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when we can identify what
the potential energy is

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for a little state.

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Now let's look at our example.

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So for our
gravitational problem we

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will choose y reference
point to be 0.

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And we'll choose the potential
energy at this reference point

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also to be 0.

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So our reference potential
at the origin is 0,

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and I'll denote it like that.

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Then the potential energy
at some initial state

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minus the reference
point-- well,

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we can use our formula
here, because this

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is between any two states.

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So this is mg y initial
minus the reference point.

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But our reference
point was 0, and so we

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see that U initial minus y
reference point, which was also

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0, is just equal to mg Yi.

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And so we have this statement
that the potential energy

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difference between our initial
state and the reference point

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is just mg where
Yi is the height

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that the initial state is
above the reference point.

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In a similar way, we have
U final is mg y final.

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And so we see we
recover what we expect.

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This is just mg y
final minus y initial.

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Now this we can generalize
just a little bit

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by saying that for
any-- let's write

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that for us our mass,
which is for any height y

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our potential energy function
for the gravitational force

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U(y) is equal to mg y.

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That's a formula that many
of you have seen before.

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But it's very important to note
that with this formula, U of 0

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equals 0, because that's
our reference point.

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And that becomes our
potential energy function

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for the gravitational force.

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Now our next step will
be to do the same thing

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for spring forces and inverse
square gravitational forces.

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And we'll also look at graphical
analysis of this function.