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We now would like to explore
the concept of potential energy

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difference for a
conservative force.

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Let's consider the
following case.

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Suppose you have
an object of mass m

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and it's located at a
certain height y i initial.

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And so this is
our initial state.

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And we would like to
move this object upwards.

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So it goes up to
a height y final.

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This is our final state.

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And in our initial
state, we might

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have some initial velocity.

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And then the final state
might have some final speed.

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And what we'd like
to do now is--

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we know that if there is
a gravitational force, mg,

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acting downwards, that
this gravitational force is

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a conservative force.

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And now what I'd like to do
is-- because the force is

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conservative, it doesn't
depend on the path

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that our object goes
to the final state.

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It just depends
on the parameters

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that describe the initial
and the final states.

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In fact, we'll see it only
depends on the initial height

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and the final height.

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So what I'd like to do is define
a potential energy difference

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in the following way.

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So as a definition, the
potential energy difference

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between these two
points is given

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by the negative of the work
done by the gravitational force

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in going from the initial
state to the final state.

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Notice this negative sign.

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Because the gravitational
force here, Fg

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is minus mg j hat-- where
we're taking j hat up--

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we've seen that
this is an example

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of a conservative force.

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And it doesn't matter how
we went from the initial

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to the final states.

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So we can generalize this
idea for potential energy.

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But first, let's just remind
ourselves of the calculation.

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When we did this
calculation before,

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we had u final minus u final
initial equals a negative sign

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in the definition.

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And when we calculated the work
done by the conservative force,

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we had negative mg y
final minus y initial.

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Notice the two minus signs.

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So we get mg y final
minus y initial.

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Now, many times people
talk about changes

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in potential energy.

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So when I write delta u, I mean
precisely the potential energy

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at the final state
minus potential energy

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of the initial state-- the
change in potential energy.

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And that's equal to mg times
the change in the displacement.

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And so we see here
for this example,

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that if delta y
is positive, that

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implies that the potential
energy is increasing.

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Now let's connect that
to our definition.

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Why is the potential
energy increasing

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when we raise something?

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Well, the gravitational
force points downwards.

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The displacement is upwards.

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So the work done is negative.

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And another minus sign means
the change in potential energy

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is positive.

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If the object is moving
with no change in height,

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that tells us that the potential
energy is change is zero.

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And finally, if delta
y is less than zero,

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that implies that the change in
potential energy is negative.

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Now again let's
examine this case.

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If we were actually lowering
an object, dropping it down.

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The object moved downward
from the initial state

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to a final state, the
gravitational force

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is downward, the
displacement is downward,

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the integral is positive,
the extra minus sign

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corresponds to the change
of potential energy

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being negative.