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Now that we've introduced
mechanical energy

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in our potential
energy functions,

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we're describing our
systems differently.

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We talk about states.

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We talk about the potential
energy of that state.

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We talk about the mechanical
energy of that state.

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Remember we're always
referring to a reference state

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for a reference potential.

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But in one dimension,
what we have

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is that the
potential energy say,

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in some final state, minus the
change of potential energies

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from some initial state
was that integral x final

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of the x component
of-- I'm going

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to put c up there for
conservative force, dx.

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And now, so the potential
energy difference

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is the integral of the
force with the minus sign.

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Now let's look at a
fundamental theorem

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of calculus, which tells
us that any time you

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take the difference of a
function between two end

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points, then by
definition that's

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the derivative integrated
with respect to dx.

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So when we compare these two
pictures, this is a map here.

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This is our physics,
how we define them.

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That when we compare
these two pictures,

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we see that we can recover
the conservative force

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by taking the derivative
minus the derivative

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of the potential function.

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Here, force does not depend
on any reference point.

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And when we differentiate
a constant, that's 0.

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So this is independent
of the reference point.

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And this enables us
to, when we think

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about the potential function
and its first derivative,

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then this tells us about forces.

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Let's look at an example.

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Suppose again we look
at our spring potential

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where we're talking about
the potential energy

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function of a spring
where at our zero

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point where it was unstretched
was our reference point.

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And if we plotted this
function-- so let's

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plot that function.

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So here is U of x versus x.

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Now we can talk about
at any given point--

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so suppose we're
at a point here.

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Maybe our energy has
some fixed value.

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Then the slope at this
point is equal to du dx,

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and the force is
minus that slope.

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So here you can see that
the slope is positive.

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So the force is negative.

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So I can write Fx like that.

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So here Fx is negative,
so our actual force

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is pointing inward.

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When we're on this side
of the potential function,

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my slope is negative.

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So the x component of
the force is positive.

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So my force is pointing
everywhere on this side

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back to the unstretched length.

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So knowledge of the
potential function,

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also by knowledge of
its first derivative,

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gives us information
about the force

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at any point, any state
that the system is in.

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So when we talk about
potential implicitly,

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we also know what the force is.

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And the potential function
is enough to tell us

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what the force is at any point.

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And let's just check
for this simple case.

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This is minus du dx.

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When you differentiate
that, you get minus kx,

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and we know that's
the spring force.

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So this is why we're
suddenly shifting our focus

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to our state, our function u of
x, and it's first derivative.