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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.