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Let's consider a
system in which there

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is no non-conservative
work and there's

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a potential energy associated
with forces in the system.

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And for an example,
let's model some system

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by the following
function b x-cubed.

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And we know that our
definition of force

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is that it's minus the
change in potential energy.

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So if we differentiate this,
our force is a function of x.

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And because of that
minus sign, you always

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have to be careful when you
take that type of derivative,

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we get as far as the force
goes, 2ax minus 3b x-squared.

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Now we ask ourselves, where
does this force vanish?

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So if we look at the 0
points for the force,

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we can see immediately that
there is one x equals 0.

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The other one, if we said
2ax equals a 3b x-squared

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tells us that we have another 0
point at 2/3 over 2/3 a over b.

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Now let's also for example,
just put some numbers in.

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For some simplicity, let's
write this as 4 joules per meter

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squared for a.

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Why do we have meter squared?

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Well, the units of
potential energy are joules,

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and we're multiplying
by x-squared.

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And so you can see
that d also we're

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going to write as 2
joules metered to minus 3.

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And if we put those
numbers in, we

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can see that we'll get 8 over 6.

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And 8 over 6 is just 4/3,
and that will be in meters

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where the other 0 point is.

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So that's nice, but we'd like
to do some properties of these 0

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points.

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If we had a particle that
we started with some energy

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at one 0 point,
where would it go?

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What would it do
at the other point?

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So in order to understand
a little bit more

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detail about these
0 points, let's

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make a plot of our potential
energy U of x verse x.

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Now to begin with, we
want to ask ourselves

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how does this function behave?

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And for big values of x, the
x cube piece will dominate.

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And if we're positive x, we
dominate with a negative value

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for potential energy.

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So our potential energy function
goes off into negative infinity

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as x goes to infinity.

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Now on the other
side of the scale,

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the x-cubed term will dominate
for negative values of x.

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But this term is negative,
and that's negative,

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so up here it will
go off to infinity

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in the positive direction.

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Now, where are the zeroes
of the potential function?

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There's obviously
one at x equals 0.

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And with these values, it's
not 100% obvious immediately,

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but 4x squared
minus 2b x-cubed--

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if you set x equal to 2, 4 times
2 is 8, b 2 times 2, 4 times

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4 is 16.

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If b is 2 and x is 2, that's 16.

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So we get a cancellation
at x equals 2.

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And so we have a point up here.

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I'm sorry, that's our
0 point at x equals 2.

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Now, we said that the force
at x equals 4/3 had a 0.

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And so if we plot
this function, it's

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going to look
something like that.

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Now, let's focus
on these two points

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right here where our force is 0.

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Remember, that this
is minus the slope.

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And so we can see
immediately that these two

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points have 0 slope and
so the force is 0 there.

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Notice, potential energy always
depends on a reference point.

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So whether the potential energy
is 0 or not is not crucial.

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But the physical fact that
the force 0 there has physical

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mean.

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So here this is one of
our points F of x is 0.

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Now, what is the
property of this point?

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Well, suppose we ask ourselves
if the particle were just

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displaced slightly
from this position

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where the force is 0, which
way would the force point?

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Well in order to answer
a question like that,

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we can look at the slopes
on either side of the 0.

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So over here the
slope is negative,

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but the force is
minus the slope.

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So F of x is positive
on this side.

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And so if a particle is
displaced a little bit

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from the 0 point, it
moves off to the right,

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and it will continue to feel
a force in this direction,

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and so it will move away from
the 0 point of the force.

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Now on the other side, we do
the same type of argument.

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And over here the
slope is positive,

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but the force is negative.

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So again, what we
see on this side,

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the particle will move
away from this 0 point.

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And so if our particle
just started there

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and it had a little
bit of kinetic energy,

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it would move one way or the
other away from that point.

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And so we call this an
unstable equilibrium point.

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And that's an abbreviation
for equilibrium.

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Now, what about this
0 point over here?

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Well, we already know
that if a particle is

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on this side of the 0 it
feels the force this way.

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But what if our particle
were on this side?

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Well, once again, we
analyze the slope.

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The slope is negative,
force is minus the slope,

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so it's positive.

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So everywhere on this
side, the particle

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is getting a restoring
force back to equilibrium.

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And so if we displace
this particle just

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on either side, if it
doesn't have enough energy

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to get over this point,
then the particle

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will stay around this area.

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And that is why we give
this point the name

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of stable equilibrium point.