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I would now like to calculate
the potential function for two

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other conservative
forces that we

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encounter all the
time, spring forces

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and gravitational forces.

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Will begin with spring forces.

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So first off, let's begin with
some type of coordinate system.

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Suppose we have a spring.

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We have a block.

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This is my point x
equilibrium, unstretched

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length of the spring.

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And now whether we stretch
it or compress it here,

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I'll stretch it.

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I'll introduce a
coordinate function x.

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Union vector i, and
my spring force, f,

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is -kxi, where x can be
positive for a stretch spring,

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force is in the negative
i-hat direction.

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When x is negative,
negative times negative,

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means the spring force is in the
positive i-hat direction when

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it's compressed.

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So this is a force
that's a restoring force,

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always back to equilibrium.

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This is an example of
a conservative force.

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And now let's calculate the
change in potential energy.

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And then introduce a
zero reference point

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and get a potential
function for this force.

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So the first calculation
is straightforward.

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So, if we took our
displacement to be dx i-hat.

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And we take the
dot product of fds,

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we get dx i-hat dot
i-hat is 1, times dx.

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And that's just the x component
of the force displacement.

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And minus sign because the x
component of the force is -kx.

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And so now, if we were
to start our system,

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so, again, for
our initial state,

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let's say that the block from
the unstretched position,

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is stretched xi.

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And our final state, whether
it's stretched or compressed,

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it won't matter, but we'll just
stretch it out a little bit

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more to x-final.

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So now let's
calculate the change

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in potential energy between our
initial and our final states.

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Now, recall there's
a minus sign,

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because it's -w conservative.

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That's minus the integral
x-initial to x-final of -kxdx.

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We have 2 minus signs.

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That's our integration
variable, if you

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want to see all the detail.

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And when you simply
integrate x-prime dx-prime,

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you get x-prime squared over 2.

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And so the change in potential
energy between these two states

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is 1/2 kx-final squared,
minus 1/2 kx-initial squared.

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And that's physically meaningful
if I take my system initially.

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And the system ends
up in the final state.

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Then what I'm calculating
is minus the work

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done by the spring
force on the block

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as it goes from the initial
state to the final state.

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I'm not talking about the
work that an external agent

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does in stretching
or compressing it.

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This is explicitly the
work done by the spring

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force on the block as it
goes from the initial state

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to the final state.

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I introduced the minus
sign in our definition

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of potential energy.

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And so this quantity
represents the negative

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of the work done
by the spring force

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as a system goes from the
initial to the final states.

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Now, you may have
already thought about it,

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but the reference point
that we're going to use

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is x equals zero.

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The unstretched
length of the spring.

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And our reference potential
at that point will be zero.

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So our reference point is
actually the zero-point

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for the potential.

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And then our arbitrary
state, we can call this

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the referent state, if you like.

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That's probably better
than the reference point.

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Our arbitrary state will
be at some arbitrary

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stretch or compress.

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And our potential
energy function,

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at that reference, minus
the potential energy

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at the arbitrary state,
minus the reference state.

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Well, if we set xi equal
to 0 and x-final equal

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to x, as we have
in this expression,

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we simply get 1/2 kx squared.

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So the potential energy function
equals our reference potential,

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plus 1/2 kx squared.

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And we have defined that our
reference potential to be zero,

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it could have been anything.

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But we're making it 0 so that
we get a nice function, 1/2 kx

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square.

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And again, it's always
worthwhile to plot

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that function.

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It's a nice parabola.

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U of x, x.

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And you can see down here,
that's our reference point.