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We're now going to introduce
the concept of potential energy.
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Let's begin by
considering a system where
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a conservative force is acting.
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So I'll consider a conservative
force, which I'll call F sub c.
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So force set to force, the work
integral is path independent.
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So the work integral for this
force is the integral of F sub
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c .ds, which is the path
going from point A to point B.
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And for a conservative force,
this integral does not depend
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upon the path from A to B.
It's independent of A and B.
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So it depends only
upon the endpoints.
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So this is a path
independent integral.
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And since it depends
only upon the endpoints,
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I can write it, since it's going
to be an integral from point A
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to point B-- this
integral must be
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equal to some function
of the final point.
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So some function of r
sub B minus some function
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of the initial point r sub A.
And to just get this integral
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from A to B, in our usual way of
evaluating a definite integral,
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it's going to be equal
to some function of r
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B minus some function of r A,
since the integral depends only
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upon the endpoints.
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Now, let's call
this function-- I'm
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going to make a sort of
funny choice here-- so let's
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call this function minus U as a
function of position factor r.
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And we'll see the reason for
this funny choice of minus sign
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in just a moment.
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So now with this definition,
my work integral, which again,
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is the integral of F sub c .ds
from point A to point B is now
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minus U of r B minus minus
U of r A. So in other words,
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that's minus U of r B-- so minus
minus gives me a plus U of r
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sub A.
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For shorthand, I can write that
as minus U sub B plus U sub A.
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And since we start out at
point A and go to point B,
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notice that I can
also write this
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as the negative of the change
in U. Since the final value of U
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is r U at B and the
initial value is U at A,
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so this minus U B plus U
A is equal to minus delta
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U, the change in U as
we go from the initial
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to the final position.
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And note that in addition
to that, given that this
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is the work integral,
I can summarize that
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by writing that--
so I'll say, note
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that delta U is equal to the
negative of the work done
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going from point
A to point B. Now,
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let's write the work kinetic
energy theorem using this newly
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introduced U function.
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So the work kinetic
energy theorem,
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which tells us that the
work done, which we've seen
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is minus U sub B plus U sub
A is equal to the change
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in kinetic energy delta k, which
I can write as K sub B minus K
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sub A. Or I could also write
that as 1/2 M V B squared
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minus 1/2 M V A squared.
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So this is just me stating that
the work done on the system
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is equal to the change
in kinetic energy.
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And I can write the work
in terms of my function U
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that I've introduced here to
minus U sub B plus U sub A.
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So I'm going to rearrange
this equation now--
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basically the one involving
U's and the one involving
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kinetic energies-- so that I
have all the terms involving
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point A on one side and
all the terms involving
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point B on the other side.
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So rearranging, I get
that at point A 1/2 M V
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A squared plus U sub A is equal
to at point B 1/2 M V B squared
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plus U sub B. Now,
notice however,
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that there is nothing
special about how
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I chose the points A and B.
They're completely arbitrary.
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So that means that
this equation must
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be true for any points A and B.
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And what that means
is that each side
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must be equal to
the same constant
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for any point in the system.
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So in fact, we can write
that K plus U for any point
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must be able to
some constant, which
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I'm going to call E sub mech.
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So K here is the kinetic energy.
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U is my function
that I introduced,
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and we're going to call
it the potential energy.
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And E sub mech-- and
remember, this E sub mech
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here is a constant.
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E sub mech is
something that we call
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the total mechanical energy.
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Now, what we've done
here is that we've
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shown that the total
mechanical energy, which
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is the sum of the kinetic
energy and the potential energy,
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is a constant under the action
of a conservative force.
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In other words, if we
look at this equation
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and look at how it
changes with time,
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the change in the kinetic
energy, plus the change
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in the potential energy
is equal to the change
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in the total mechanical energy.
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And this is 0 for our
conservative force.
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So in other words, the
change in kinetic energy
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is balanced by the change
in potential energy, such
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that the sum is 0 when the
force acting is conservative.
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Now, we've now introduced
the very important concept
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of the potential energy
that is associated
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with the conservative force.
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And we see that the change
in the potential energy,
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the way we defined it, the
change in potential energy
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is equal to the negative
of the work integral
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for our conservative
force going from point A
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to point B. Now in
fact, it's actually
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only the change in
the potential energy
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that has physical significance.
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We'll be concerned with
potential energy differences
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or changes.
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The actual value of
the potential energy
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itself doesn't matter.
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We're free to choose
any convenient reference
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point, or 0 point, for
measuring the potential energy.
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It's equivalent to choosing
a coordinate origin when
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we're talking about positions.
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Now, the potential
energy change is
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related to the work done
by conservative forces.
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But we know that
in general, work
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can also be done by
non-conservative forces.
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Although, that work by
non-conservative forces
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will depend upon the path
taken from point A to point B.
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So in general, the
total work is given
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by the sum of the
conservative work--
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the work done by
conservative forces, which
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we can relate to a
potential energy change,
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and the non-conservative
work done.
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And it's this total
work that tells us
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what the change in
the kinetic energy is.
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Now we'll soon see
that in the presence
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of non-conservative forces, the
total mechanical energy, which
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is K plus U, is not a constant.