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Let's review what
we've done this week.

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We've seen that the work done
by a force along a trajectory

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from point A to point
B can be written

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as the integral of the
dot product of that force,

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with the displacement
going from point A

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to point B, where we note that
this integral, in general,

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must be evaluated over
the particular path taken

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from point A to point
B. This is called a line

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integral or a path integral.

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And again, I want to
emphasize the value

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of this integral depends, in
general, on the path taken

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from point A to point B.
And you could understand

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that if you considered the fact
that, in general, the force is

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a function of position.

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And so if you go
along different paths,

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the dot product of the
force with a particular path

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may be different, even
though you're going

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between the same end points.

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Now, the reason that the
work is a useful concept

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is that it allows
us to understand

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how the kinetic energy of an
object-- so the kinetic energy

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is 1/2mv squared.

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The work allows us to understand
how the kinetic energy evolves

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under the action of a force.

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And we see that through what's
called the work-kinetic energy

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theorem, which states that
the work done-- so again,

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the integral of F
dot ds from point A

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to point B, evaluated
along the specific path--

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is equal to the change
in the kinetic energy.

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So that's 1/2mvB
squared, since B

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is the final point,
minus 1/2mvA squared,

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where A is the initial point.

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And this is equal to the
change in the kinetic energy.

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And we derived this theorem
from Newton's second law,

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F equals ma.

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So this is essentially
just a restatement

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of Newton's second law.

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Now, two things to note
about the work and this work

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integral-- the first is that the
work integral, when evaluated,

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it just gives a number, because
a dot product is a scalar.

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And that number can be
either positive, or negative,

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or even zero, depending upon
whether the velocity increases,

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decreases, or stays the same
under the action of the force,

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as you go from
point A to point B.

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The second point to
emphasize is that, again,

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this integral, in
general, must be evaluated

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along the specific
path taken from point A

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to point B, which means that in
order to evaluate the integral,

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we need to know
what that path is.

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Now, if we have a situation
where we don't know in advance

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what the path is, then we
can't evaluate the integral.

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So in a way, it might seem
like a not very useful concept,

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that we haven't
really made progress,

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because we need to know
the answer about what

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the trajectory is before we
can calculate the work done.

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And it's not clear
why that's useful.

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However, what I
want to point out

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is that there are two
special cases that

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are particularly useful.

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The first special case is
that of constrained motion.

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Sometimes we know in advance
what the path is going to be,

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because the motion
is constrained.

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We know that it's going
to be moving along--

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the object is going to be
moving along a circular path

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or along some particular track
of some particular shape.

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And as long as we can
describe that trajectory,

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we can evaluate our work
integral along that trajectory.

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So that's a case where
even though the work is

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a path-dependent integral,
we know the path,

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and so we can
evaluate the integral.

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The second special
case we've seen

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is that there is a
special class of forces

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called conservative forces.

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And conservative forces have the
property that the work integral

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is independent of the path.

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So for a conservative force,
to work integral only depends

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upon the endpoints A and
B, and not upon the path

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you take to get from A
to B. So for any path,

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you'll always get the same
value of the work integral

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if the force is conservative.

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OK, so those are the
two cases in which

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we can usefully apply work
and the work-energy theorem.

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Now in general, one
might have a combination

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of conservative and
non-conservative forces

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acting on a system.

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And in that case,
we can actually

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write the work as consisting
of two separate terms.

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The work done by conservative
forces, what I'll write as Wc,

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plus the work done by the
non-conservative forces, Wnc.

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And so the conservative
work is just

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the work integral evaluated
for the conservative forces,

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so Fc, for conservative force,
dot ds, going from A to B.

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And because this
force is conservative,

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this integral is
path independent.

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Let me emphasize that here.

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This is a
path-independent integral.

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It's not a line integral.

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It only depends
upon the endpoints,

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because the force
is conservative.

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The second term is the
non-conservative work.

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And that is the work done by the
non-conservative force, F sub

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mc, dotted with ds, between
the same end points, A and B.

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But this integral does depend
upon the specific path.

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The path matters for this
term, because there the force

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is non-conservatives.

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So in general, you
can compute the work

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by separating out the
conservative force

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and the non-conservative
force acting on the system,

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if both are acting.

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If the force is
only conservative,

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then you only have
to worry about

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the path-independent
integral, which in general is

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much simpler to evaluate.

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If the force is
non-conservative,

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then you need to specify
the path in order

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to evaluate the integral.

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Again, a path-dependent
integral is often

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called a path integral
or a line integral.