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The total mechanical
energy, E sub mech

2
00:00:09,780 --> 00:00:13,410
is defined as the sum
of the kinetic energy

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00:00:13,410 --> 00:00:15,720
and the potential
energy, so the sum

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00:00:15,720 --> 00:00:23,830
of K plus U, where U is the
potential energy function

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00:00:23,830 --> 00:00:26,110
associated with the
conservative force--

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00:00:26,110 --> 00:00:29,260
with an appropriate
choice of zero point.

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00:00:29,260 --> 00:00:31,510
If there are multiple
conservative forces acting

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on the system, then
U will be the sum

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of individual potential
energy functions

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for each conservative force with
an appropriate choice of zero

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00:00:39,910 --> 00:00:43,240
point for each
individual function.

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00:00:43,240 --> 00:00:50,530
We've also seen that the
change in the kinetic energy

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plus the change in
the potential energy

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is equal to the change in
the total mechanical energy.

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And that's just basically the
derivative of the equation

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that I wrote above.

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And this change in the
total mechanical energy

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will be 0 for
conservative forces.

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In addition, we've seen that
the change in potential energy

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is related to the conservative
work done in the system

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so that W sub C, the
conservative work done,

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is equal to the negative of the
change in the potential energy.

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But we also know that
the total work done

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includes both conservative
and non-conservative forces.

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So the total work done is
equal to the sum of W sub C,

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the conservative work done, plus
the non-conservative work done,

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and that that total work
is equal to the change

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in kinetic energy.

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Now, I can rewrite that
equation by rewriting

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the conservative work in
terms of the potential energy.

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So I could write this, now,
as the non-conservative work

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minus the change in
the potential energy

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is equal to the change
in the kinetic energy.

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This is, again, just a
restatement of the work

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kinetic energy theorem.

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Now, by rearranging
this equation,

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this means that the
non-conservative work is equal

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to delta K, the change
in kinetic energy,

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plus delta U, the change
of potential energy.

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But that's equal to the change
in the total mechanical energy.

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So the non-conservative
work is equal to the change

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in the total mechanical energy.

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This is a sufficiently
important result

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that I'm going to write
it by itself in a box.

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So the non-conservative
work is equal to the change

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in the total mechanical energy.

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So what we've shown
here is that the result

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of any non-conservative
work on the system

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is to change the total
mechanical energy

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of the system.

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Now, if there is no
non-conservative work,

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if the forces acting on the
system are all conservative,

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then the total mechanical
energy remains unchanged

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now that Emech is 0.

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And we say to the total
mechanical energy is conserved.

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Now if there is a
non-conservative force,

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it turns out that in
most cases the work done

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by non-conservative
force is negative,

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resulting in a
negative change in

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the total mechanical
energy-- in other words,

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in the removal of
total mechanical energy

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from the system.

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It's lost from the system.

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Now we've been talking here only
about mechanical energy, which

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we've defined as the
sum of kinetic energy

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and potential energy.

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But one can also talk about
non mechanical energy.

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The most common example of which
is loosely referred to as heat.

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As an example, in
the case of friction,

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which is a
non-conservative force,

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we know that the action of
friction generates heat.

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What's happening is that the
non-conservative work done

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by friction is negative
and, therefore, is reducing

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the mechanical energy, the
total mechanical energy,

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according to this expression,
but where does that energy go?

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We said that it's removed
from the system-- in terms

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of mechanical
energy, that's true.

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But we can think of it
as, at the same time,

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causing an increase in the non
mechanical energy, or the heat

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energy, of the system.

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00:05:01,710 --> 00:05:05,150
So if one were to keep
track of both mechanical and

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00:05:05,150 --> 00:05:07,760
non-mechanical energy,
then it turns out

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00:05:07,760 --> 00:05:10,910
that the total energy,
mechanical and non-mechanical,

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00:05:10,910 --> 00:05:14,180
the total energy of a
system is conserved,

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even if non-conservative
forces are acting.

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However, it's
difficult to recover

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the non-mechanical
energy and change it back

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into mechanical energy.

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These non-conservative
processes are not reversible.

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This is a complicated
topic, and it's

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the province of more advanced
courses in thermodynamics

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or statistical mechanics,
which you may encounter later.

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In this course, we will
confine our attention

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to mechanical energy,
the sum of kinetic energy

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and potential energy, and will
treat non-conservative force

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as removing, or adding,
mechanical energy

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to the system.