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Now we'd like to
analyze in more depth

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our result that for a
system of particles--

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so let's indicate our system.

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We had particle 1.

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We have our jth particle.

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And we have a
particle N. So here's

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our system of particles
where the total force caused

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the momentum of the system
of particles to change.

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Now, I'd like to examine that
concept of the total force.

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Before we said that our total
force on the jth particle--

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we just wrote it like this.

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And I'm going to put a little
t up here for the moment.

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Because when we examine
what force we mean here--

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and I also want to put a little
boundary around our system.

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And let's now consider
another particle

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internal to the system.

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And let's try to
identify the types

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of forces on the jth particle.

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We can really have two
types of forces here.

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Our first force can be an
interaction between these two

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particles.

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So what I'll write is the
force on the jth particle

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due to the interaction between
the k and the jth particle.

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And I'm going to put
a little sign up here.

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I'm going to write
this internal.

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What do I mean by internal?

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This is a force strictly
between the internal particles

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in the system.

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Now, of course,
we know that there

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must be a force on
the kth particle

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due to the interaction between
the jth and the kth particle.

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And we can call that internal.

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So what we have
here is that we can

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divide-- there can still be
other forces acting on the jth

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particle.

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And we'll do a
decomposition like this.

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We'll say that the total
force on the jth particle

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can come from some
external forces.

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There could be an object
outside our system.

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If these were interacting
gravitationally,

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there could be a
planet outside here.

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And this could be a moon.

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And our system is
just the moons.

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That would be an external
gravitational force,

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plus the total internal forces.

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So I'm going to keep
this same color.

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Internal on the jth particle.

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Now how do we write this
total internal force?

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Well, we're interested in the
force on the jth particle.

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But the internal forces can come
from all of the other particles

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in the system.

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So what we're looking at
here is for a sum over all

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of the possible interactions
where the other particles, k,

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go from 1 to N. And we have
to be very careful here that

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in this sum k cannot
be equal to j.

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Now this sum-- again, because
it's a little bit tricky

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to understand-- is the internal
force on the jth particle.

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Here's the kth one.

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But this could be a sum.

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I'll just draw one here.

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This is the internal
force on the jth particle

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due to particle number 1.

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And so we're adding
as k goes from 1 to N

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all of these internal forces.

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But we're excluding
the case when

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k equals j, because
that would be

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a force of an object on itself.

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And this quantity
here, we can write

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as the total internal
force on the jth particle.

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So in summary, we see that the
total force on the jth particle

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is equal to the total
external forces.

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I didn't say total there.

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I'm assuming there could
be many different types

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of internal forces plus
the total internal force.

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A little bit later on,
we can drop the T's

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for simplicity of notation.

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But this is our big idea, that
a force on the jth particle,

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external plus internal.

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And now when we
look at this sum,

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and we want to now
apply our main idea,

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we have that the force, which
we're writing as-- let's

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explore this.

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Our total force is the sum of
the forces on the jth particle.

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And we've now done
this decomposition.

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I'm going to drop total.

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So it's the sum of the external
forces on the jth particle.

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j goes from 1 to N.

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And here, we have a sum
of the internal forces.

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So we have our sum j.

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It goes from 1 to N of the
internal forces on the jth

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particle.

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Now we want to apply
Newton's second law.

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And the concept is
very straightforward.

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But the mathematical expression
can be a little bit messy.

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We know by Newton's
second law that the sum

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of a pair of internal
forces is zero-- third law,

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by Newton's third law.

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So what we're saying
here is, as an example,

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for Newton's third
law-- let's just focus

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on this particular pair-- that
F internal kj plus F internal jk

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is zero.

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So this is the statement
that internal forces

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cancel in pairs.

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And so when I look at this
total internal force, which

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is the sum of all of these
pairs of internal forces,

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I can see that the total
internal force has to be zero.

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So internal force
cancel in pairs.

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Now here, we can see
it another way if we

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want to look at this notation.

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We took the sum.

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j goes from 1 to
N of F internal j.

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Now we use our definition
for F internal.

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This where things get
a little bit messy.

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k goes from 1 to N.
k not equal to j.

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j goes from 1 to
N. F internal kj.

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This looks terribly messy.

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But what we're saying is this
sum is just a sum of pairs.

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And every single pair
in this adds to 0.

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So what we have for
our statement now

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is that the total force is the
sum of the external forces,

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plus the sum of
the internal forces

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which we've now said
that cancels in pairs.

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So let's rewrite that as the
total force-- now instead

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of writing this
sum, let's write it

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as the sum of the
external forces.

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And the internal
forces cancel in pairs.

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And so this is now our
force on our system.

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It's only the external force.

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And now we can recast
our Newton's second law

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for a system of particles
with the following statement

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that the external force causes
the momentum of the system

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to change.

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And this becomes our expression
for Newton's second law

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when we apply it to
a system of particles

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where the beauty of this idea is
that no matter how complicated

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the interaction is
inside the system,

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all of those interactive
pairs sum to zero.

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And so only thing that
matters is the external force

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in terms of changing the
momentum of the system.

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And now we'll look at
some applications of that.