Now we'd like to analyze in more depth
our result that for a system of particles--
so let's indicate our system.
We had particle 1.
We have our jth particle.
And we have a particle N. So here's
our system of particles where the total force caused
the momentum of the system of particles to change.
Now, I'd like to examine that concept of the total force.
Before we said that our total force on the jth particle--
we just wrote it like this.
And I'm going to put a little t up here for the moment.
Because when we examine what force we mean here--
and I also want to put a little boundary around our system.
And let's now consider another particle
internal to the system.
And let's try to identify the types
of forces on the jth particle.
We can really have two types of forces here.
Our first force can be an interaction between these two
particles.
So what I'll write is the force on the jth particle
due to the interaction between the k and the jth particle.
And I'm going to put a little sign up here.
I'm going to write this internal.
What do I mean by internal?
This is a force strictly between the internal particles
in the system.
Now, of course, we know that there
must be a force on the kth particle
due to the interaction between the jth and the kth particle.
And we can call that internal.
So what we have here is that we can
divide-- there can still be other forces acting on the jth
particle.
And we'll do a decomposition like this.
We'll say that the total force on the jth particle
can come from some external forces.
There could be an object outside our system.
If these were interacting gravitationally,
there could be a planet outside here.
And this could be a moon.
And our system is just the moons.
That would be an external gravitational force,
plus the total internal forces.
So I'm going to keep this same color.
Internal on the jth particle.
Now how do we write this total internal force?
Well, we're interested in the force on the jth particle.
But the internal forces can come from all of the other particles
in the system.
So what we're looking at here is for a sum over all
of the possible interactions where the other particles, k,
go from 1 to N. And we have to be very careful here that
in this sum k cannot be equal to j.
Now this sum-- again, because it's a little bit tricky
to understand-- is the internal force on the jth particle.
Here's the kth one.
But this could be a sum.
I'll just draw one here.
This is the internal force on the jth particle
due to particle number 1.
And so we're adding as k goes from 1 to N
all of these internal forces.
But we're excluding the case when
k equals j, because that would be
a force of an object on itself.
And this quantity here, we can write
as the total internal force on the jth particle.
So in summary, we see that the total force on the jth particle
is equal to the total external forces.
I didn't say total there.
I'm assuming there could be many different types
of internal forces plus the total internal force.
A little bit later on, we can drop the T's
for simplicity of notation.
But this is our big idea, that a force on the jth particle,
external plus internal.
And now when we look at this sum,
and we want to now apply our main idea,
we have that the force, which we're writing as-- let's
explore this.
Our total force is the sum of the forces on the jth particle.
And we've now done this decomposition.
I'm going to drop total.
So it's the sum of the external forces on the jth particle.
j goes from 1 to N.
And here, we have a sum of the internal forces.
So we have our sum j.
It goes from 1 to N of the internal forces on the jth
particle.
Now we want to apply Newton's second law.
And the concept is very straightforward.
But the mathematical expression can be a little bit messy.
We know by Newton's second law that the sum
of a pair of internal forces is zero-- third law,
by Newton's third law.
So what we're saying here is, as an example,
for Newton's third law-- let's just focus
on this particular pair-- that F internal kj plus F internal jk
is zero.
So this is the statement that internal forces
cancel in pairs.
And so when I look at this total internal force, which
is the sum of all of these pairs of internal forces,
I can see that the total internal force has to be zero.
So internal force cancel in pairs.
Now here, we can see it another way if we
want to look at this notation.
We took the sum.
j goes from 1 to N of F internal j.
Now we use our definition for F internal.
This where things get a little bit messy.
k goes from 1 to N. k not equal to j.
j goes from 1 to N. F internal kj.
This looks terribly messy.
But what we're saying is this sum is just a sum of pairs.
And every single pair in this adds to 0.
So what we have for our statement now
is that the total force is the sum of the external forces,
plus the sum of the internal forces
which we've now said that cancels in pairs.
So let's rewrite that as the total force-- now instead
of writing this sum, let's write it
as the sum of the external forces.
And the internal forces cancel in pairs.
And so this is now our force on our system.
It's only the external force.
And now we can recast our Newton's second law
for a system of particles with the following statement
that the external force causes the momentum of the system
to change.
And this becomes our expression for Newton's second law
when we apply it to a system of particles
where the beauty of this idea is that no matter how complicated
the interaction is inside the system,
all of those interactive pairs sum to zero.
And so only thing that matters is the external force
in terms of changing the momentum of the system.
And now we'll look at some applications of that.