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Let's now extend our
concept of momentum
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to a system of particles.
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Again, we need to choose
a reference frame.
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So we'll have a ground frame.
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And let's consider N particles.
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Now when we have a
lot of particles,
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we need some type of notation.
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So let's use the symbol j.
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And it will goes
from 1 to N. And then
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our arbitrary j
particle will be moving.
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This particle will have mass mj.
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And it will be moving
with a velocity vj.
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Now recall in our system, we
have many other particles.
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We can call that one 1.
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This is one n.
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We have lots of different
particles in the system.
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And this just represents
an arbitrary particle
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in that system.
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And the momentum
of the jth particle
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is just the mass, mj,
times the velocity, vj.
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And again, we're assuming
some fixed reference frame.
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So the total momentum
of this system,
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we now have to add
up the momentum
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of all the particles, all the
way up to the nth particle.
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Now, when we make
a sum like this,
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there is a standard
mathematical summation notation,
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which we'll write like this.
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We'll do the sum, this capital
sigma sin of j goes from 1 to j
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goes to N of the momentum
of the jth particle.
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And that represents the sum
j goes from 1 to n of mj vj.
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And this is what we call
the momentum of the system.
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This is a vector sum.
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And now let's see how
Newton's second law applies
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to the momentum of the system.
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Suppose that acting
on our particles--
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for instance, here's our jth
particle-- we have a force
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Fj acting on the jth particle.
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Then we know that
from Newton's law
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that the force will be also
the sum of the forces on all
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of the particles, F1, F2,
plus dot, dot, dot, plus FN.
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So once again, we can
write this as a sum
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j goes from 1 to N of the
force on the jth particle.
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And that's the
force on the summing
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over all the forces on all
the particles in the system.
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But now, we can apply
Newton's second law.
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So Newton's second
law is the statement
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that the force on
the jth particle
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causes the momentum of the
jth particle to change.
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And when we write that now,
the total force on the system,
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j goes from 1 to N, is just the
sum of the change in momentum.
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Because every single term--
let's just look at that.
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T1 plus dP2/dt plus dot,
dot, dot, plus dPN/dt,
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that's what we mean by the sum.
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We can rewrite this as
d/dt of P1 plus P2 plus P3
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plus dot, dot, dot, plus PN.
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And what we see is
that the total force is
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the derivative of the sum j goes
from 1 to N of the momentum.
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But recall, this
sum we've defined
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as the momentum of the system.
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So our conclusion
is the total force
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causes the momentum of
the system to change.
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Now so far, all
we've done is we've
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recast Newton's second
law in this form.
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Our next step is to
analyze the forces
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on the individual
particles we have
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and apply Newton's third law.
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So we'll do that next.