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I'd now like to talk about the
velocity of the center of mass
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for a system of particles.
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So let's take a system, which
I'll just outline by this.
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And in that system, we have a
bunch of particles, particle 1,
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particle 2-- let's refer to
this as the j-th particle
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and some point xcm.
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And if I want to talk about the
position of the center of mass,
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I can choose a point s.
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And if I want to
define that vector Rcm,
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then what I have to do is draw
a vector to each object Rsj.
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And we saw that this
velocity, the position
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of the center of mass with
respect to this origin
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s is the sum and mjrsj.
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And that's divided by the
total mass and m total.
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And j goes from 1 to n,
where n is the number
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of particles in the system.
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Now if I want to find the
velocity of the center of mass,
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then I can just
differentiate this.
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And I'm dropping the
point s for the moment,
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but let's just
differentiate 1 to n.
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And you'll see why.
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And when I differentiate
the position vector
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of the object, that's the
velocity of the object divided
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by j goes from 1 to
n, the total mass.
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Now why did I drop the position?
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Because if you
have any two fixed
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points-- so if I chose another
fixed point, say, over here p,
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then this distance R--
we'll call it vector from s
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to t Rsp-- this is a constant.
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And if I draw position vector
with respect to p-- now
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the point here is that this
is a constant distance,
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because this is a fixed--
these are fixed points.
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Then if you were to draw
your vector triangle, which
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is the position of the
object with respect to s--
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that's this vector-- is
equal to that fixed position
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vector from s to p, plus
the vector from p to j,
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and I differentiate
this, drs jdt.
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Well, this derivative of
a constant vector, this
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is 0 plus drp jdt.
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And so we see that
the velocity j
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is independent of the
choice of point s.
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You choose any other
fixed point and you
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get that velocities
drs jdt equals
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drp jpt for all fixed points p.
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And that's why in
this expression, when
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we differentiate the velocity,
even though we had an index s,
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we dropped that.
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And so our conclusion
is that we can
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treat that we have the
velocity of the center of mass
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of this system is
equal to the sum mj vj.
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j from 1 to n divided
by the total mass.
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Now what's interesting
here is, why
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is this an important quantity?
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Let's just add that
if we want to talk
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about the acceleration
of the center of mass,
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I do exactly the same
type of calculation.
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I just differentiate.
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And I get the mass
of the j-th particle
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times the acceleration of
the j-th particle divided
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by the total mass.
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And our next step
is to understand
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why this is an
important quantity
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for a system of particles.