WEBVTT
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Today we'd like to
explore the idea
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of the center of mass,
or the center of gravity,
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of a rigid object.
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For instance, take this rod.
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And if I try to
balance it on my finger
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at a certain point
in that rod, I'm
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balancing it via the
gravitational force.
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And this point is often
referred to as the center
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of gravity of the rod.
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Now, if we were in empty space
with no gravitational field,
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then center of gravity
doesn't make any sense.
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But this point still
coincides with what
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we call the center of
mass of the object.
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And now I'd like to
define center of mass.
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So let's consider
our rigid body.
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And we'll just describe
it as some object--
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we'll make it idealized--
and that there's
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going to be trying to find
some point in this object.
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And we'll identify that
point as the center of mass.
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Now, let's imagine that
this rigid body is made up
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of a bunch of little pieces.
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So we have m, j.
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And this j piece is
located from the center
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of mass, a vector rcm.
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Now how we want to define
this particular point
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is that when we
make the sum from j
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equals 1 to n over every
single point in this body,
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then that will be 0.
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And this will be the definition
of the center of mass,
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that when you add up the
position vector with respect
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to this point
weighted by the mass,
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and you add up all of those
vectors, you'll get 0.
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Now, if you don't know where
the center of mass is then
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this is difficult to calculate.
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So let's find a way where you
choose an arbitrary point,
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and that's write that arbitrary
point, say, over here.
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We'll write it like this-- s.
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And we'll treat
that as our origin.
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And I'll draw a vector, rsj.
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And here, I'll draw
the vector Rcm.
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And now what we have from
our vector relationship
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is that the vector rsj
equals the vector Rcm--
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and that's what I want to
find-- plus the vector rcmj.
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And now let's add up-- multiply
each of these by the mass
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and make a sum.
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So we have mj rsj
equals the sum of mj.
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Now, the vector
Rcm, this vector,
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no matter where we picked
a point in this object,
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the vector's always the same.
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And that's why I pulled
it out of the sum.
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And over here, I
have the sum from j
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goes from 1 to n, j
1 to n, of mj rcmj.
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Now, recall, this
is precisely how
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we define the center of
mass point, that this is 0.
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And so we can now conclude
that the center of mass--
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so you pick an arbitrary
point, and if you want
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to find that vector to the
center of mass, what you do
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is you make the sum from j
goes from 1 to n of mj rsj,
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and you divide that by
j goes from 1 to n, mj.
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And this is what we
call the center of mass.
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So conceptually,
the center of mass
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is the point in
the object where,
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if you take a vector to any
of the little mass elements,
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and you weight it by the mass
element, and you add them up,
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you get 0.
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If you wanted to calculate the
center of mass about any point,
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you choose a point, s,
you draw the vector from s
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to the object.
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You sum up those vectors.
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We see by our
vector triangle rule
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that we can now calculate
that center of mass vector
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by this equation.
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And now let's just rewrite this
because this is the total mass.
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And what we see is that the
Rcm equals j goes from 1 to n,
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mj rsj divided by
the total mass.