Center of Mass

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Center of Mass

• of a
• Solid of Revolution

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See-Saws
We all remember the fun see-saw of our youth.
But what happens if . . .
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Balancing Unequal Masses
Moral Both the masses and their positions affect
whether or not the see saw balances.
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Balancing Unequal Masses
M1
M2
d1
d2
Need M1 d1 M2 d2
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Changing our Point of View
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In other words. . .
We can think of leaving the masses in place and
moving the fulcrum.
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In other words. . .
M1
M2
d1
d2
(We still) need M1 d1 M2 d2
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What happens if there are many things trying to
balance on the see-saw?
Where do we place the fulcrum?
Mathematical Setting First we fix an origin and a
coordinate system. . .
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Mathematical Setting
And place the objects in the coordinate system. .
.
Except that now d1, d2, d3, d4, . . . denote the
placement of the objects in the coordinate
system, rather than relative to the fulcrum.
(Because we dont, as yet, know where the
fulcrum will be!)
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Mathematical Setting
And place the objects in the coordinate system. .
.
M4
M1
M3
M2
d3
d4
0
d2
d1
We want to place the fulcrum at some coordinate
that will balance the system. is called
the center of mass of the system.
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Mathematical Setting
And place the objects in the coordinate system. .
.
In order to balance 2 objects, we needed M1 d1
M2 d2 OR M1 d1 - M2 d2 0 For a system with n
objects we need
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Finding the Center of Mass of the System
Now we solve for .
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The Center of Mass of the System
In the expression
The numerator is called the first moment of the
system
The denominator is the total mass of the system
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The Center of Mass of a Solid of Revolution.

• For the goblet project, you will need to
  calculate the position of the center of mass of
  your goblet (which is a solid of revolution!)

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The Center of Mass of a Solid of Revolution.

• Some preliminary remarks

First, I will ask you to believe the following (I
think) plausible fact Due to the circular
symmetry of a solid of revolution, the center of
mass will have to lie on the central axis.
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The Center of Mass of a Solid of Revolution.

• Some preliminary remarks

Next, in order to approximate the location of
this center of mass, we slice the solid into
thin slices, just as we did in approximating
volume.
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The Center of Mass of a Solid of Revolution
We can treat this as a discrete, one-dimensional
center of mass problem!
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Approximating the Center of Mass of a Solid of
Revolution

• What is the mass of each bead?
• Assume that the solid is made of a single
  material so its density is a uniform ?
  throughout.
• Then the mass of a bead will simply be ? times
  its volume.

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Approximating the Center of Mass of a Solid of
Revolution
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Approximating the Center of Mass of a Solid of
Revolution
f
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Approximating the Center of Mass of a Solid of
Revolution
Summarizing The mass of the ith bead is The
position of the ith bead is
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The Center of Mass of a Solid of Revolution
Both the numerator and denominator are Riemann
sums, and as we subdivide the solid more and more
finely, they approach integrals.
. . . And the fraction approaches the center of
mass of the solid!
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The Center of Mass of a Solid of Revolution
In the limit as the number of slices goes to
infinity, we get the coordinate of the center of
mass of the solid . .
where a and b are the endpoints of the region
over which the solid is sliced.
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If the cross sections are washers. . .
The derivation is more or less the same, except
that when we compute the area of the little
cylinder, we get as we did when we computed
the volume of a solid of revolution. So the
coordinate of the center of mass will be
where a and b are the endpoints of the region
over which the solid is sliced.