Center of Massof aSolid of Revolution

See-SawsWe all remember the fun see-saw of our youth.But what happens if . . .

Balancing Unequal MassesMoralBoth the masses and their positions affect whether or not the see saw balances.

Balancing Unequal MassesIf the heavy mass and the light mass are equidistant from the fulcrum, the seesaw will not balance. The heavier object must be closer to the fulcrum than the lighter object.

Balancing Unequal MassesNeed:M1 d1 = M2 d2

M1M2d1d2

Changing our point of viewWe can think of leaving the masses in place and moving the fulcrum.

In other words. . .We (still) need:M1 d1 = M2 d2

M2d1d2M1

What happens if there are many things trying to balance on the see-saw?Where do we place the fulcrum?Mathematical SettingFirst we fix an origin and a coordinate system. . .

Mathematical SettingAnd 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!)

Mathematical SettingAnd place the objects in the coordinate system. . .0M2M1M3M4d2d1d3d4We want to place the fulcrum at some coordinate that will balance the system. is called the center of mass of the system.

Mathematical SettingAnd 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:

Finding the Center of Mass of the SystemNow we solve for .

The Center of Mass of the SystemIn the expression The numerator is called the first moment of the systemThe denominator is the total mass of the system

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!)

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.

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.

The Center of Mass of a Solid of RevolutionWe can treat this as a discrete, one-dimensional center of mass problem!

Approximating the Center of Mass of a Solid of RevolutionWhat 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.

Approximating the Center of Mass of a Solid of Revolution

Approximating the Center of Mass of a Solid of Revolutionf

Summarizing:The mass of the ith bead isThe position of the ith bead is

Approximating the Center of Mass of a Solid of Revolution

The Center of Mass of a Solid of RevolutionBoth 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!

The Center of Mass of a Solid of Revolutionwhere a and b are the endpoints of the region over which the solid is sliced.In the limit as the number of slices goes to infinity, we get the coordinate of the center of mass of the solid . .

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:If the cross sections are washers. . .where a and b are the endpoints of the region over which the solid is sliced.

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