Center of Mass of a Solid of Revolution

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

Balancing Unequal Masses Moral Both the masses and their positions affect whether or not the “see saw” balances. The great Greek mathematician Archimedes said, “give me a place to stand and I will move the Earth,” meaning that if he had a lever long enough he could lift the Earth by his own effort.

Balancing Unequal Masses If 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 Masses Need: M 1 d 1 = M 2 d 2 M1M1 M2M2 d1d1 d2d2

Changing our point of view We can think of leaving the masses in place and moving the fulcrum. It would have to be a pretty long see-saw in order to balance the school bus and the race car, though!

In other words... We (still) need: M 1 d 1 = M 2 d 2 M2M2 d1d1 d2d2 M1M1

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

Mathematical Setting And place the objects in the coordinate system... 0 M2M2 M1M1 M3M3 M4M4 d2d2 d1d1 d3d3 d4d4 Except that now d 1, d 2, d 3, d 4,... denote the placement of the objects in the coordinate system, rather than relative to the fulcrum. (Because we don’t, as yet, know where the fulcrum will be!)

Mathematical Setting And place the objects in the coordinate system... 0 M2M2 M1M1 M3M3 M4M4 d2d2 d1d1 d3d3 d4d4 We want to place the fulcrum at some coordinate that will balance the system. is called the center of mass of the system.

Mathematical Setting And place the objects in the coordinate system... 0 M2M2 M1M1 M3M3 M4M4 d2d2 d1d1 d3d3 d4d4 In order to balance 2 objects, we needed: M 1 d 1 = M 2 d 2 OR M 1 d 1 - M 2 d 2 =0 For a system with n objects we need:

Finding the Center of Mass of the System Now we solve for.

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

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 Revolution We can treat this as a discrete, one- dimensional center of mass problem!

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.

Approximating the Center of Mass of a Solid of Revolution

f R h

Summarizing: The mass of the i th bead is The position of the i th bead is Approximating the Center of Mass of a Solid of Revolution

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!

The Center of Mass of a Solid of Revolution where 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.”