Chapter 7 Extra Topics Crater Lake, Oregon Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998

Lake Superior, Washburn, WI Photo by Vickie Kelly, 2004 Centers of Mass: Torque is a function of force and distance. (Torque is the tendency of a system to rotate about a point.)

If the forces are all gravitational, then If the net torque is zero, then the system will balance. Since gravity is the same throughout the system, we could factor g out of the equation. This is called the moment about the origin.

If we divide M o by the total mass, we can find the center of mass (balance point.)

For a thin rod or strip:  = density per unit length moment about origin: (  is the Greek letter delta.) mass: center of mass: For a rod of uniform density and thickness, the center of mass is in the middle.

x y strip of mass dm For a two dimensional shape, we need two distances to locate the center of mass. distance from the y axis to the center of the strip distance from the x axis to the center of the strip x tilde (pronounced ecks tilda) Moment about x-axis:Moment about y-axis:Mass:Center of mass:

x y For a two dimensional shape, we need two distances to locate the center of mass. Vocabulary: center of mass= center of gravity= centroid constant density  = homogeneous= uniform For a plate of uniform thickness and density, the density drops out of the equation when finding the center of mass.

coordinate of centroid = (2.25, 2.7)

Note: The centroid does not have to be on the object. If the center of mass is obvious, use a shortcut: square rectangle circle right triangle

Theorems of Pappus: When a two dimensional shape is rotated about an axis: Volume = area. distance traveled by the centroid. Surface Area = perimeter. distance traveled by the centroid of the arc. Consider an 8 cm diameter donut with a 3 cm diameter cross section:

We can find the centroid of a semi-circular surface by using the Theorems of Pappus and working back to get the centroid. 