8.4 Density and Center of Mass Crater Lake, Oregon Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, 1998

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

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 Mo by the total mass, we can find the center of mass (balance point.)

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

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

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

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: right triangle square circle rectangle

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: 1.5 2.5

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