1. 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

2. 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

3. 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.

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

5. 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.

6. 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:

7. 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

8. coordinate of
centroid =
(2.25, 2.7)

9. 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

10. 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

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