Title: Calculus Ch. 7 Extra Topics
18.4 Density and Center of Mass
Crater Lake, Oregon
2Centers of Mass
Lake Superior, Washburn, WI Photo by Vickie
Kelly, 2004
3If 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.
4If we divide Mo by the total mass, we can find
the center of mass (balance point.)
5For 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.
6For a two dimensional shape, we need two
distances to locate the center of mass.
y
x
x tilde (pronounced ecks tilda)
7For a two dimensional shape, we need two
distances to locate the center of mass.
y
x
Vocabulary
center of mass
center of gravity
centroid
constant density d
homogeneous
uniform
8coordinate of centroid (2.25, 2.7)
9Note The centroid does not have to be on the
object.
If the center of mass is obvious, use a shortcut
10Theorems of Pappus
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
11We can find the centroid of a semi-circular
surface by using the Theorems of Pappus and
working back to get the centroid.
p