Calculus Ch. 7 Extra Topics - PowerPoint PPT Presentation

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Calculus Ch. 7 Extra Topics

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Title: Calculus Ch. 7 Extra Topics Subject: Centers of Mass and Theorems of Pappus Author: Gregory Kelly Last modified by: jon bannon Created Date – PowerPoint PPT presentation

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Title: Calculus Ch. 7 Extra Topics


1
8.4 Density and Center of Mass
Crater Lake, Oregon
2
Centers of Mass
Lake Superior, Washburn, WI Photo by Vickie
Kelly, 2004
3
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
For a two dimensional shape, we need two
distances to locate the center of mass.
y
x
x tilde (pronounced ecks tilda)
7
For 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
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
10
Theorems 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
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
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