Chapter 7 Extra Topics
Crater Lake, OregonGreg Kelly, Hanford High School, Richland, WashingtonPhoto 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.)
d
gF
If the forces are all gravitational, then torque mgx
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.
O k kM m x This is called themoment about the origin.
1m g 2m g
If we divide Mo by the total mass, we can find the center of mass (balance point.)
O k kM m xk k
O
k
x mM
xM
m
For a thin rod or strip:
= density per unit length
moment about origin: b
O aM x x dx
( is the Greek letter delta.)
mass: b
aM x dx
k kO
k
x mM
xM
m
center of mass: OMxM
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.
y
x
x distance from the y axis to the center of the strip
y distance from the x axis to the center of the strip
x tilde (pronounced ecks tilda)Moment about x-axis: xM y dm
yM x dmMoment about y-axis:
Mass: M dm
Center of mass:
y xM M
x yM M
x
y
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 = homogeneous = uniform
For a plate of uniform thickness and density, the density drops out of the equation when finding the center of mass.
2y x
2.5x
x
x x21
2y x
243
10xM
3 2 2
0
1
2xM x x dx 3 4
0
1
2xM x dx5 31
010xM x
81
4yM
3 2
0yM x x dx 3 3
0yM x dx
4 31
04yM x
8194
9 4yMx
M
2432710
9 10xMy
M
coordinate ofcentroid =(2.25, 2.7)
3 2 3
0
319
03M x dx x
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 triangle3
b3
h
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:
3111cmV
2 areaV r
22 2.5 1.5V
211.25V
2.5
1.5