7.5 moments, centers of mass and centroids. if the forces are all gravitational, then if the net...
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7.5 Moments, Centers of Mass
And Centroids
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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
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For a thin rod or strip:
d = density per unit length
moment about origin: b
O aM x x dx
(d is the Greek letter delta.)
mass: b
aM x dx
k kO
k
x mM
xM
m
center of mass: OM
xM
For a rod of uniform density and thickness, the center of mass is in the middle.
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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
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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 d = homogeneous = uniform
For a plate of uniform thickness and density, the density drops out of the equation when finding the center of mass.
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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 4yM
xM
2432710
9 10xM
yM
coordinate ofcentroid =(2.25, 2.7)
3 2 3
0
319
03M x dx x
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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
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We can find the centroid of a semi-circular surface by using the Theorems of Pappus and working back to get the centroid.
2 31 4 V=
2 3A r r
4
3
ry
2 31 42
2 3y r r y
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Examples
• Find the mass of a rod that has a length of 5 meters and whose density is given by
at a distance of x meters away from the left end
gm/m23)( xx
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Continuous Mass Density
• Instead of discrete masses arranged along the x-axis, suppose there is an object lying on the x-axis between x = a and x = b– Divide it into n pieces of length Δx– On each piece the density is nearly constant so the
mass of each piece is given by density times the length
– Mass of ith piece is xxm ii )(