moment (m ) mass ( mteachers.oregon.k12.wi.us/debroux/calc/6.6lessonkey.pdf · planar lamina ‐ a...
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6.6 Moments, Centers of Mass, and Centroids
mass (m) p459 Def. Table (Gram, Kilogram, Slug, . . .)
Moment (M) = (force) (distance from the point or line of rotation)
Def.
A seesaw will balance when the left and the right moments are equal!
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Linear (one dimensional) Mass Distributionnot continuous ‐ "discrete"
ie. teeter totter
m1 m2xaxis
heavier
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m1 m2xaxis
x1 x2
‐ Center of Massx(relative to the origin)
mass = system
m = m1 + m2
M1 = m1 x1. M2 = m2 x2.
M .M1 + M2 = .m1 x1= system
+ m2 x2
(concentrate the total mass at one point with coordinate )xM x.m= O
...
x m= MO
(Moment about the origin)
MO
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Examples:
1.)
Linear (One‐Dimensional) Mass Distribution
Find the center of mass of the system consisting of a 60 lb. girl and a 90 lb. boy at opposite ends of a teeter board 10 feet long.
60 lb. 90 lb.
x = ?!
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2.)
Linear (One‐Dimensional) Mass Distribution ‐ Application
The road from City A (population 35,000) to City B (population 80,000) is 40 miles long, and has two towns along the way that are 12 miles from City A and 10 miles from City B, respectively. The populations of the towns are 10,000 and 15,000. Where is the best place to locate an airport serving these communities?
City A City Bm1 = 35,000 m2 = 80,000m3 = 10,000 m4 = 15,000
x = ?!
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Center of Mass in a Two‐Dimensional Systemnot continuous ‐ "discrete"
x
y
m1
m2
mn
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x
y
m1
m2
mn(x1 , y1)
(x2 , y2)
(xn , yn)
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x
y
m1
m2
mn(x1 , y1)
(x2 , y2)
(xn , yn)
Moment about the y ‐axis:
= M y
..m1 x1 + m2 x2 + mn xn.
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x
y
m1
m2
mn(x1 , y1)
(x2 , y2)
(xn , yn)
Moment about the x ‐axis:
= M x
..m1 y1 + m2 y2 + mn yn.
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(x, y)Center of Massin a Two‐Dimensional System
= . . + m2 + mn.m1 +mwhere
is the total mass of the system.
andM y
mx = M x y = m
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3.)
The Center of Mass of a Two‐Dimensional SystemExample 3 p462
not continuous ‐ "discrete"
Find the center of mass of a system of point masses m1 = 6, m2 = 3, m3 = 2, and m4 = 9, located at
(3, 2), (0, 0), (5, 3), and (4, 2) as shown. m3 = 2
m1 = 6
m2 = 3
m4 = 9(5, 3)
(0, 0) (3, 2)
(4, 2).
..
.
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Assignmentp467 #1, 3-8 (one-dimensional), #9, 11 (two-dimensional)
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Center of Mass ("balancing point")of a Planar Lamina
Defs:
continuous mass throughout!
planar lamina ‐ a thin, flat plate of material of constant density.
density ‐ ( "rho") a measure of mass per unit area.p
massdensity = area
. mass density = area.. .
= .m Ap
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Moments and Center of Mass of a Planar Lamina
"vertical strips", on [a, b] f (x) g(x)
g(x)
f (x)
x
y
a b| |
g(x)
f (x)
x
y
a b| |
. .x f (x) g(x)+
2
aM y
= .pb
x [ f (x) g(x) ] dx
aM x = .p
b
[ f (x) g(x) ] dx f (x) g(x)+2[ [
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"vertical strips", on [a, b] f (x) g(x)
Center of Mass or Centroid
(x, y) ,= M x
mM y
m( (where,
= m p [ f (x) g(x) ] dxa
b
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The Centroid of a Plane RegionExample 5 p465
continuous mass distribution
Find the centroid of the region bounded by the graphs of f (x) = 4 x2 and g(x) = x + 2.4.)
f (x)
g(x)
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What if ?!!
"horizontal strips", on [c, d] f ( y) g( y)
g(y)y
f (y)
x
c
d
|
|
.
f (y)
x
c
d
|
|
y
f (y) g(y)+2
g(y)
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The Centroid of a Plane Region horizontal strips
continuous mass distribution
Find the centroid of the region bounded by the graphs of x = 4 y2 and x = 1.5.)
x = 1
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Assignmentp467 #13, 15, 17, 22, 24, 30
planar laminas
Set up! Use fnInt!