Transcript Moments, Center of Mass, Centroids Lesson 7.6

Moments, Center of Mass,
Centroids
Lesson 7.6
Mass
• Definition: mass is a measure of a body's
resistance to changes in motion
 It is independent of a particular gravitational
system
 However, mass is sometimes equated with
weight (which is not technically correct)
 Weight is a type of force … dependent on gravity
Mass
• The relationship is Force  mass  acceleration
• Contrast of measures of mass and force
System
Measure of
Mass
Measure of
Force
U.S.
Slug
Pound
International
Kilogram
Newton
C-G-S
Gram
Dyne
Centroid
• Center of mass for a system
 The point where all the mass seems to be
concentrated
 If the mass is of constant density this point is
called the centroid
4kg
10kg
•
6kg
Centroid
• Each mass in the system has a "moment"
 The product of the mass and the distance from
the origin
4kg
•
10kg
6kg
 "First moment" is the sum of all the moments
The centroid is x  m1 x1  m2 x2
m1  m2
Centroid
• Centroid for multiple points
n
x
m x
i 1
n
i i
 mi
First moment of
the system
Also notated My,
moment about
y-axis
n
i 1
Total mass of the
system
Also notated m,
the total mass
• Centroid about x-axis
Also
notated Mx,
moment
about
x-axis
y
m y
i 1
n
i
i
m
i 1
i
Centroid
• The location of the centroid is the ordered
pair ( x , y )
x
My
m
Mx
y
m
• Consider a system with 10g at (2,-1), 7g at
(4, 3), and 12g at (-5,2)
 What is the center of mass?
Centroid
• Given 10g at (2,-1), 7g at (4, 3), and 12g
at (-5,2)
7g
12g
10g
M y  10  (2)  7  4  12  ( 5)
M x  10  (1)  7  3  12  2
x ? y ?
m  10  7  12
Centroid
• Consider a region under
a curve of a material of
uniform density
 We divide the region into
•
a
x
b
rectangles
 Mass of each considered to be centered at
geometric center
 Mass of each is the product of the density, ρ and
the area
 We sum the products of distance and mass
Centroid of Area Under a Curve
• First moment with respect
b
to the y-axis
M y     x  f ( x) dx
a
• First moment with respect
b
to the x-axis
1
Mx 
   f ( x)

2
2
a
• Mass of the region
b
m     f ( x ) dx
a
dx
Centroid of Region Between
Curves
• Moments
f(x)
b
M y     x   f ( x)  g ( x)  dx
g(x)
a
b
1
Mx    
2a
 f ( x)   g ( x)  dx
2
2
• Mass
Centroid
b
m      f ( x)  g ( x)  dx
a
x
My
m
Mx
y
m
Try It Out!
• Find the centroid of
the plane region
bounded by
y = x2 + 16 and
the x-axis over
the interval 0 < x < 4
 Mx = ?
 My = ?
 m=?
Theorem of Pappus
• Given a region, R, in the plane and L a line in
the same plane and not intersecting R.
• Let c be the centroid and r be the distance
from L to the centroid
L
r
R
c
Theorem of Pappus
• Now revolve the region about the line L
• Theorem states that the volume of the solid
of revolution is V  2    r  A
where A is the area of R
L
r
R
c
Assignment
• Lesson 7.6
• Page 504
• Exercises 1 – 41 EOO
also 49