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

• Contrast of measures of mass and force

Force mass acceleration

System Measure ofMass

Measure ofForce

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 6kg10kg •

Centroid

• Each mass in the system has a "moment" The product of the mass and the distance from

the origin

"First moment" is the sum of all the moments

• The centroid is

4kg 6kg10kg

1 1 2 2

1 2

m x m xx

m m

Centroid

• Centroid for multiple points

• Centroid about x-axis

1

1

n

i iin

ii

m xx

m

First moment of the system

Also notated My, moment about

y-axis

First moment of the system

Also notated My, moment about

y-axis

Total mass of the system

Total mass of the system

1

1

n

i iin

ii

yy

m

m

Also notated Mx,

moment about x-axis

Also notated Mx,

moment about x-axis

Also notated m, the total mass

Centroid

• The location of the centroid is the ordered pair

• Consider a system with 10g at (2,-1), 7g at (4, 3), and 12g at (-5,2) What is the center of mass?

( , )x y

y xM M

x ym m

Centroid• Given 10g at (2,-1), 7g at (4, 3), and 12g

at (-5,2)

10g

7g12g

10 (2) 7 4 12 ( 5)

10 ( 1) 7 3 12 2

10 7 12

y

x

M

M

m

? ?x y

Centroid

• Consider a region under a curve of a material of uniform density We divide the region into

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

a bx

•

Centroid of Area Under a Curve

• First moment with respectto the y-axis

• First moment with respectto the x-axis

• Mass of the region

( )b

y

a

M x f x dx

21( )

2

b

x

a

M f x dx

( )b

a

m f x dx

Centroid of Region Between Curves

• Moments

• Mass

f(x)

g(x) ( ) ( )b

y

a

M x f x g x dx

2 21( ) ( )

2

b

x

a

M f x g x dx

( ) ( )b

a

m f x g x dx y xM M

x ym m

Centroid

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

c

r

Theorem of Pappus

• Now revolve the region about the line L

• Theorem states that the volume of the solid of revolution iswhere A is the area of R

L

c

r

2V r A

Assignment

• Lesson 7.6

• Page 504

• Exercises 1 – 41 EOO also 49