7.5 Moments, Centers of Mass

And Centroids

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:

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.

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 d = 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 4yM

xM

2432710

9 10xM

yM

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

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

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

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 )(