Chapter 7 Extra Topics

Crater Lake, OregonGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998

Centers of Mass:

Torque is a function of force and distance.(Torque is the tendency of a system to rotate about a point.)

d

gF

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:

= density per unit length

moment about origin: b

O aM x x dx

( is the Greek letter delta.)

mass: b

aM x dx

k kO

k

x mM

xM

m

center of mass: OMxM

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 = 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 4yMx

M

2432710

9 10xMy

M

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

Theorems of Pappus:

When a two dimensional shape is rotated about an axis:

Volume = area . distance traveled by the centroid.

Surface Area = perimeter . distance traveled by the centroid of the arc.

Consider an 8 cm diameter donut with a 3 cm diameter cross section:

3111cmV

2 areaV r

22 2.5 1.5V

211.25V

2.5

1.5

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