Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Physics 111 Lecture 09

Rotation, Moment of Inertia

SJ 8th Ed.: Chap 10.1 –10.5

•Introduction and Overview

•Particles versus Rigid Bodies

•Rotational Variables

–Angular Position and Displacement

–Angular Velocity and Acceleration

•Relating Linear and Angular Variables

Angular Kinematics (constant angular

acceleration)

•Kinetic Energy of Rotation

•Calculating Moment of Inertia

–Definitions

–Parallel Axis Theorem

–Examples

10

.1A

ng

ula

r P

os

itio

n,

Ve

loc

ity,

an

d A

cc

ele

rati

on

10

.2R

ota

tio

na

l M

oti

on

wit

h C

on

sta

nt

An

gu

lar

Ac

ce

lera

tio

n (

Ro

tati

on

al

Kin

em

ati

cs

)

10

.3A

ng

ula

r a

nd

Tra

ns

lati

on

al

Qu

an

titi

es

10

.4R

ota

tio

na

l K

ine

tic

En

erg

y

10

.5C

alc

ula

tio

n o

f M

om

en

ts o

f In

ert

ia

Add

: e

xam

ple

10

.5

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

2

2

dt

rd

dtv

da

am

Fext

≡ ≡≡≡≡ ≡≡≡

= ===∑ ∑∑∑

)z,

y,x(

r= ===

r

2

21m

vK

tran

s= ===

•P

AR

TIC

LE

S (

LIN

EA

R M

OT

ION

)• •

RO

TA

TIO

N I

S M

EA

NIN

GL

ES

S•

Ze

ro s

ize

, s

ha

pe

un

de

fin

ed

–T

ran

sla

tio

n o

nly

fo

r p

art

icle

s

–C

M f

or

syste

ms o

f p

art

icle

s

•S

pe

cif

y p

os

itio

n b

y 3

sp

ati

al c

oo

rdin

ate

s.

• • •3

ve

loc

ity a

nd

3 a

cce

lera

tio

n c

om

po

ne

nts

•F

orc

es

an

d i

ne

rtia

go

ve

rn m

oti

on

•M

ass

m m

eas

ure

s in

ert

ia –

acce

lera

tio

n i

s

the

res

po

ns

e t

o f

orc

es

•T

ran

sla

tio

na

l k

ine

tic e

ne

rgy

•E

ne

rgy c

on

se

rve

d i

n i

so

late

d s

ys

tem

s•

Mo

me

ntu

m p

= m

v c

on

se

rve

d w

he

n n

et

ex

tern

al

fo

rce

= 0

.

Particles versus “Rigid Bodies”

A “rigid body’s”size and shape don’t change

POINT BODIES UNTIL NOW

2

2

dt

d

dt

d

I

ext

θ θθθ≡ ≡≡≡

ω ωωω≡ ≡≡≡

α αααα ααα

= ===Τ ΤΤΤ

∑ ∑∑∑r

rr

r

IK

KK

K

ro

tro

ttr

an

sto

t2

21ω ωωω

= ===+ +++

= ===

•R

IGID

BO

DIE

S

(AD

D R

OT

AT

ION

)

•R

OT

AT

ION

CR

ITIC

AL

TO

MO

TIO

N•

Sh

ap

e, s

ize

, &

ori

en

tati

on

aff

ec

t m

oti

on

–C

en

ter

of

Mass (

CM

) m

oves lik

e a

part

icle

–C

M t

ran

sla

tio

n +

ro

tati

on

ab

ou

t axis

•S

pe

cif

y p

os

itio

n +

ori

en

tati

on

:

A

dd

3

rota

tio

na

l c

oo

rdin

ate

s:

θ θθθa

bo

ut

rota

tio

n a

xis

+ 2

an

gle

s t

o

sp

ec

ify a

xis

dir

ec

tio

n

•3

an

gu

lar

ve

loc

ity a

nd

acce

lera

tio

n c

om

po

ne

nts

•T

orq

ue

an

d r

ota

tio

na

l in

ert

ia a

lso

go

ve

rn m

oti

on

.

To

rqu

e m

eas

ure

s t

wis

tin

g a

bo

ut

an

ax

is:

de

pe

nd

s

on

fo

rce

& a

pp

lic

ati

on

po

int.

•R

ota

tio

na

l in

ert

ia I

(m

om

en

t o

f in

ert

ia)

mea

su

res

re

sp

on

se

to

to

rqu

es

(“ta

u”).

•R

ota

tio

na

l k

ine

tic

en

erg

y i

n a

dd

itio

n t

o t

ran

sla

tio

na

l K

E

•E

ne

rgy s

till

co

ns

erv

ed

in

is

ola

ted

sys

tem

s•

An

gu

lar

mo

me

ntu

m L

= I

ω ωωωa

lso

co

ns

erv

ed

wh

en

ne

t ex

tern

al

torq

ue

= 0

.

FINITE SIZE BODIES

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Angular Variables for Rotation

rad

ian

s)

(i

n

r

sle

ng

th

arc

θ θθθ= ===

≡ ≡≡≡rs

rad

≡ ≡≡≡θ θθθ

•Exam

ple:

r =

10 c

m,

θ θθθ=

100 r

ad

ian

s � ���

s =

1000 c

m =

10 m

.

Defi

nit

ion

:•

1 r

ad

ian

= a

ng

le t

hat

cu

ts o

ff a

rc len

gth

s =

rad

ius r

•

2π πππ

rad

ian

s =

fu

ll c

ircle

oo

o

.π

π

rad

ian

3

57

180

2

360

1= ===

= ==== ===

s

θ θθθ

rθ θθθ’

An

gle

s in

rad

ian

s sim

ple

ro

tati

on

fo

rmu

las

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Angular displacement and arc length (rigid body)

Angular displacementis the n

et

ch

an

ge

in

th

e a

ng

ula

r c

oo

rdin

ate

ra

dia

ns

)

in

(an

gle

sin

ita

lfi

na

l

θ θθθ

− −−−θ θθθ

≡ ≡≡≡θ θθθ

∆ ∆∆∆

Arc length

∆ ∆∆∆s m

easu

res d

ista

nce

co

vere

d b

y

a p

oin

t as i

t m

ove

s t

hro

ug

h ∆

θ∆

θ∆

θ∆

θ(c

on

sta

nt

r)

yR

efe

ren

ce lin

e r

ota

tin

g w

ith

bo

dy

x∆ ∆∆∆s = r ∆

θ∆

θ∆

θ∆

θ

θ θθθo

θ θθθf

rr

arc

)

cir

cu

lar

a

alo

ng

d

ista

nce

(a

rs

θ θθθ∆ ∆∆∆

≡ ≡≡≡∆ ∆∆∆

Ch

oo

se a

ro

tati

on

axis

•E

ach

po

int

of

the b

od

y m

oves a

rou

nd

th

e a

xis

in

a c

ircle

wit

h a

sp

ecif

ic r

ad

ius

•O

ne a

ng

ula

r co

ord

ina

te θ θθθ

sp

ecif

ies

po

sit

ion

of

the w

ho

le b

od

y a

bo

ut

this

rota

tio

n a

xis

.

•θ θθθ

is m

eas

ure

d C

CW

fro

m t

he x

-axis

, can

incre

ase p

as

t 2

π

π

π

π o

r can

be n

eg

ati

ve

.

x

y

rigid bod

yrotation

axis “o”

parallel to z-axis

Refe

renc

e

line

rotates

with bod

y

θ θθθ

An

gu

lar

dis

pla

cem

en

t:

•T

he n

et

an

gle

a r

igid

bo

dy r

ota

tes t

hro

ug

h d

uri

ng

so

me t

ime

in

terv

al…

..

•…

…a

lso

th

e a

ng

le s

wep

t o

ut

by a

re

fere

nce l

ine f

ixed

a b

od

y

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Rigid body rotation: angular & tangential velocity

Angular velocity

ω

ω

ω

ω

( (((“o

meg

a”):

•R

ate

of

ch

an

ge

of

the

an

gu

lar

dis

pla

ce

men

t

d

t

d t

tL

im

t

in

st

ave

θ θθθ

∆ ∆∆∆

θ θθθ∆ ∆∆∆

→ →→→∆ ∆∆∆

≡ ≡≡≡ω ωωω

∆ ∆∆∆

θ θθθ∆ ∆∆∆

≡ ≡≡≡ω ωωω

≡ ≡≡≡

0

•U

nit

s:

rad

ian

s/s

ec. P

os

itiv

e i

n C

CW

sen

se

•S

am

e ω ωωω

for

every

po

int

on

th

e b

od

y

•If

ω ωωωis

CO

NS

TA

NT

(un

ifo

rm c

ircu

lar

mo

tio

n)

tf

∆ ∆∆∆ω ωωω

= ===θ θθθ

− −−−θ θθθ

= ===θ θθθ

∆ ∆∆∆0

•F

req

uen

cy f

=

# o

f co

mp

lete

revo

luti

on

s/u

nit

tim

e

•f

= 1

/T T

= p

eri

od

(ti

me f

or

1 c

om

ple

te r

evo

luti

on

)

/2f

/T2

fπ πππ

ω ωωω= ===

π πππ= ===

π πππ= ===

ω ωωω2

Tangential velocityvT:

•U

nit

s:

dis

tan

ce

/ t

ime

•R

ate

at

wh

ich

a p

oin

t sw

eep

s o

ut

arc

len

gth

alo

ng

cir

cu

lar

pa

th

•P

rop

ort

ion

al

to r

, sa

me ω ωωω

rv

Tω ωωω

= ===

rs

θ θθθ∆ ∆∆∆

= ===∆ ∆∆∆

rdd

sco

nsta

nt)

(r

θ θθθ= ===

r

dt

dr

dt

ds

vT

ω ωωω= ===

θ θθθ= ===

≡ ≡≡≡

xvT

θ =

θ

=

θ =

θ

= ω

∆τ

ω∆

τω

∆τ

ω∆

τr

For

any

poi

nt,

r is

the p

erp

end

icul

ar

dis

tanc

e t

o th

e r

otati

on a

xis

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

�9.1

. T

he

peri

od

of

a r

ota

tin

g w

heel is

12.5

7 s

ec

on

ds.

T

he

rad

ius o

f th

e w

heel is

3 m

ete

rs.

It’s

an

gu

lar

sp

eed

is

clo

sest

to:

Angular velocity, period, and linear velocity

A.

79 rpm

B.

0.5 rad/s

C.

2.0 rad/s

D.

.08 rev/s

E.

6.28 rev/s

�9.2

. A

po

int

on

th

e r

im o

f th

e s

am

e w

heel h

as a

tan

gen

tia

l sp

eed

clo

ses

t to

:

A.

12.57 rev/s

B.

0.8 rev/s

C.

0.24 m/s

D.

1.5 m/s

E.

6.28 m/s

/T

2f

π πππ= ===

π πππ= ===

ω ωωω2

rv

Tω ωωω

= ===

r

sθ θθθ

∆ ∆∆∆≡ ≡≡≡

∆ ∆∆∆

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Centripetal, angular, tangential accelerations

Centripetal (radial) acceleration a

cor a

r:

•B

od

y r

ota

tes a

t ra

te ω ωωω

.

•C

en

trip

eta

l accele

rati

on

co

mp

on

en

t, p

oin

ts t

ow

ard

ro

tati

on

axis

•C

on

sta

nt

mag

nit

ud

e if

ω ωωωis

co

nsta

nt

(UC

M).

U

nit

s:

len

gth

/tim

e2

•C

han

gin

g if

ω ωωωn

ot

co

nsta

nt

(an

gu

lar

accele

rati

on

no

t zero

)

r)T

v (u

se

Tc

r

rv

aω ωωω

= ===ω ωωω

= ==== ===

2

2

So

far:

not

hin

g abou

t ang

ular

velo

city

chang

ing

ma

Fc

lc

en

trip

eta

= ===

xvT

ω ωωω

rac

Angular acceleration α

:α

:α

:α

:

•R

ate

of

ch

an

ge o

f th

e a

ng

ula

r velo

cit

y

t

Lim

tt

inst

ave

∆ ∆∆∆ω ωωω∆ ∆∆∆

≡ ≡≡≡α ααα

∆ ∆∆∆ω ωωω∆ ∆∆∆

≡ ≡≡≡α ααα

→ →→→∆ ∆∆∆

0

•U

nit

s:

rad

ian

s/s

ec/s

ec.

•C

CW

co

nsid

ere

d p

osit

ive

•for C

ON

ST

AN

T α

:α

:α

:α

:

t

f∆ ∆∆∆

α ααα= ===

ω ωωω− −−−

ω ωωω= ===

ω ωωω∆ ∆∆∆

0

t

dd

d

t

d2θ θθθ

= ===ω ωωω

≡ ≡≡≡α ααα

2

x

vT ω,α

ω,α

ω,α

ω,α

rac

aT

Tangential acceleration a

T:

•T

an

gen

tial accele

rati

on

co

mp

on

en

t

•P

rop

ort

ion

al to

an

gu

lar

accele

rati

on

a a

nd

als

o t

o

rad

ius r

•U

nit

s:

len

gth

/ t

ime 2

ra

Tα ααα

= ===

r

dt

dr

dt

dv

aT

Tα ααα

= ===ω ωωω

= ===≡ ≡≡≡

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

The whole set of angular and linear motion variables

x

r

θ,ω

,αθ

,ω,α

θ,ω

,αθ

,ω,α

vT

ac

aT

a to

tal

Angular variables ∆

θ∆

θ∆

θ∆

θ,

ω,

α,

ω,

α,

ω,

α,

ω,

α:

•apply to the whole rotating body

Linear variables r, s, vT, aT, ac:

•apply to a specific point on a rotating

body & are signed magnitudes

•values all proportional to r –distance

from the rotation axis

r

ac

2ω ωωω

= ===

ra

Tα ααα

= ===

Total linear acceleration of a point (rotation):

•vector sum of radial and tangential

components (normal to each other)

aa

aT

cto

tal

22

+ +++= ===

Implications:

•IF

ω ωωωis constant, then α ααα=0, aT=0. But ac= a

ris not 0

•IF

ω ωωωis not constant, then α αααand aTare not 0, a

cand VTvary with time

•IF α αααis constant, then aTis constant for a particular point, different for different r

∆ω

∆ω

∆ω

∆ωis proportional to ∆ ∆∆∆t(angular kinematics)

vTis not constant (so this is not UCM) }

2

2

dt

d

dt

d

dt

d

θ θθθ= ===

ω ωωω≡ ≡≡≡

α αααθ θθθ

≡ ≡≡≡ω ωωω

θ θθθ∆ ∆∆∆

ωr

vr∆

θ θθθs

T= ===

= ===

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Rotational variables are vectors, having direction

•T

he a

ng

ula

r d

isp

lacem

en

t, s

pe

ed

, an

d a

ccele

rati

on

•(

θ,

θ,

θ,

θ,

ω ωωω, , , ,

α ααα)

•are

th

e m

ag

nit

ud

es o

f ro

tati

on

al

dis

pla

ce

men

t,

ve

locit

y,

an

d

accele

rati

on

ve

cto

rs

•T

he d

irecti

on

s a

re g

iven

by t

he

rig

ht-

han

d r

ule

Poin

t th

umb a

long

+ a

xis

fing

ers

cur

l CCW

(po

siti

ve)

sens

e

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Linear and Angular Kinematics Equations

(Same mathematical forms)

dt

dv

a= ===

dt

dx

v= ===

Linear motion

•Constant acceleration a

x(t

), v

(t),

a(t

)

Angular motion

•Constant angular acceleration α ααα

Variables

Definitions

Kinematic

Equations

at

v)t(

vf

+ +++= ===

0

2

21

00

at

tv

x)t(

xf

+ ++++ +++

= ===

]x

x[a

v)t(

vf

f0

2 0

22

− −−−+ +++

= ===

Check by differentiating:

at

vat

)(

vd

t

dx

v+ +++

= ===+ +++

= ==== ===

021

02

Both sets of kinematic equations follow from the definitions of

velocity and acceleration by integrating

tv

vx

)t(x

)f

o(

f+ +++

+ +++= ===

21

0

θ(

θ(

θ(

θ(t

),

),

),

), ω

(ω

(ω

(ω

(t),

),

),

),

α(

α(

α(

α(t

) )))

dt

dθ θθθ

= ===ω ωωω

dt

dω ωωω

= ===α ααα

t)t(

fα ααα

+ +++ω ωωω

= ===ω ωωω

0

2

21

00

tt

)t(f

α ααα+ +++

ω ωωω+ +++

θ θθθ= ===

θ θθθ

][

)t(f

f0

2 0

22

θ θθθ− −−−

θ θθθα ααα

+ +++ω ωωω

= ===ω ωωω

at

t)

(d

t

d+ +++

ω ωωω= ===

α ααα+ +++

ω ωωω= ===

θ θθθ= ===

ω ωωω0

21

02

t)t(

)f

o(

fω ωωω

+ ++++ +++

θ θθθ= ===

θ θθθω ωωω

21

0

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Hints for solving rotational kinematics problems

•The r

otati

onal ki

nem

ati

cs e

quati

ons

have

the s

am

e m

ath

em

ati

cal

form

as

the lin

ear

kine

mati

cs e

quati

ons

but

wit

h d

iffe

rent

sym

bol

s.

•Tech

niqu

es

are

sim

ilar

to t

hos

e u

sed in

line

ar

mot

ion

prob

lem

s•

Con

stant

(or

zero

) ang

ular

acc

ele

rati

on.

•Velo

citi

es

and

acc

ele

rati

ons

(lin

ear

and

ang

ular)

are

resp

ect

ively

firs

t and

seco

nd d

eri

vati

ves

of d

ispl

ace

ment

s.

Som

e d

iffe

renc

es

for

Rot

ati

onal Kin

em

ati

cs:

•A r

otati

on a

xis

is

invo

lved.

Onc

e s

et,

axis

choi

ce m

ust

rem

ain

the s

am

e f

or t

he p

roble

m•

In

som

e p

roble

ms

there

may b

e a

n ob

viou

s, n

atu

ral axis

.•

The o

bje

ct k

eeps

retu

rnin

g to

its

ori

gina

l or

ient

ati

on (

if t

here

is n

o tr

ans

lati

on).

•

Dis

place

ment

can

be s

peci

fied a

s th

e n

umber

of r

evo

luti

ons

made b

y t

he b

ody,

or b

y t

he t

otal ang

le.

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Angular kinematics example:

A t

urn

tab

le i

nit

iall

y a

t re

st

reach

es a

sp

ee

d o

f 33 1

/3 r

pm

in

15

seco

nd

s.

D

ete

rmin

e t

he a

vera

ge a

ng

ula

r accele

rati

on

.

t

t0

fave

∆ ∆∆∆

ω ωωω− −−−

ω ωωω= ===

∆ ∆∆∆ω ωωω∆ ∆∆∆

≡ ≡≡≡α ααα

Use t

he d

efi

nit

ion

or

a k

inem

ati

c f

orm

ula

t

0f

ave

∆ ∆∆∆

ω ωωω− −−−

ω ωωω= ===

α ααα

ave

an

dco

nsta

nt

is

f

t)t(

α ααα= ===

α ααα∆ ∆∆∆

α ααα+ +++

ω ωωω= ===

ω ωωω0

Eit

her

wa

y:

Un

its:

rad

/sec

ra

d/s

re

v/m

in

3

.49

/60

2 x

10

0/3

1/3

3

3= ===

π πππ= ===

2ra

d/s

ec

ave

0

.23

3

.

= ===− −−−

= ===α ααα

15

04

93

Su

bsti

tute

:

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

at

t =

0:

Angular kinematics example:

A g

rin

dsto

ne is

ro

tati

ng

wit

h c

on

sta

nt

an

gu

lar

accele

rati

on

ab

ou

t a f

ixed

axis

in

sp

ace.

Ini

tial co

ndit

ions

Use

kin

em

ati

cs f

orm

ula:

t

t)t(

f2

21

00

α ααα+ +++

ω ωωω+ +++

θ θθθ= ===

θ θθθ

Po

sit

ive

dir

ec

tio

ns

:

rig

ht

ha

nd

ru

le

0

rad

/s

4.6

-

rad

/s

0.3

5

0

2= ===

θ θθθ= ===

ω ωωω= ===

α ααα0

a) When is the displacement

∆ ∆∆∆θ θθθ= 5 rev.

2

21f

t 0.3

5

t

4.6

-

rad

10

rev

5.0

)t(

+ +++= ===

π πππ= ===

= ===θ θθθ

− −−−θ θθθ

0

0

10

t

4.6

t

.2

= ===π πππ

− −−−− −−−

175

0Sol

ve q

uadra

tic

equ

ati

on:

ne

ga

tive

)

isro

ot

(se

co

nd

s

ec

3

2

t

= ===

init

ially r

otati

ng in

nega

tive

sens

e b

ut w

ith p

osit

ive a

ngul

ar

acc

ele

rati

on

b) When is ∆ ∆∆∆

θ θθθ= 0 again?

Set:

t

t2

21

00

α ααα+ +++

ω ωωω+ +++

= ===

Two

root

s are

:{

co

nd

itio

n)

in

itia

l

the

(ju

st

0

t

= ===

s 26.3

2-

t

0 t

2 0

= ===α αααω ωωω

= ===⇒ ⇒⇒⇒

= ===α ααα

= ===ω ωωω

0

Reve

rses direction,

rotates back and

th

roug

h origina

l po

sition

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

�9.3

. A

t t

= 0

, a

wh

eel ro

tati

ng

ab

ou

t a f

ixed

ax

is a

t a

co

ns

tan

ta

ng

ula

r accele

rati

on

ha

s a

n a

ng

ula

r velo

cit

y o

f 2

.0 r

ad

/s. T

wo

seco

nd

s la

ter

it h

as

tu

rne

d t

hro

ug

h 5

.0 c

om

ple

te r

evo

luti

on

s.

Fin

d t

he a

ng

ula

r acc

ele

rati

on

of

this

wh

eel?

Wheel rotating and accelerating

A.

17 rad/s2

B.

14 rad/s2

C.

20 rad/s2

D.

23 rad/s2

E.

13 rad/s2

t)t(

fα ααα

+ +++ω ωωω

= ===ω ωωω

0

2

21

00

tt

)t(f

α ααα+ +++

ω ωωω+ +++

θ θθθ= ===

θ θθθ

][

)t(f

f0

2 0

22

θ θθθ− −−−

θ θθθα ααα

+ +++ω ωωω

= ===ω ωωω

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Mom

ent

of

Ine

rtia

I

measu

res B

OT

H m

ass a

nd

its

dis

trib

uti

on

dep

en

ds o

n t

he c

ho

ice o

f ro

tati

on

axis

r pis

perp

en

dic

ula

r d

ista

nce f

rom

mass p

oin

t to

axis

masses o

naxis

have r

p=

0, n

o c

on

trib

uti

on

to

I

m

as

se

s

po

int

fo

r2

i,i

P

rm

I ∑ ∑∑∑

⊥ ⊥⊥⊥= ===

sam

e m

ass

small I

larg

er

I

“M

om

en

t o

f in

ert

ia”

pla

ys t

he r

ole

of

mass

“T

orq

ue p

lays t

he r

ole

of

forc

e

Rot

ati

onal 2

ndLaw:

I

ne

tα ααα

= ===τ τττ

ma

F net= ===

Ro

tati

on

al w

ork

:

Po

wer

used

wh

en

d

oin

g r

ota

tio

nal w

ork

:

θ θθθτ τττ

= ===d

dW

rot

Fd

xd

W= ===

Pd

t

dW

rot

rot

τω τωτωτω= ===

≡ ≡≡≡v

FP

o

dt

dW

= ===≡ ≡≡≡

Tor

que τ τττ

TW

IST

-m

easu

res a

pp

lied

fo

rce A

ND

wh

ere

it

acts

MO

ME

NT

AR

M(l

ever

arm

) r p

= p

erp

en

dic

ula

r d

ista

nce

fro

m a

xis

to

lin

e o

f acti

on

of

forc

e

Fr

)sin

(rF

arm

le

ver

x

forc

e

× ×××= ===

θ θθθ= ===

= ===τ τττ

F

r p

r

θF

r pr

θ

Key Angular Dynamics Concepts

bo

die

s

co

nti

nu

ou

s

for

P

dm

xd

mr

I

∫ ∫∫∫∫ ∫∫∫

= ===≡ ≡≡≡

⊥ ⊥⊥⊥

22

the

ore

m

ax

is

pa

rall

el

c

mP

M

h

I

I2

+ +++= ===

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Rotational Kinetic Energy

∆m

i

ω

•T

ota

l K

E c

an

ha

ve t

wo

part

s:

Kto

t=

Kro

t+

Kcm

•C

on

sid

er

wh

eel ro

tati

ng

in

x-y

pla

ne.

K

cm

= 0

•C

om

po

sed

of

mass c

hu

nks ∆ ∆∆∆

mi

mo

vin

g i

n c

ircle

s•

All

ch

un

ks h

ave s

am

e a

ng

ula

r sp

eed

ω ωωω

axis

ro

tati

on

to

m

fro

m

dis

tan

ce

la

rp

erp

en

dic

ui

i

r ∆ ∆∆∆

⊥ ⊥⊥⊥= ===

• •••

v

mK

ET

i i

rot

h

as

ch

un

k

mass

E

ach

2

21∆ ∆∆∆

= ===• •••

im

o

f

velo

cit

y

an

gen

tial

ti

Ti

r

v ∆ ∆∆∆

⊥ ⊥⊥⊥= ===

ω ωωω= ===

• •••

[ [[[] ]]]

2i

i2

i i

rot

:to

tal

get

to

ch

un

ks

th

e

all

fo

r

en

erg

ies

d

dA

r

m

2

rm

KE

ω ωωω∆ ∆∆∆

= ===ω ωωω

∆ ∆∆∆= ===

∑ ∑∑∑∑ ∑∑∑

⊥ ⊥⊥⊥

• •••

22

1

2

∆ ∆∆∆m

i

ω

ir ⊥ ⊥⊥⊥r

Ti

v

y

xz

∑ ∑∑∑⊥ ⊥⊥⊥

∆ ∆∆∆≡ ≡≡≡

2

i ir

m I

Mo

men

t o

f In

ert

iam

easu

res r

esis

tan

ce

to t

orq

ue (

twis

t)2

1ω ωωω

= ===I

K

E2

rot

Ro

tati

on

al

KE

in

term

s o

f I

-re

ca

ll 1

/2m

v2

for

lin

ear

mo

tio

n

Ι ΙΙΙd

ep

en

ds o

n w

here

yo

u p

ick t

he a

xis

:

vs vs

Moment of Inertia -Definition

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Example: Find moments of inertia for a crossed dumbbell

•Fou

r id

ent

ical balls

as

show

n:

m =

2 k

g•C

onne

cted b

y m

ass

less

rods:

le

ng

th d

= 1

m.

m m

mm

d d

dd

AC

Bd 2

A)

Ch

oo

se r

ota

tio

n a

xis

perp

en

dic

ula

r to

fig

ure

th

rou

gh

po

int

“A

”

m

as

se

sp

oin

t

for

2 i,

i

rm

I ∑ ∑∑∑

⊥ ⊥⊥⊥= ===

22

2

2 i,

iA

m kg

1

2

4

4m

d

r

mI

8= ===

× ×××× ×××

= ==== ===

= ===∑ ∑∑∑

⊥ ⊥⊥⊥

B)

No

w c

ho

ose a

xis

perp

en

dic

ula

r to

fig

ure

th

rou

gh

po

int

“B

”

22

22

2

Bm

kg

6

8m

d

d

)

2m

(

d

)

2m

(

m(2

d)

I

1= ===

= ===+ +++

+ +++= ===

C)

Let

rota

tio

n a

xis

pass t

hro

ug

h p

oin

ts “

B”

an

d “

C”

22

22

Cm

kg

2m

d

m

d

m

d

m(0

)

m(0

)

I 4

= ==== ===

+ ++++ +++

+ +++= ===

Rot

ati

onal in

ert

ia I

depe

nds

on a

xis

chos

en

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Parallel Axis Theorem

para

llel

axis

th

eo

rem

descri

be

s v

ari

ati

on

, b

as

ed

on

Ic

m

ap

pli

es t

o p

oin

t m

asses a

nd

to

co

nti

nu

ou

s b

od

ies

Th

e m

om

en

t o

f in

ert

ia I

d

ep

en

ds w

here

th

e

rota

tio

n a

xis

P i

s

M = the total mass of the body

CM = mass center

h = perpendicular distance from CM to axis through P

2M

h

I

Icm

P+ +++

= ===

mom

ent

of

inert

ia a

bou

t axis

thro

ugh p

oint

P

rota

tion

al in

ert

ia o

f to

tal m

ass

M if

at

the C

M p

oint

, ta

ken

abou

t axis

thro

ugh P

mom

ent

of

inert

ia a

bou

t axis

thro

ugh C

M p

ara

llel

to a

xis

thro

ugh P

para

llel axes

thro

ugh

P and

CM

“rot

ati

onal in

ert

ia”

and

“m

oment

of

inert

ia”

are

syno

nym

s

Ch

oo

sin

g t

he

ax

isT

hro

ug

h t

he

CM

min

imiz

es

th

e m

om

en

t o

f in

ert

ia (

h =

0)

CM

P

h

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

An example showing that the parallel axis theorem works

•Rot

ati

on a

xis

perp

end

icul

ar

to p

lane

of

figu

re

•M

ass

es

on t

he c

orne

rs o

f a r

ect

ang

le,

sides

a &

b

•Abou

t an

axis

thro

ugh t

he C

M: [ [[[

] ]]]2

22

22

44

44

ba

m

b

a

mh

m I C

M+ +++

= ===

+ +++

= ===≡ ≡≡≡

•Abou

t an

axis

“P”

thro

ugh a

cor

ner:

[ [[[] ]]]

[ [[[] ]]]

22

22

0b

a

2m

ba

m

b m

ma

I 2

2P

+ +++= ===

+ ++++ +++

+ ++++ +++

= ===

h2= (a/2

)2+ (b/2

)2

X cm

b

a

h

h

mm m

mP

2

i,i

r

mI ∑ ∑∑∑

⊥ ⊥⊥⊥= ===

•U

sing

the P

ara

llel Axis

Theor

em

dir

ect

ly f

or t

he s

am

e c

orne

r axis

:

[ [[[] ]]]

[ [[[] ]]]

22

22

2

44

4b

a

2m

ba

m

ba

m

hM

I

I 2

2to

tcm

P+ +++

= ===

+ +++

+ ++++ +++

= ===+ +++

= ===

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

�9.4

T

he f

igu

re s

ho

ws f

ou

r p

art

icle

s j

oin

ed

to

ge

ther

by m

assle

ss, ri

gid

ro

ds.

A

ssu

me a

= b

an

d M

> m

.

Ab

ou

t w

hic

h o

f th

e f

ollo

win

g a

xes d

oes

the

sys

tem

ha

ve t

he s

ma

lles

t m

om

en

t o

f in

ert

ia?

Moment of inertia for another crossed dumbbell

A.

the x axis

B.

the y axis

C.

the z axis

D.

the moment of inertia is the same for

the first two choices above

E.

the moment of inertia is the same for

all of the first three choices above

2

i,i

r

mI ∑ ∑∑∑

⊥ ⊥⊥⊥= ===

�9.5

. A

bo

ut

wh

ich

of

the

axes d

oes t

he

sys

tem

ha

ve t

he

larg

est

mo

men

t o

f in

ert

ia?

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Moment of Inertia for Continuous Mass Distributions

∫ ∫∫∫∑ ∑∑∑

→ →→→∞ ∞∞∞

→ →→→→ →→→

∆ ∆∆∆

,

n

,d

m

m

i

dm

rd

m

s

i

i i

22

⊥ ⊥⊥⊥= ===

→ →→→∆ ∆∆∆

= ===∆ ∆∆∆

II

Con

tinu

um lim

it:

let

chun

k si

ze � ���

0

∑ ∑∑∑∑ ∑∑∑

= ==== ===

∆ ∆∆∆= ===

∆ ∆∆∆≡ ≡≡≡

n,i

i

n,i

ii

m

s

11

2I

I

∆ ∆∆∆m

is

i

P

The p

ara

llel axis

theor

em

hol

ds,

as

befo

re

2M

h

I

Ic

mP

+ +++= ===

mom

ent of inertia abou

t ax

is throug

h point P

mom

ent of inertia abou

t mass ce

nter

total

mass

perpend

icular

distanc

e

betw

een ax

es

The int

egr

al ca

n of

ten

be e

valu

ate

d,

esp

eci

ally f

or c

onst

ant

dens

ity.

V

d )r(

rd

mr

I P

32

2

ρ ρρρ= ===

≡ ≡≡≡∫ ∫∫∫

∫ ∫∫∫⊥ ⊥⊥⊥

⊥ ⊥⊥⊥

dens

ity

volu

me

d

m r

M

1

r

vo

lto

tc

m∫ ∫∫∫

≡ ≡≡≡r

v

Com

pare

wit

h M

ass

Cent

er

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Exam

ple:

Fin

d t

he m

oment

of

inert

ia o

f a u

nifo

rm t

hin

…ri

gid r

od a

bou

t an

axis

thro

ugh its

cent

er

•The s

haded a

rea h

as

a m

ass

:

dm

= λ λλλ

dx

λ λλλ=

mass/u

nit

len

gth

= M

/Ldm

xd

mr

I P

∫ ∫∫∫∫ ∫∫∫

= ===≡ ≡≡≡

⊥ ⊥⊥⊥

22

tota

l m

ass M

-L

/ 2

L /

2

y

•Then

the m

oment

of

inert

ia

abou

t y-axis

is:

( ((() )))

L

L

31 LM

/L

/L

31 LM

L/2

L/2

-

L/2

L/2

-y

x

d

xx

dm

x

I

− −−−− −−−

+ +++ − −−−

+ ++++ +++

= ==== ===

λ λλλ= ===

≡ ≡≡≡∫ ∫∫∫

∫ ∫∫∫8

8

2 2

32

23

3

2M

L

I

121

y

= ===∴ ∴∴∴

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Moments of Inertia of Various Rigid Objects

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Exam

ple:

Mom

ent

of

inert

ia f

or a

rod

ro

tati

ng a

roun

d o

ne e

nd

Use Parallel Axis Theorem

cm

2M

h

I

Ic

mP

+ +++= ===

M

LL

M

ML

I

121

P2

312

2

2= ===

+ +++= ===

M

L

I

121

cm

2= ===

There

fore

,

The m

oment

of

inert

ia o

f th

e r

od a

bou

t it

s ce

nter

is

h =

½L

Co

pyri

gh

t R

. J

an

ow

–S

pri

ng

2012

Application of the parallel axis theorem to a wheel and a sphere

Pr

cm

�Cir

cula

r pl

ate

or

cylind

er:

Fin

d I

abou

t axis

thro

ugh P

Fro

m T

ab

le:

M

r

I21

cm

2= ===

“cm

”axis

is n

orm

al

to p

lan

e t

hro

ug

h c

en

ter

of

wh

eel

r

h

Mh

I

I2

cm

P= ===

+ +++= ===

“P

”axis

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