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Rotation of Rigid Object

• A rigid object is one in which the relative positions

of all the parts is fixed

• What happens when we rotate this object?– All points move in a circle

about the axis

– How far do they move?

– Note that θ is unitless (ratio

of distances) but “measured”

in radians

θrs =

r

s=θ

Rotation of Rigid Object

• What if the object is “spinning”– It turns thru a given angle

every second

– Define the “angular velocity”

– Angular velocity measured

in radians per second (rad/s

or s-1)

• What if it’s “spinning up”– Angular acceleration

dt

dθω =

2

2

dt

d

dt

d θωα ==

Rotation of Rigid Object

• While θ is a scalar, ωωωω and αααα are really vectors

• Use “right hand rule” to

determine ωωωω

• For αααα, it’s in the same

direction if the magnitude of ωωωω is increasing

Linear to Angular and Back

• For each linear

quantity (x,v,a) there is

a corresponding

angular quantity

connected by r

• Note that centripetal

acceleration is now:

ar

vr

=

=

α

ω

22

ωrr

vac ==

Linear to Angular and Back

• This angular-linear correlation extends to the

kinematic equations as well

Rotational Kinetic Energy

• What is the kinetic energy of a spinning object?– Just the sum of it’s parts!

– Call I the Moment of Inertia

( )( )2

2

2122

21

22

212

21

ii i

ii i

ii iii ii iR

rmI

Irm

rmvmKK

∑∑

∑∑∑

=

==

===

ωω

ω

2

Rotational Kinetic Energy

• What is the kinetic energy of a rolling object?

– Note that for a given energy, a larger I gives a

smaller v.• Hoops on a ramp go slower than disks

( ) 2

21

2

212

21

roll

2 vM

MvIKKK

R

I

CMR

+=

+=+= ω

Rotational Kinetic Energy

• What is the kinetic energy of a spinning object?– Just the sum of it’s parts!

( )( )2

2

2122

21

22

212

21

ii i

ii i

ii iii ii iR

rmI

Irm

rmvmKK

∑∑

∑∑∑

=

==

===

ωω

ω

If M >> m

Moment of Inertia

• What is the moment of inertia of an extended object– Break it up into little pieces

∫

∫

∑

=

=

∆=→∆

V

V

i iim

dVrr

dmr

mrIi

)(

lim

2

2

2

0

ρ

Moments of Inertia

• Moment of inertia of a

hoop or a thin cylinder– All the mass is at the

same R!

2

2

2

MR

dmR

dmrI

V

V

=

=

=

∫

∫

Moments of Inertia

• Moment of inertia of a

disk or solid cylinder– Consider ring at r, with

volume 2πrLdr

( )

( )2

21

22

21

4

21

0

4

41

0

3

0

2

22

2

22

MR

RLR

LRrL

drrLrLdrr

dVrdmrI

R

RR

VV

=

=

==

==

==

∫∫

∫∫

ρπ

πρπρ

πρπρ

ρ

3

Moments of Inertia

• Moment of inertia of a

thin rod about CM

( ) ( )[ ]( )

2

121

2

1213

12

3

2

3

23

3

31

22

2

2

2

2

ML

LLL

x

dxxdmrI

LL

V

L

L

L

L

=

==

−==

==

−

−

−∫∫

λ

λ

λ

λ

λ

Moments of Inertia

Moments of Inertia

• What about an

arbitrary axis?– Use “Parallel Axis

Theorem”

I = ICM + MD2

– Moment of inertia about

any axis is just moment of inertia about center of mass plus moment of inertia of “CM” about

the axis

Moments of Inertia

• What about an

arbitrary axis?– “Parallel Axis Theorem”

( )

( ) ( )[ ]( )

( ) ( )

( )

CM

2

CMCM

222

CM

2

CM

2

CM

2

CM

2

CM

2

CM

2

CM

2

CM

222

2

22

IMD

dmyyxx

dmyxdmyx

dmyyyyxxxx

dmyyxx

dmyxdmrI

+=

′+′+

′+′++=

′+′++′+′+=

′++′+=

+==

∫

∫∫

∫

∫

∫∫

Torque

• If angular acceleration

(α) is analogous to acceleration (a), what

is analogous to force

(F)?– Since α = rac, use

τ = rFc =rF sinφ– Call it “Torque”

(like α it too is a vector)

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Torque and Angular Acceleration

• Consider a particle at

position r– What are the

kimematics?

• Analogous to F=ma

(linear) we have τ=Iα

(angular)

( )

α

α

α

τ

I

mr

rrm

rmarF tt

=

=

=

==

2

Torque and Angular Acceleration

• What about for

extended object?

– Net torque gives rise to angular acceleration

α

α

ατ

I

mr

mr

i i

i iii i

=

=

=

∑∑∑

2

2

Work and Power

• Work and power for a rotating object

( )22

21

21

2

sin

ab

b

adt

db

adtd

b

a

b

a

b

a

I

dtIdtI

dtIdtPdtW

dt

d

dt

dWP

d

rdF

FdsFdxdW

ωω

ω

αωτω

τωθτ

θτ

θφ

ωω

−=

==

===

===

=

⋅=

==

∫∫

∫∫∫

Example: Trapdoor

• What is acceleration of

tip when horizontal?

• What is speed of tip

when vertical?

ggLa

L

gMgL

MLI

>==

=

==

23

tip

2 2

3

2

3

α

τα

( )

gLLv

MLI

MgUK

L

g

L

3 ,3

22

31

212

21

2horizvert

===

==

==

ωω

ωω

Rolling Motion

• For a rolling object the

point in contact with

the surface in

momentarily stationary– Thus, center (also CM)

moves thru a distance s=rθ, at a velocity v=rω

– Top moves at speed 2vCM

• Kinetic energy is

rotation plus

translation2

CM212

CM21 MvIK += ω

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T, a = ?

∞→?

M

∞→?

I

0?→M

0?

→I

TTTT === 321

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