PHY 711 Fall 2014 -- Lecture 25 110/24/2014

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 25:Rotational motion (Chapter 5)

1. Rigid body motion in body fixed frame

2. Conversion between body and inertial reference frames

PHY 711 Fall 2014 -- Lecture 25 210/24/2014

PHY 711 Fall 2014 -- Lecture 25 310/24/2014

r

w

22

2

1 1Kinetic energy: 2 2

1 21 2

inertial

p p p pp p

p p pp

p p p pp

ddt

T m v m

m

m

r ω r

ω r

ω r ω r

ω ω r r r ω

Summary of previous results describing rigid bodies rotating about a fixed origin

ω I ω� 2

p p p pp

m r I 1 r r�

PHY 711 Fall 2014 -- Lecture 25 410/24/2014

2

Matrix notation:

xx xy xz

yx yy yz ij p ij p pi pjp

zx zy zz

I I II I I I m r r rI I I

I�

Moment of inertia tensor

1 1 2 2 3 3

1For general coordinate system: 2

For (body fixed) coordinate system that diagonalizesˆ ˆ moment of inertia tensor: 1,2,3

1ˆ ˆ ˆ 2

ij i jij

i i i

T I

I i

T I

ww

w w w

I e e

ω e e e

�

2i i

i

w

PHY 711 Fall 2014 -- Lecture 25 510/24/2014

r

w

Angular momentum:

inertial

p p p p pp p

p p p pp

p

p

ddt

m m

m

r ω r

L r v r ω r

ω r r r r ω

Summary of previous results describing rigid bodies rotating about a fixed origin

I ω� 2

p p p pp

m r I 1 r r�

PHY 711 Fall 2014 -- Lecture 25 610/24/2014

Descriptions of rotation about a given origin -- continued

1 1 2 2 3 3

1 1 1 2 2 2 3 3 3

For (body fixed) coordinate system that diagonalizes moment of inertia tensor:

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

Time derivative:

i i i

body

II I I

d ddt dt

ddt

w w ww w w

I e e ω e e eL e e e

L L ω L

L

�

1 1 1 2 2 2 3 3 3

2 3 3 2 1 3 1 1 3 2 1 2 2 1 3

ˆ ˆ ˆ

ˆ ˆ ˆ

I I I

I I I I I I

w w w

w w w w ww

e e e

e e e

PHY 711 Fall 2014 -- Lecture 25 710/24/2014

Descriptions of rotation about a given origin -- continued

1 1 1 2 2 2 3 3 3

Note that the torque equation

is very difficult to solve directly in the body fixed frame.For 0 we can solve the Euler equations:

ˆ ˆ ˆ0

body

d ddt dt

d I I Idt

w w w

L L ω L τ

τL e e e

2 3 3 2 1 3 1 1 3 2 1 2 2 1 3ˆ ˆ ˆ I I I I I Iw w w w ww e e e

1 1 2 3 3 2

2 2 3 1 1 3

3 3 1 2 2 1

0

0

0

I I I

I I I

I I I

w w w

w w w

w ww

PHY 711 Fall 2014 -- Lecture 25 810/24/2014

0~~~

0~~~0~~~

:frame fixedbody in rotation for equationsEuler

122133

311322

233211

III

III

III

www

www

www

1

133

333

311321

133211

12

~ :Define

(constant)~ 0~0~~~0~~~

: -- topsymmetricfor Solution

IIIΩ

I

III

III

II

w

ww

www

www

12

21

~~

~~

ww

ww

PHY 711 Fall 2014 -- Lecture 25 910/24/2014

Solution of Euler equations for a symmetric top -- continued

)sin()(~ )cos()(~ :Solution

~ where

~~ ~~

2

1

1

133

1221

ww

w

wwww

tAttAt

IIIΩ

233

21

2 ~21

21~

21 ww IAIIT

iii

333211

333222111

ˆ~ˆsinˆcos ˆ~ˆ~ˆ~

eeeeeeL

wwww

IttAIIII

PHY 711 Fall 2014 -- Lecture 25 1010/24/2014

0~~~

0~~~0~~~

:frame fixedbody in rotation for equationsEuler

122133

311322

233211

III

III

III

www

www

www

2

1332

1

23313

123

~ :Define

~ :Define 0~ :Suppose

: -- topasymmetricfor Solution

IIIΩ

IIIΩ

III

w

ww

PHY 711 Fall 2014 -- Lecture 25 1110/24/2014

1 1 2 3 3 2

2 2 3 1 1 3

3 3 1 2 2 1

3 2 3 13 1 3 2 3

1 2

0

0

0

If 0, Define:

I I I

I I I

I I I

I I I IΩ ΩI I

w w w

w w w

w ww

w w w

unstable. issolution signs, opposite have and If

sin)(~

cos)(~:negativeboth or positiveboth are and If

~~ ~~

21

211

22

211

21

122211

w

w

wwww

tAt

tAt

Euler equations for asymmetric top -- continued

PHY 711 Fall 2014 -- Lecture 25 1210/24/2014

Transformation between body-fixed and inertial coordinate systems – Euler angles

http://en.wikipedia.org/wiki/Euler_angles

inertial

body

y

xy

x

PHY 711 Fall 2014 -- Lecture 25 1310/24/2014

03e

y

xy

x '2e

3e

3'2

03 ˆ ˆ ˆ ~ eeeω

Need to express all components in body-fixed frame:

PHY 711 Fall 2014 -- Lecture 25 1410/24/2014

03e

y

xy

x '2e

3e

3'2

03 ˆ ˆ ˆ ~ eeeω

0cossin

010

1000cossin0sincos

ˆ

:tionrepresentaMatrix ˆ cosˆ sinˆ

'2

21'2

e

eee

PHY 711 Fall 2014 -- Lecture 25 1510/24/2014

3'2

03 ˆ ˆ ˆ ~ eeeω

03 1 3 1 2 3

03

ˆ ˆ ˆ ˆ ˆ ˆ sin ' cos ' cos sin sin sin cos Matrix representation:

cos sin 0 cos 0 sin 0ˆ sin cos 0 0 1 0 0

0 0 1 sin 0 cos 1

sin cos sin sin

cos

e e e e e e

e

PHY 711 Fall 2014 -- Lecture 25 1610/24/2014

3'2

03 ˆ ˆ ˆ ~ eeeω

332211 ˆ~ˆ~ˆ~ ~100

0cossin

cossinsincossin

~

eeeω

ω

www

ww

w

cos~cossinsin~

sincossin~100

0cossin

cossinsincossin

~

3

2

1

ω

PHY 711 Fall 2014 -- Lecture 25 1710/24/2014

03e

y

xy

x '2e

3e

3'2

03 ˆ ˆ ˆ ~ eeeω

3

2

1

ˆcos ˆcossinsin

ˆsincossin ~

ee

eω

PHY 711 Fall 2014 -- Lecture 25 1810/24/2014

Rotational kinetic energy

23

22

21

233

222

211

cos21

cossinsin21

sincossin21

~21~

21~

21,,,,,

www

I

I

I

IIIT

23222

1

21

cos21 sin

21,,,,,

: If

IIT

II

PHY 711 Fall 2014 -- Lecture 25 1910/24/2014

Transformation between body-fixed and inertial coordinate systems – Euler angles

http://en.wikipedia.org/wiki/Euler_angles

inertial

body

y

xy

x

PHY 711 Fall 2014 -- Lecture 25 2010/24/2014

General transformation between rotated coordinates – Euler angles

http://en.wikipedia.org/wiki/Euler_angles

inertial

body

y

xy

x

1000cossin0sincos

cos0sin010

sin0cos

1000cossin0sincos

'

RRRRR VVV