10/22/2014phy 711 fall 2014 -- lecture 241 phy 711 classical mechanics and mathematical methods...

PHY 711 Fall 2014 -- Lecture 24 110/22/2014

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 24:Rotational motion (Chapter 5)

1. Rigid body motion2. Moment of inertia tensor

PHY 711 Fall 2014 -- Lecture 24 210/22/2014

PHY 711 Fall 2014 -- Lecture 24 310/22/2014

PHY 711 Fall 2014 -- Lecture 24 410/22/2014

PHY 711 Fall 2014 -- Lecture 24 510/22/2014

The physics of rigid body motion; body fixed frame vs inertial frame:

PHY 711 Fall 2014 -- Lecture 24 610/22/2014

Comparison of analysis in “inertial frame” versus “non-inertial frame”

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eVeV

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i

iii

i

ii

i

i

inertial

ii

iii

i

i

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ˆ

ˆ :Define

ˆˆˆ

ˆˆ

system coordinate moving a ˆby Denote

system coordinate fixed a ˆby Denote

3

1

3

1

3

1

3

1

03

1

0

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0

0

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V

V

V

PHY 711 Fall 2014 -- Lecture 24 710/22/2014

Properties of the frame motion (rotation):

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PHY 711 Fall 2014 -- Lecture 24 810/22/2014

VωVV

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i

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dtd ˆ3

1

Effects on acceleration:

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bodybodyinertial

bodybodyinertial

22

2

2

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PHY 711 Fall 2014 -- Lecture 24 910/22/2014

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bodyinertial dtd

dtd

:body rigid ofenergy Kinetic

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PHY 711 Fall 2014 -- Lecture 24 1010/22/2014

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notation) (dyad

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:notationMatrix

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PHY 711 Fall 2014 -- Lecture 24 1110/22/2014

x

z

ya

c

b

: tensorinertia ofMoment

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Example:

PHY 711 Fall 2014 -- Lecture 24 1210/22/2014

Properties of moment of inertia tensor: Symmetric matrix real eigenvalues I1,I2,I3

orthogonal eigenvectors3,2,1 ˆˆ iI iii eeI

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: tensorinertia ofMoment

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PHY 711 Fall 2014 -- Lecture 24 1310/22/2014

Changing origin of rotation ''''

2

2

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ppjpipijpij

rrrmI

rrrmI

:mass ofcenter theDefine

'

M

m

m

mp

pp

pp

ppp

CM

pp

rrr

Rrr

2' 2CMjijCMiijCMjiijijij rRRrMRRRMII Rr

x

z

ya

c

b

z’

y’

x’

Rrp r’p

PHY 711 Fall 2014 -- Lecture 24 1410/22/2014

2' 2CMjijCMiijCMjiijijij rRRrMRRRMII Rr

RrzyxR

cba

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and ˆˆˆ that Suppose

' 2jiijijij RRRMII

'

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PHY 711 Fall 2014 -- Lecture 24 1510/22/2014

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z

ya

c

b

z’

y’

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PHY 711 Fall 2014 -- Lecture 24 1610/22/2014

Descriptions of rotation about a given origin

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ˆ~ˆ~ˆ~3,2,1 ˆˆ

: tensorinertia ofmoment esdiagonaliz that system coordinate fixed)(body For

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system coordinate generalFor

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PHY 711 Fall 2014 -- Lecture 24 1710/22/2014

Descriptions of rotation about a given origin -- continued

3122123113

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: tensorinertia ofmoment esdiagonaliz that system coordinate fixed)(body For

momentumangular of change of rate Time

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PHY 711 Fall 2014 -- Lecture 24 1810/22/2014

Descriptions of rotation about a given origin -- continued

3122123113

12332333222111

333222111

332211

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: tensorinertia ofmoment esdiagonaliz that system coordinate fixed)(body For

momentumangular of change of rate Time

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PHY 711 Fall 2014 -- Lecture 24 1910/22/2014

Descriptions of rotation about a given origin -- continued

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:equationsEuler thesolvecan we0For frame. fixedbody in thedirectly solve todifficult very is

equation torque that theNote

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PHY 711 Fall 2014 -- Lecture 24 2010/22/2014

0~~~

0~~~0~~~

:frame fixedbody in rotation for equationsEuler

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www

1

133

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~ :Define

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

: -- topsymmetricfor Solution

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I

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12

21

~~

~~

ww

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PHY 711 Fall 2014 -- Lecture 24 2110/22/2014

Solution of Euler equations for a symmetric top -- continued

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

~ where

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2

1

1

133

1221

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w

wwww

tAttAt

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2 ~21

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PHY 711 Fall 2014 -- Lecture 24 2210/22/2014

0~~~

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:frame fixedbody in rotation for equationsEuler

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2

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~ :Define

~ :Define 0~ :Suppose

0~~~0~~~0~~~

: -- topasymmetricfor Solution

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