1

Class #28 of 31

Inertia tensor Kinetic energy from inertia

tensor. Eigenvalues w/ MATLAB

Tops and Free bodies using Euler equations Precession Lamina theorem Free-body wobble

:02

2

Rest of course

:60

  11/28 THANKSGIVING    

28 12/3 Tops, Gyros and Rotations #14 –Supplement / Taylor

-

29 12/5 Ch. 13 Hamiltonian and Quantum Mechanics or

Chaos

   

30 12/10 Test #4 Inertia Tensor / Tops / Euler equations /Chaos

12/10

31 12/12 Review for Final    

FINAL EXAM

12/16 MONDAY – 9AM-12NOON WORKMAN 310  

3

Angular Momentum and Kinetic Energy

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We derived the moment of inertia tensor from the fundamental definitions of L, by working out the double cross-product

Do the same for T (kinetic energy)

1 1( )

2 21

( ) ( )21

2

T mv v r mv

T r mv vector identity

T L

( )

( )

p mv m r

L r m r

1

2T

L I

T I

4

L 28-2 Angular Momentum and Kinetic Energy

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1) A square plate of side L and mass M is rotated about a diagonal.

2) In the coordinate system with the origin at lower left corner of the square, the inertia tensor is.

20

31 0

4 13

10 0 13 4 2

00 0 2

MLI

.Calculate L and T1

2T

L I

T I

5

Symmetrical top

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3 3 3 1 2 1 2

1 2 3

3 33

( )

0

0 con

a d

s

n

t

Euler equation

6

Precession

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ˆsinCM

dLr F

dt

dLR mg

dt

Ignore in limit

2 22 2 2 23 3z zL L L L L

3p sin

L

L

3 3

sin1

sin sinCM CM

p

R mgd dL

dt L dt L

R mg

7

Lamina Theorem

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zzIyyI

xxI

2 2 2 2

2 2

( ) ( )

( )

( 0)

xx yy

zz

zz xx yy

I I y z x z dxdydz

I x y dxdydz

I I I for laminar objects z

yyIzzI

xxI

( 0)yy xx zzI I I for laminar objects y

8

Euler’s equations for symmetrical bodies

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1 1 2 3 2 3

2 2 3 1 3 1

3 3 1 2 1 2

( )

( )

( )

zzIyyI

xxI

2

2

1

41

22

xx yy

zz xx yy

For Disk I I MR

I I I MR

1 2 3

2 3 1

3 1 2

( 2 )

(2 )

2 ( )

1 2 3

2 3 1

3 0

Note even for non-laminar symmetrical tops AND even for

3 0 1 2, 0

9

3 zL

p

Euler’s equations for symmetrical bodies

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1 2 3

2 3 1

3 0

21 3 1

22 3 2

2 23p

Precession frequency=rotation frequency for symmetrical lamina

10

Euler’s equations for symmetrical bodies

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3 z

L

p

3 1 3 1

3 zL

p

11

L28-1 – Chandler Wobble

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1) The earth is an ovoid thinner at the poles than the equator.

2) For a general ovoid,

3) For Earth, what are

2 21( )

5xxI M a b

, ?yy zz pI and I and

2b

2a

yyIzzI

xxI

1 1 1 3 2 3

1 2 3 1 3 1

3 3

223 1

1 3 121

( )

( )

0

( )

2 2 2a b a b b 6400

20

a km

km

12

L 28-2 Angular Momentum and Kinetic Energy

:02

1) A square plate of side L and mass M is rotated about a diagonal.

2) In the coordinate system with the origin at lower left corner of the square, the inertia tensor is.

20

31 0

4 13

10 0 13 4 2

00 0 2

MLI

.Calculate L and T1

2T

L I

T I

13

Lecture 28 windup

:02

1

2T

L I

T I

3 3

CMp

R mgfor top

minzz xx yyI I I for la a

2 21( )

5xxI M a b for ellipsoid

14

Angular Momentum and Kinetic Energy

:02

L 1) A complex arbitrary system is

subject to multi-axis rotation.

2) The inertia tensor is

3) A 3-axis rotation is

applied

15 6 1

6 10 5

1 5 20

A

5.0

8.2

3.0

.Calculate L and T1

2T

L I

T I