1

Handout #18

Inertia tensor Inertia tensor for a continuous

body Kinetic energy from inertia tensor.

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

:02

2

Inertia Tensor For continuous body

2 2

2 2

2 2

( ) ( ) ( )

( ) ( ) .

... ( ) ( )

y z dxdydz xy dxdydz xz dxdydz

I yx dxdydz x z dxdydz etc

zy dxdydz x y dxdydz

3

Lamina Theorem

:60

zzIyyI

xxI

2 2 2 2

2 2

( ) ( )

( )

0

( 0)

xx yy

zz

zz xx yy xz yz

I I y z x z dxdydz

I x y dxdydz

I I I AND I I

for laminar objects z

yyIzzI

xxI

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

4

L 18-1 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?

3) What are the eigenvalues and eigenvectors for this square plate?

L

5

Angular Momentum and Kinetic Energy

:02

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

6

L 18-2 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

6.6 2.6 0.44

-2.6 4.4 2.2

0.44 2.2 8.8

I

5.0

8.2

3.0

.

?

.

Calculate L

What is special about

Calculate T

1

2T

L I

T I

7

Symmetrical top

:02

3 3 3 1 2 1 2

1 2 3

3 33

( )

0

0 con

a d

s

n

t

Euler equation

8

Precession

:02

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

9

Euler’s equations for symmetrical bodies

:60

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

10

3 zL

p

Euler’s equations for symmetrical bodies

:60

1 2 3

2 3 1

3 0

21 3 1

22 3 2

2 23p

Precession frequency=rotation frequency for symmetrical lamina

11

Euler’s equations for symmetrical bodies

:60

3 z

L

p

3 1 3 1

3 zL

p

12

L18-3 – Chandler Wobble

:60

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

13

Handout #18 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