1. Handout #18 Inertia tensor Tops and Free bodies using Euler equations
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
I is symettric and also I is hermitian (Equal to the conjugate of the transpose).

3. Lamina Theorem
:60

4. L 18-1 Angular Momentum and Kinetic Energy
A square plate of side L and mass M is rotated about a diagonal.
In the coordinate system with the origin at lower left corner of the square, the inertia tensor is?
What are the eigenvalues and eigenvectors for this square plate? L
Matrix
| |
ML^2 / 12 *| |
| |
T=(ML^2)*(w-0^2)/12
L=ML^2w-0/12sqrt(2) * (1 1 0)
Eigenvectors w=w/root 2 (1 1 0), (1 –1 0), w(0 0 1)
I= (ML^2/12)[4 –3 0 ]
:02

5. Angular Momentum and Kinetic Energy
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)
:02

6. L 18-2 Angular Momentum and Kinetic Energy
A complex arbitrary system is subject to multi-axis rotation.
The inertia tensor is
A 3-axis rotation is
applied
Eigenvalue is 2 for vector
Energy is 203
:02

7. Symmetrical top
Euler equation
:02

8. Precession Ignore in limit :02
Three principal axes. Do demo. Show how things go other direction if spin disk backwards.
Gyro 0) Balance it
1) Precession --- with w reversed
2) Remove weight – rap the gyro
3) Now rap the gyro with a weight on it.
:02

9. Euler’s equations for symmetrical bodies
Note even for non-laminar symmetrical tops AND even for
W-3 is general. Constant may not be 1 for non-laminar
:60

10. Euler’s equations for symmetrical bodies
3-axis balancing!
Plates in air
Precession frequency=rotation frequency for symmetrical lamina
:60

11. Euler’s equations for symmetrical bodies
Free body – L fixed (space frame)
Body frame (w-3 fixed, L and w precess around it).
:60

12. L18-3 – Chandler Wobble
The earth is an ovoid thinner at the poles than the equator.
For a general ovoid,
For Earth, what are
W1-dotdot=-(w-3)^2)(2/5M-eps-b)^2 / (2/5M-b)^2=(eps/b)^2*(w-3)^2-> w-p=20/6400 * w-3
:60

13. Handout #18 windup
:02