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Physics 321

Physics 321. Hour 29 Principal Axes. Center of mass. cm. Center of Mass. A useful result: +. More Conclusions. Angular Momentum and Angular Velocity where is the component of perpendicular to . A Little Math. =.

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Physics 321

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  1. Physics 321 Hour 29 Principal Axes

  2. Center of mass cm Center of Mass • A useful result: • +

  3. More Conclusions

  4. Angular Momentum and Angular Velocity • where is the component of perpendicular to A Little Math =

  5. In the instantaneous rotation of a solid body about an axis, is always perpendicular to so . A Little Math

  6. Angular Momentum and Angular Velocity • Conclusions: • is perpendicular to , but not necessarily parallel to . • Therefore is not always valid. • If is perpendicular to : A Little Math

  7. y Find vectors about the origin: x An Example

  8. y Find vectors about the origin: x Another Example

  9. Cross Products and the Antisymmetric Tensor

  10. Angular Momentum of a Point Mass II

  11. The Inertia Tensor of a Point Mass I

  12. The Inertia Tensor of a Point Mass II

  13. The Inertia Tensor of an Extended Object

  14. Lamina a Example a

  15. What if ? Furthermore, if Diagonalizing the Inertia Tensor

  16. Find the eigenvalues: • For each λ, find the eigenvectors: • The three eigenvectors define the “principle” axes. Diagonalizing the Inertia Tensor

  17. principal axes.nb Examples

  18. Physics 321 Hour 30 Euler’s Equations

  19. Space and Body Coordinates • Body coordinates are on principal axes • If possible, use c.m. as origin in both frames • If not possible, c.m. motion is easy

  20. Start with in body coordinates Relating Coordinates

  21. Transform to space coordinates Relating Coordinates

  22. Relating Coordinates Real external torque In space coordinates “Perceived” body torque

  23. Relating Coordinates Real external torque In body coordinates “Perceived” body torque

  24. Euler’s Equations It’s usually very hard to find the actual torques in terms of the rotating body coordinates!

  25. Euler’s Equations – No Torques We can do these!

  26. Rotating a book - eulerseqs.nb Example

  27. Letting , then The Method of Ellipsoids

  28. These are two ellipsoids with equations Note that the ellipsoids are in ω-space. They don’t predict motion. Possible values of ωare given by the intersection of the ellipsoids. The Method of Ellipsoids

  29. ellipsoids.nb Example

  30. is a constant Euler’s Equations – No Torques, I11=I22

  31. Euler’s Equations – No Torques, I11=I22

  32. Space and Body Cones

  33. Prolate object: Ωb<0, Ωs>0 In the body axes:

  34. football.nb Example

  35. Euler’s Angles

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