1 Handout #18 Inertia tensor Inertia tensor for a continuous body Kinetic energy from inertia tensor. Tops and Free bodies using Euler equations Precession.

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Presentation transcript:

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

3 Lamina Theorem :60

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)

6 L 18-2 Angular Momentum and Kinetic Energy :02 1)A complex arbitrary system is subject to multi-axis rotation. 2)The inertia tensor is 3)A 3-axis rotation is applied

7 Symmetrical top :02 Euler equation

8 Precession :02 Ignore in limit

9 Euler’s equations for symmetrical bodies :60 Note even for non-laminar symmetrical tops AND even for

10 Euler’s equations for symmetrical bodies :60 Precession frequency=rotation frequency for symmetrical lamina

11 Euler’s equations for symmetrical bodies :60

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

13 Handout #18 windup :02