Handout 18 Inertia tensor n n Inertia tensor

Handout #18 Inertia tensor n n Inertia tensor for a continuous body Kinetic energy from inertia tensor. Tops and Free bodies using Euler equations n n n Precession Lamina theorem Free-body wobble 1 : 02

Inertia Tensor For continuous body 2

Lamina Theorem 3 : 60

L 18 -1 Angular Momentum and Kinetic Energy 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? 4) L 4 : 02

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) 1) 5 : 02

L 18 -2 Angular Momentum and Kinetic Energy 1) A complex arbitrary system is subject to multi-axis rotation. 2) The inertia tensor is 3) A 3 -axis rotation is 4) applied 5) 6 : 02

Symmetrical top Euler equation 7 : 02

Precession Ignore in limit 8 : 02

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

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

Euler’s equations for symmetrical bodies 11 : 60

L 18 -3 – Chandler Wobble 1) The earth is an ovoid thinner at the poles than the equator. 2) For a general ovoid, 3) For Earth, what are 12 : 60

Handout #18 windup 13 : 02