p09fl22 - Lecture 22: Rigid body rotations (23 Oct 09)...

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Unformatted text preview: Lecture 22: Rigid body rotations (23 Oct 09) Kepler ellipse averages: Angle average h r i = a √ 1- 2 . t-averages : h r i ≥ 1 / h (1 /r ) i = a A. Review/complete–moment of inertia 1. Start with case of no net translation, but with a rotation relative to an inertial frame. For the position r , the relation between inertial and body frame time derivatives is v = d r dt | inertial = d r dt | body + ~ω × r 2. Then if the vector r is fixed in the body frame coordinates, the total (rotational) kinetic energy is T = 1 2 X p m p v 2 p = 1 2 X p m p ( ~ω × r p ) · ( ~ω × r p ) 3. Vector cross product identities: a · ( b × c ) = ( a × b ) · c ; a × ( b × c ) = ( a · c ) b- ( a · b ) c 4. Rotational kinetic energy term: ( ~ω × r ) · ( ~ω × r ) = [( ~ω × r ) × ~ω ] · r =- [ ~ω × ( ~ω × r )] · r = ω 2 r 2- ( ~ω · r ) 2 = ω 2 r 2 ⊥ 5. The rotational kinetic energy and the moment of inertia tensor are: T = 1 2 ~ω · I · ~ω = 1 2 X i,j I ij ω i ω j ; I ij...
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This note was uploaded on 12/14/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at Wisconsin.

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p09fl22 - Lecture 22: Rigid body rotations (23 Oct 09)...

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