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

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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 + × 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 ( × r p ) · ( × 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: ( × r ) · ( × r ) = [( × r ) × ] · r = - [ × ( × r )] · r = ω 2 r 2 - ( · r ) 2 = ω 2 r 2 5. The rotational kinetic energy and the moment of inertia tensor are: T = 1 2 · I · = 1 2 X i,j I ij ω i ω j ; I ij X p m p ( r 2 p δ i,j - x pi x pj ) 6. The angular momentum is ~ L = I · [and T = 1 2 · ~ L ], using ~ L i = X p [ r p × ( m p × r p )] i = X p m p [ i
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