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# p09fl26 - Lecture 26 Top with gravity(2 Nov 09 A Review...

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Lecture 26: Top with gravity (2 Nov 09) A. Review: Euler angles 1. Three successive rotations to get to the principal axes of the moment of inertia tensor: (1) by angle α around the original ˆ z = ˆ e 0 3 , (2) rotation by angle β around the new “y” = “2” axis, (3) rotate by angle γ around the new (and final) ˆ z axis ˆ e 3 . 2. The final result after matrix multiplication for the successive rotations C ( γ ) B ( β ) A ( α ) is CBA = C γ C β C α - S γ S α C γ C β S α + S β C α - C γ S β - S γ C β S α - C γ S α - S γ C β S α + C γ C α S γ S β S β C α S β S α C β using a notation C α = cos α , S α = sin α , etc. 3. Express rotation axes in terms of the principal axes (a) Rotation for γ (= symmetry axis of the symmetric top) is ˆ e γ ˆ e 3 . (b) Rotation for α ˆ e α = ˆ e 0 3 and can express that in terms of the final axes by the transform: (so ˆ e 0 3 · ˆ e 3 = C β ) ˆ e α = CBA · 0 0 1 = - C γ S β ˆ e 1 + S γ S β ˆ e 2 + C β ˆ e 3 (c) ˆ e β : ˆ e β = sin γ ˆ e 1

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