HW11_prob2_Space-Body Cone Geometry

# HW11_prob2_Space-Body Cone Geometry - Problem Space and...

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Problem: Space and Body Cones for the torque free top (a) One way to do this is as follows. Designate the body frame unit vectors as i , j ,and k , and assume they lie along the principal axes of the body. In the body principal axis frame we can write the angular momentum vector L as L = I 1 ω 1 i + I 2 ω 2 j + I 3 ω 3 k , (1) so that L · k = L 3 = I 3 ω 3 = L cos θ, (2) and | L × k | = L sin θ = | L 2 i L 1 j | = I 1 q ω 2 1 + ω 2 2 = I 1 A, (3) where θ is the Euler angle between the space and body z axes and the compo- nents of the angular velocity vector are ω 1 = A cos t and ω 2 = A sin t .T h e s e two results give us tan θ = I 1 A/ ( I 3 ω 3 ) . (4) If we repeat the same operations with the vectors ω and k ,we f nd ω · k = ω cos θ 00 = ω 3 , (5) and | ω × k | = ω sin θ 00 = | ω 2 i ω 1 j | = q ω 2 1 + ω 2 2 = A, (6) from which we have tan θ 00 = A/ω 3 . (7) From the ratio of Eqs.(4) and (7), we f nd the desried result tan θ

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HW11_prob2_Space-Body Cone Geometry - Problem Space and...

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