HW11_prob4 - Heavy Top Angular Momentum Problem (a)Long Way...

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Heavy Top Angular Momentum Problem (a)Long Way Designate the body frame unit vectors as i , j ,and k , and assume they lie along the principal axes of the body. The space frame unit vectors will be designated as x , y ,and z . Since it is easiest to write L in the body frame as L = I 1 ( ω 1 i + ω 2 j )+ I 3 ω 3 k , we need to evaluate the components of the gravitational torque N in the body frame. The torque can be written as N = Mgl k × z , (1) and it’s not di cult to express z in terms of the body frame unit vectors, so that one can easily work out the cross product. Equivalently, we see from Eq.(1) and Figure 5.7 that N always lies along the line of nodes (the intermediate x axis used in de f ning the Euler angles). Since the line of nodes always lies in the body i - j plane and makes the (Euler) angle ψ with i ,wecanwr itebyinspect ion N 1 = N · i = N cos ψ, (2) N 2 = N · j = N sin ψ, (3) and N 3 =0 , (4) where
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HW11_prob4 - Heavy Top Angular Momentum Problem (a)Long Way...

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