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HW11_prob5_GPS10_G23 - Problem Goldstein 5-23(3rd ed 5.10...

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Problem Goldstein 5-23 (3 rd ed 5.10) We will 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 . In order to use Euler’s equations of motion, 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 Fig. 5.7 that N always lies along the line of nodes (the intermediate x axis used in de fi 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 , we can write by inspection N 1 = N · i = N cos ψ , (2) N 2 = N · j = N sin ψ , (3) and N 3 = 0 , (4) where N , the magnitude of the torque, is easily obtained from Eq.(1) as N = Mgl sin θ . (5) The Euler equations of motion can be written as N i = I i · ω i ( I j
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