Solutions_Part37 - Problem Goldstein 5-19(3rd ed 5.6(a We...

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Problem Goldstein 5-19 (3 rd ed 5.6) (a) We will designate the body frame unit vectors as i , j , and k , and assume they lie along the principal axes of the body. Now we want to show that the angular momentum vector L rotates about the body symmetry axis k with the same angular frequency that the angular velocity vector ω rotates with about k . First we need to establish that the projection of L on k is a constant. In the body principal axis frame we can write 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 = const. , (2) because the principal moments I i are constant in the body frame and the so- lution of Euler’s equations for the torque-free symmetric top ( I 1 = I 2 ) showed that ω 3 is constant. This means that the time dependence of L resides entirely in its i and j components, L 0 = I 1 ω 1 i + I 2 ω 2 j = I 1 ( ω 1 i + ω 2 j ) . (3) After substituting the solution of the Euler equations for the components of the angular velocity vector ( ω 1 = A cos t , ω 2 = A sin t ) we fi nd L 0 = I 1 A (cos t i + sin t j ) , (4) from which we see that L 0 and, hence, L rotate in the i - j plane with angular frequency . Next we wish to show that in the space frame the body axis k rotates about L with the angular frequency · φ . Since L is fi

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