HW11_prob1_GPS6_G19

# HW11_prob1_GPS6_G19 - Problem Goldstein 5-19(3rd ed 5.6(a...

This preview shows pages 1–2. Sign up to view the full content.

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 xed in space, we are free to locate

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern