Dynamics_Part75

Dynamics_Part75 - Problem Set 9 Problem 8 Problem A sphere of mass m and radius c is attached to a cord AB of length as shown The sphere

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Unformatted text preview: Problem Set 9: Problem 8. Problem: A sphere of mass m and radius c is attached to a cord AB of length as shown. The sphere experiences steady precession at a constant rate . The cord makes an angle with the vertical direction, while the sphere spins at a constant rate about its diameter BC. Do your computations using a sphere-fixed coordinate system with z pointing from G to B and x pointing to the left at an angle to the horizontal. (a) Compute the angular momentum vector about the sphere's center of mass, HG , and verify that HG = 2 mc2 sin j, where j is a unit vector pointing out of the page. 5 (b) Determine the acceleration of the sphere's center of mass, a, the magnitude of the tension, T , in cord AB and verify that the moment about the sphere's center of mass is MG = cT sin( - ) j. (c) Determine the angle as a function of , , , c and g, the acceleration of gravity. Solution: (a) We select xyz to be a rectangular Cartesian coordinate system centered at the sphere's center of mass as shown. The y coordinate is out of the page. In terms of standard gyroscope notation, the absolute and precession angular-velocity vectors are = K + k and = K where K is a unit vector in the vertical direction. In terms of the unit vectors in the xyz coordinate frame, K = -i sin + k cos Substituting this expression for K in the equation for the absolute angular-velocity vector, we obtain = (-i sin + k cos ) + k = - sin i + ( + cos ) k The diagonal moments of inertia for a sphere of radius c are Ix = Iy = Iz = 2 mc2 in its center-of-mass 5 based principal axis system. Thus, the angular-momentum vector relative to the center of mass is 2 2 0 0 - sin - 2 mc2 sin 5 mc 5 2 2 0 0 0 = HG = [I] = 0 5 mc 2 2 2 2 0 0 + cos 5 mc 5 mc ( + cos ) ...
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This note was uploaded on 05/12/2010 for the course AME 301 taught by Professor Shiflett during the Spring '06 term at USC.

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