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HW8_prob4_GPS20_G9

# HW8_prob4_GPS20_G9 - Problem GPS 4.20(2nd ed 4-9 In general...

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Problem GPS 4.20 (2 nd ed. 4-9) In general, we would need six generalized coordinates to describe the motion of the sphere. Three of these would be for the center of mass ( X , Y , Z ) and three for the orientation of a body fi xed frame rotating with the sphere (the three Euler angles). Because the sphere is con fi ned to a plane, one COM coordinate, say Z , is always a constant. So our task comes down to fi nding suitable relationships among ( X , Y , ψ , θ , φ ) to govern the constraint of rolling without slipping. Let’s start by imagining that we are sitting at the origin of a space fi xed Cartesian frame (SFF) watching the sphere rolling in the x y plane located at z = 0 . The COM vector R runs from the origin of the SFF to the center of the sphere. Another vector r runs from the center of the sphere to its surface. This vector is fi xed in the sphere, so it rotates as the sphere rolls. Finally, a third vector r 0 connects the SFF origin with the end of vector r on the sphere’s surface.

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HW8_prob4_GPS20_G9 - Problem GPS 4.20(2nd ed 4-9 In general...

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