HW13_prob5_GPS25_G15

HW13_prob5_GPS25_G15 - Goldstein 8-15 (3rd ed. 8.25) This...

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Goldstein 8-15 (3 rd ed. 8.25) This problem di f ers from 8-14 because the cylinder is now driven to rotate at a constant angular speed ω , and its kinetic energy is no longer variable. Since the cylinder’s angular motion is prescribed, we only need to consider the motion of the particle of mass m . Its kinetic energy is T = m 2 [ a 2 · θ 2 + · z 2 ] , (1) where θ is measured from a reference axis f xed in space. The potential energy of the system is simply V = mgz . (2) The vertical displacement z of the particle is measured from its initial position at the top of the cylinder, z =0 . Since the helical track that the particle moves on is rigidly fastened to the cylinder it is convenient to express z in terms of the particle’s angular displacement α on the cylinder, z = b α , (3) where b is the pitch of the helix. It should be clear that α is related to the angular displacements in the f xed reference frame by the simple equation, α = θ ω t, (4) where ω > 0 if the cylinder is rotating counterclockwise as viewed from above. Similar conventions apply to
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HW13_prob5_GPS25_G15 - Goldstein 8-15 (3rd ed. 8.25) This...

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