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HW13_prob4_GPS24_G14

# HW13_prob4_GPS24_G14 - Goldstein 8-14(3rd ed 8.24 This view...

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Goldstein 8-14 (3 rd ed. 8.24) This view is looking down on the cylinder from above. The z axis is coming up out of the page through the center of the circular cylinder. The x axis is fixed in space. The x ± axis is fixed in the cylinder. The angle α measures the angular position of the particle on the cylinder. The angle θ measures the angular position of the particle in the fixed frame. The angle φ measures the angular displacement of the cylinder in the fixed frame. Initially, the particle was located on the x axis, which then coincided with the x ± axis.

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Goldstein 8-14 (3 rd ed 8.24) Start by writing the kinetic energy as the sum of contributions from the cylinder of mass M and from the particle of mass m , T = T cyl + T m , (1) and express T cyl and T m using the generalized coordinates (angles) measured from a reference axis f xed in space, as illustrated in the drawing on the previous page. We have T cyl = 1 2 I · φ 2 , (2) where I = Ma 2 / 2 is the moment of inertia of the cylinder about its rotation axis, the z axis in this case. The particle’s kinetic energy has two components, T m = m 2 [ a 2 · θ 2 + · z 2 ] . (3) The potential energy of the system is simply V = mgz . (4) The vertical displacement z of the particle is measured from its initial position
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HW13_prob4_GPS24_G14 - Goldstein 8-14(3rd ed 8.24 This view...

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