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Unformatted text preview: Problem Set 2: Problem 5.
Problem: A lighthouse keeper's son enjoys sliding down the spiral staircase railing from the light to the ground floor. His path is a spiral of constant radius, R. The cylindrical coordinates of his position as measured from the top of the staircase are r = R, = t and z = -wt, where t is time, is his angular rotation rate and w is his vertical speed. Both and w are constant. Determine his velocity and acceleration components. If his speed along the railing is 5 w, what is his angular-rotation rate? 4 Solution: Because the stairway has cylindrical symmetry and motion occurs at constant radius r = R, we compute the velocity-vector components by differentiating as follows. vr v vz The acceleration-vector components are d2 r d -r = -R2 dt2 dt d2 dr d =0 a = r 2 + 2 dt dt dt d2 z az = =0 dt2 Therefore, the boy's velocity and acceleration vectors are ar = Finally, his speed, v, is So, if v = 5 w, we have 4 v = R e - w k v= and a = -R2 er R2 2 + w2 r2 2 = 9 2 w 16
2 dr =0 dt d = R = r dt dz = -w = dt = 2 2 2 vr + v + vz = 25 2 w = r2 2 + w2 16 Therefore, his angular-rotation rate is = = 3w 4R ...
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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.
- Spring '06