Terminal Trouble

A long slipway is inclined at an angle α\alpha to the horizontal and is fitted with a series of identical rollers of mass mm and radius r,r, a distance dd apart along the slipway. A plank of length LdL\gg d and mass M=20mM = 20m is released from the top of the slipway, and eventually reaches some terminal velocity v.v_\infty. Find vv_\infty and report your answer to the nearest integer.

Assumptions

  • When the plank is at terminal velocity, the rollers obtain ω=v/r\omega_\infty = v_\infty/r just before losing contact with the plank.
  • sinα=12,\sin \alpha = \frac12, r=0.2 m,r = \SI{0.2}{\meter}, d=1 m,d=\SI{1}{\meter}, g=9.81 m/s2.g=\SI[per-mode=symbol]{9.81}{\meter\per\second\squared}.
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