Terminal Trouble

A long slipway is inclined at an angle \(\alpha\) to the horizontal and is fitted with a series of identical rollers of mass \(m\) and radius \(r,\) a distance \(d\) apart along the slipway. A plank of length \(L\gg d\) and mass \(M = 20m\) is released from the top of the slipway, and eventually reaches some terminal velocity \(v_\infty.\)

Find \(v_\infty\) and report your answer to the nearest integer.

Assumptions

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