A long slipway!

There is long slipway, inclined at an angle \(\alpha\) with the ground, has many identical rollers, fitted at a a distance \(d\) between them. The rollers have horizontal axles and consist of rubber-covered solid steel cylinders each of mass \(m\) and radius \(r\). A plank of mass M, is released at the top of the slipway.

Find terminal velocity \(\large v_{max}\) of the plank. Submit your answer to 3 decimal digits.

Details and Assumptions

  • The rollers are covered with rubber so that there is friction between the plank and the block
  • The rollers start pure rolling (no slipping) just as they lose contact with the plank.
  • Initially, there is slipping between the plank and the rollers
  • Take the length of the plank to be much greater than \(d\), i.e. \(d\ll L\)
  • Take \(M = 10kg,~m = 500g,~ d = 10cm,~ r = 3cm, ~\alpha = \frac{\pi}{3}\).

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