# A long slipway!

Classical Mechanics Level 5

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.

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.
• At first, 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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