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A wooden plank of mass 1 kg and length 1 m is kept 3m away from a rigid wall. 1m of the surface is rough and rest is smooth as shown in the figure. Coefficient of friction between plank and rough part is $μ = 0.5$ and collision between the plank & wall is perfectly elastic. The minimum velocity for which plank comes to its original position is $v_{0} = 2 \sqrt{x} m / s$ . Find the value of $x .$ $(g = 10 m / s^{2})$

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answer is 5.

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Detailed Solution

Friction force being proportional to normal reaction must very linearly with length of the planck which is on the rough area

For(1): To come to original position again the planck passes two times on the rough area so

$\frac{1}{2} m v_{0}^{2} = 2 \times (\frac{1}{2} \times 5 \times 2) \Rightarrow v_{0} = 2 \sqrt{5} m / s$

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