A plank of mass $M$ and length $L$ is hinged at its mid point from a fixed support in vertical plane as shown. Plank is free to rotate in vertical plane about the hinge. Two persons of equal mass running on the plank with same speed v relative to the plank so that angular velocity $ω$ of the plank remains constant.Initially one person is at the centre of the plank and the other person is at a distance d from the first person . Assuming they run till running is possible then $v$ is

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a constant

independent of the separation between the persons.

independent of the distance of the persons from the hinge.

independent of their mass

answer is B, C, D.

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Detailed Solution

$\begin{array}{rcl} m g [x \cos ω t + (x + d) \cos ω t] & = & ω \frac{d I}{d t} \\ \Rightarrow & m g \cos ω t (2 x + d) = ω \frac{d}{d t} [\frac{M L^{2}}{12} + m x^{2} + m {(x + d)}^{2}] \\ = & m ω [2 x v + 2 (x + d) v] \\ = & 2 m ω (2 x + d) v \\ \Rightarrow & v = \frac{g \cos ω t}{2 ω} \end{array}$