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A plank of mass M is placed on a rough horizontal surface and a constant horizontal force F is applied on it. A man of mass m runs on the plank. Find the range of acceleration of the man so that the plank does not move on the surface. Coefficient friction between the plank and the surface is mu. Assume that the man does not slip on the plank

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$\frac{F}{m} - \frac{μ (M + m) g}{m} \leq a \leq \frac{F}{m} + \frac{μ (M) g}{m}$

$\frac{F}{m} - \frac{μ (m) g}{m} \leq a \leq \frac{F}{m} + \frac{μ (M + m) g}{m}$

$\frac{F}{m} - \frac{μ (M + m) g}{m} \leq a \leq \frac{F}{m} + \frac{μ (M + m) g}{m}$

$\frac{F}{m} - \frac{μ (M) g}{m} \leq a \leq \frac{F}{m} + \frac{μ (M) g}{m}$

answer is D.

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Detailed Solution

Here $f_{1} =$ force of friction between man and plank and $f_{2} =$ force of friction between plank and surface.

For the plank not to move

$F - {(f_{2})}_{\max} \leq f_{1} \leq F + {(f_{2})}_{\max}$

or $F - μ (M + m) g \leq m a \leq F + μ (M + m) g$

or a should lic between $\frac{F}{m} - \frac{μ (M + m) g}{m}$

and $\frac{F}{m} + \frac{μ (M + m) g}{m}$

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