A block of mass m rests on rough horizontal surface as shown in the figure. Coefficient of friction between the block and the surface is μ. A force F = mg acting at an angle θ with the vertical side of the block pulls it. the block can be pulled along the surface when

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$\tan \frac{θ}{2} \geq μ$

$\tan θ \geq μ$

$\cot \frac{θ}{2} \geq μ$

$\cot θ \geq μ$

answer is D.

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Detailed Solution

Block can be pulled when the horizontal component of the force is greater than the force of friction. The vertical component will decrease the weight as well as the normal reaction.

$\begin{array}{l} ∴ M g \sin θ \geq μ (M g - M g \cos θ) \\ 2 \sin \frac{θ}{2} \cos \frac{θ}{2} \geq μ \times \sin^{2} \frac{θ}{2} \\ i.e. \cot \frac{θ}{2} \geq μ \end{array}$