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Question

A block of mass m rests on a 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 angle θ with the vertical side of the block pulls it. In which of the following cases can the block be pulled along the surface?

A
tanθμ
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B
cotθμ
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C
tanθ2μ
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D
cotθ2μ
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Solution

The correct option is D cotθ2μ
It is clear from the figure that vertical component of force will decrease the normal reaction, i.e., N=mgmgcosθ

and frictional force =μN
=μ(mgmgcosθ)
Now block can be pulled when, horizontal component of force frictional force
i.e., mgsinθμ(mgmgcosθ)
or 2sinθ2cosθ2μ(1cosθ)
or 2sinθ2cosθ22μsin2θ2
or cotθ2μ
Hence, the correct answer is option (d).

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