First slide

Static friction

Question

A block of mass M rests on a rough horizontal surface as shown. 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, the block can be pulled along the surface?

Moderate

A

$\tan θ \geq μ$
B

$\tan (\frac{θ}{2} \geq μ)$
C

$\cos θ \geq μ$
D

$\cot (\frac{θ}{2}) \geq μ$

Solution

$N = M g - F \cos θ = M g - M g \cos θ = 2 M g \sin^{2} \frac{θ}{2}$
Further, block can be pulled, if F $sinθ \geq μ N$
$\Rightarrow 2 M g \sin \frac{θ}{2} \cdot \cos \frac{θ}{2} \geq 2 μ M g \sin^{2} \frac{θ}{2}$
$\Rightarrow \cot (\frac{θ}{2}) \geq μ$

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