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4-2.Friction
medium
The force acting on the block pushes it, then pushing of the block will be possible along the surface if
A$\tan \,\theta \ge \mu $
B$\cot \,\theta \ge \mu $
C$\tan \,\frac{\theta }{2} \ge \mu $
D$\cot \,\frac{\theta }{2} \ge \mu $
Solution
$\mathrm{N}=\mathrm{mg}+\mathrm{F} \cos \theta=\operatorname{mg}(1+\cos \theta)=2 \mathrm{mg} \cos ^{2} \frac{\theta}{2}$
Pushing of the block will be possible if
$F \sin \theta \geq \mu N$
$\Rightarrow \operatorname{mg}\left(2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\right) \geq \mu\left(2 \operatorname{mg} \cos ^{2} \frac{\theta}{2}\right)$
$\Rightarrow \tan \frac{\theta}{2} \geq \mu$
Pushing of the block will be possible if
$F \sin \theta \geq \mu N$
$\Rightarrow \operatorname{mg}\left(2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\right) \geq \mu\left(2 \operatorname{mg} \cos ^{2} \frac{\theta}{2}\right)$
$\Rightarrow \tan \frac{\theta}{2} \geq \mu$
Standard 11
Physics
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