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 the block can be pulled along the surface

$m g = F$

• A. $\tan \theta \geq \mu$
• B. $\cot \theta \geq \mu$
• C. $\tan \theta / 2 \geq \mu$
• D. $\cot \theta / 2 \geq \mu$

Answer 1

Answer: (D)

$\cot \theta / 2 \geq \mu$

Answer 2

For pulling of block $P \geq f$

⇒ $m g \sin \theta \geq \mu R$ ⇒ $m g \sin \theta \geq \mu ( m g - m g \cos \theta )$

⇒ $\sin \theta \geq \mu ( 1 - \cos \theta )$

⇒ $2 \sin \frac { \theta } { 2 } \cos \frac { \theta } { 2 } \geq \mu \left( 2 \sin ^ { 2 } \frac { \theta } { 2 } \right)$ ⇒ $\cot \left( \frac { \theta } { 2 } \right) \geq \mu$