question_answer

A horizontal force F is applied to a small  object P of mass m on a smooth plane inclined to the horizontal at an angle \[\theta .\]If F is just enough to keep P in equilibrium, then F is equal to

A)  \[mg{{\cos }^{2}}\theta \]                          

B)         \[mg{{\sin }^{2}}\theta \]                           

C)         \[mg\cos \theta \]                         

D)         \[mg\tan \theta \]

Correct Answer: D

Solution :

By applying Lamis theorem at P, we have \[\frac{R}{\sin {{90}^{o}}}=\frac{F}{\sin ({{180}^{o}}-\theta )}=\frac{mg}{\sin ({{90}^{o}}+\theta )}\] \[\Rightarrow \]\[\frac{R}{1}=\frac{F}{\sin \theta }=\frac{mg}{\cos \theta }\] \[\Rightarrow \]\[F=mg\tan \theta \]