Question

A horizontal force $F$ is applied to a small object $P$ of mass $m$ on a smooth plane inclined to the horizontal at 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. $mg\tan \theta$

Answer 1

Answer: (D)

$mg\tan \theta$

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