% subject: physics
% type: mcq_mc
% has_diagram: false

\item A block of mass $m$ is placed on a rough inclined plane of angle $\theta$ with the horizontal. The coefficient of static friction is $\mu_s$. Which of the following statements is/are correct?

\begin{tasks}(1)
    \task The normal force on the block is $mg \cos\theta$ \ans
    \task The block will slide if $\tan\theta > \mu_s$ \ans
    \task The friction force is always $\mu_s mg \cos\theta$
    \task The net force along the incline is $mg \sin\theta - \mu_s mg \cos\theta$ when the block is about to slide \ans
\end{tasks}

\begin{solution}
\begin{align*}
\intertext{Resolving forces perpendicular to the incline}
N &= mg \cos\theta
\intertext{This confirms option (a). Along the incline, the block slides when}
mg \sin\theta &> f_s = \mu_s N \\
mg \sin\theta &> \mu_s mg \cos\theta \\
\tan\theta &> \mu_s
\intertext{This confirms option (b). Option (c) is incorrect because friction equals $\mu_s mg \cos\theta$ only at the point of sliding, not always. At the point of sliding, the net force along the incline is}
F_{\text{net}} &= mg \sin\theta - \mu_s mg \cos\theta
\end{align*}
\intertext{This confirms option (d). Therefore, the correct options are (a), (b), and (d).}
\end{solution}
