Collar $A$ is connected as shown to a $50lb$ load and can slide on a frictionless horizontal rod. For a given horizontal force $P=30lb,$

$($a$)$ draw the appropriate free$-$body diagram;

$($b$)$ apply the condition of equilibrium and write the governing equations;

$($c$)$ solve the governing equations and determine the distance $x$ for which the collar is in equilibrium.