Problem $6$: Slippage $(8$ pts$)$

A mass $\backslash ($ M$\_\{1\}\backslash )$ is supported by two distinct ropes. One rope, of length L $,$ is attached to a massless ring at a point R that can slide on a rod with friction. The other rope is passed through a massless frictionless pulley attached to the rod at point P $,$ and a mass $\backslash ($ M$\_\{2\}\backslash )$ can be hung from its end. The value of $\backslash ($ M$\_\{2\}\backslash )$ is increased until the ring is about to slide on the rod. Assume that the coefficient of static friction between the ring and the rod is $\backslash (\backslash$mu$\_\{$s$\}\backslash ),$ the distance between R and P is D $,$ and denote the angle between the two ropes when the ring starts sliding by $\backslash (\backslash$theta $\backslash ).$ The setup is illustrated below; note that $\backslash ($ m$=0\backslash ).$

$($a$)(2$ pt$)$ Right before the ring starts to slide, what are the tensions in the two ropes connected to $\backslash ($ M$\_\{1\}\backslash )($the one connected to the ring at $\backslash ($ R $\backslash )$ and the one going through the pulley at $\backslash ($ P$,$ T$\_\{$R$\}\backslash )$ and $\backslash ($ T$\_\{$P$\}\backslash )$ respectively$)?$

$($b$)(2$ pt$)$ Right before the ring starts to slide, what is the angle $\backslash (\backslash$theta $\backslash )$ between the two ropes connected to $\backslash ($ M$\_\{1\}\backslash )?$

$($c$)(4$ pt$)$ Find the minimum value of $\backslash ($ M$\_\{2\}\backslash )$ for which the ring starts to slide, as a function of some or all of the the variables provided.