$3.$ The draw bridge $\backslash ($ A B $\backslash )$ relies on friction to hold itself in static equilibrium. The counterweight weighs $50$ kg and rests on a slope of $\backslash (30^\{\backslash$circ$\}\backslash )$ with a static friction coefficient of $0\mathrm{.}4.$ A rope attached at C $(\backslash ($ L $/3\backslash )$ away from one end of the bridge$)$ is used to suspend the bridge AB$.$ The bridge is uniform and has a weight force, W $,$ acting in its centre $($i$.$e$.$ at $\backslash (\backslash$mathrm$\{$L$\}/2\backslash )).$

You can assume the pulleys at the top are frictionless and are only used to change the direction of the rope.

a$.$ Draw FBDs of the bridge $\backslash ($ A B $\backslash )$ as well as the counterweight

b$.$ If the counterweight is about to slip, determine the normal and friction force

c$.$ Determine the tension in the rope

d$.$ Calculate the maximum weight of the bridge that can be suspended without the weight slipping

$[15$ marks$]$