A $10\mathrm{.}0$ kg monkey climbs a uniform ladder with weight $1\mathrm{.}20*$ and length L$=3\mathrm{.}00$m as shown in Figure P$12\mathrm{.}14.$ The ladder rests against the wall and makes an angle of $0=60\mathrm{.}0$ with the ground. The upper and lower ends of the ladder rest on frictionless surfaces. The lower end is connected to the wall by a horizontal rope that is frayed and can support a maximum tension of only $80\mathrm{.}0$ N$.$

$($a$)$

Draw a force diagram for the ladder.

$($b$)$

Find the normal force exerted on the bottom of the ladder.

$($c$)$

Find the tension in the rope when the monkey is two$-$thirds of the way up the ladder.

$($d$)$

Find the maximum distance that the monkey can climb up the ladder before the rope breaks.

$($e$)$

If the horizontal surface were rough and the rope were removed, how would your analysis of the problem change? What other information would you need to answer parts $($c$)$ and $($d$)?$