91Ó°ÊÓ

Weight of the climber,$W=533.8\text{ N}$

$\mathrm{θ}={40.0}^0$

$\mathrm{Ï•}={30.0}^0$

To find coefficient of friction, first find the tension in the wire using equilibrium conditions. Then, find the friction force and normal force. Once you know friction force and normal force, you can find the coefficient of friction force using the basic definition of friction force. The equations used are given below.

$\begin{array}{c}∑\mathrm{τ}=0\\ ∑{\text{F}}_{\text{x}}=0\\ ∑{\text{F}}_{\text{y}}=0\\ {\text{F}}_{\text{s}}=\mathrm{μ}{\text{F}}_{\text{n}}\end{array}$

Free body diagram:

Horizontal force:

$F_N=T\text{ s¾±²Ô }\mathrm{Ï•}$

Vertical force:

$\begin{array}{c}{F}_s+T\text{ c´Ç²õ }\mathrm{Ï•}=W\\ F_s=W−T\text{ c´Ç²õ }\mathrm{Ï•}\end{array}$

Torque:

$\begin{array}{c}WL\text{ s¾±²Ô }\mathrm{θ}−TL\text{ s¾±²Ô}(190−\mathrm{θ}−\mathrm{Ï•})\\ T=\frac{W\text{ s¾±²Ô }\mathrm{θ}}{\sin (180−\mathrm{θ}−\mathrm{Ï•})}\\ =\frac{533.8\text{ s¾±²Ô }40}{\sin (180−30−40)}\\ =365.14\text{ N}\end{array}$

So,

$\begin{array}{c}{F}_N=365.14\text{ s¾±²Ô }30\\ =182.6\text{ N}\\ F_s=533.8−365.14\text{ c´Ç²õ }30\\ =217.6\text{ N}\end{array}$

Hence,

$\begin{array}{c}\mathrm{μ}=\frac{F_s}{F_N}\\ =\frac{217.6}{182.6}\\ =1.19\end{array}$

Hence, the coefficient of static friction between the wall and the shoes is 1.19.