In Fig. 12-41, a climber with a weight of 533.8 N is held by a belay rope connected to her climbing harness and belay device; the force of the rope on her has a line of action through her center of mass. The indicated angles are $\mathit{\theta }\mathbf{=}\mathbf{}\mathbf{40}\mathbf{.0}\mathbf{°}$and$\mathbf{}\mathit{\varphi }\mathbf{=}\mathbf{30}\mathbf{.0}\mathbf{°}$. If her feet are on the verge of sliding on the vertical wall, what is the coefficient of static friction between her climbing shoes and the wall?

Weight of the climber,$W=533.8\text{\hspace{0.17em}N}$

$\theta ={40.0}^0$

$\varphi ={30.0}^0$

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.

$\begin{array}{c}\sum \tau =0\\ \sum {\text{F}}_{\text{x}}=0\\ \sum {\text{F}}_{\text{y}}=0\\ {\text{F}}_{\text{s}}=\mu {\text{F}}_{\text{n}}\end{array}$

Free body diagram:

Horizontal force:

$F_N=T\text{\hspace{0.17em}sin\hspace{0.17em}}\varphi$

Vertical force:

$\begin{array}{c}{F}_s+T\text{\hspace{0.17em}cos\hspace{0.17em}}\varphi =W\\ F_s=W-T\text{\hspace{0.17em}cos\hspace{0.17em}}\varphi \end{array}$

Torque:

$\begin{array}{c}WL\text{\hspace{0.17em}sin\hspace{0.17em}}\theta -TL\text{\hspace{0.17em}sin}(190-\theta -\varphi )\\ T=\frac{W\text{\hspace{0.17em}sin\hspace{0.17em}}\theta }{\sin (180-\theta -\varphi )}\\ =\frac{533.8\text{\hspace{0.17em}sin\hspace{0.17em}}40}{\sin (180-30-40)}\\ =365.14\text{\hspace{0.17em}N}\end{array}$

So,

$\begin{array}{c}{F}_N=365.14\text{\hspace{0.17em}sin\hspace{0.17em}}30\\ =182.6\text{\hspace{0.17em}N}\\ F_s=533.8-365.14\text{\hspace{0.17em}cos\hspace{0.17em}}30\\ =217.6\text{\hspace{0.17em}N}\end{array}$

Hence,

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

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