A uniform beam of mass $m$ and length $L$ leans against a rough wall as shown below. The coefficients of static friction between the beam and the floor, and between the beam and the wall, are $\mu_{f}=0.50$ and $\mu_{w}=0.30$ , respectively. Assume the beam is on the verge of slipping and stability is ensured if $\theta \geq \theta_{\min }$, given by

Calculate the true value of $\theta_{\min }$. Then, assuming the wall is frictionless ($\mu_{w}=0$), the expected value of $\theta _{\min }\text{ is }45^{\circ }$. Determine the percentage error between the true value and the frictionless wall approximation.

• A. $\theta _{\min }=43^{\circ }\text{, Percentage error = }4.4\%$
• B. $\theta _{\min }=45^{\circ }\text{, Percentage error}\ =6.7\%$
• C. $\theta _{\min }=49^{\circ }\text{, Percentage error}\ =9.8\%$
• D. $\theta _{\min }=51^{\circ }\text{, Percentage error}\ =14\%$