Let's break down the solution step by step: ## 1. **Given Data** - Length of ladder, \( L = 4\, \text{m} \) - Angle with ground, \( \theta = 74^\circ \) - Height of wall, \( h = 3.5\, \text{m} \) - Mass of ladder, \( m_1 = 18\, \text{kg} \) - Mass of worker, \( m_2 = 65\, \text{kg} \) - Coefficient of static friction, \( \mu_s = 0.29 \) - Gravity, \( g = 9.8\, \text{m/s}^2 \) --- ## 2. **Forces on the Ladder** - **Weight of ladder** at center: \( W_1 = m_1 g \) - **Weight of worker** at top: \( W_2 = m_2 g \) - **Normal force at ground**: \( N \) - **Friction force at ground**: \( f \) - **Normal force at wall (horizontal)**: \( F_w \) --- ## 3. **Equilibrium Conditions** ### (a) **Vertical Forces** \[ N = W_1 + W_2 \] ### (b) **Horizontal Forces** \[ f = F_w \] ### (c) **Torque about the ground point** Taking moments about the base (ground contact): - Perpendicular distance from base to ladder top: \( L \sin\theta \) - Perpendicular distance from base to ladder center: \( \frac{L}{2}\sin\theta \) - Wall force acts at height \( h \) (or \( L \sin\theta \)) #### **Moment equation:** \[ F_w \cdot h = W_1 \cdot \left(\frac{L}{2}\cos\theta\right) + W_2 \cdot (L \cos\theta) \] --- ## 4. **Plug in the Numbers** - \( W_1 = 18 \times 9.8 = 176.4\, \text{N} \) - \( W_2 = 65 \times 9.8 = 637\, \text{N} \) - \( N = 176.4 + 637 = 813.4\, \text{N} \) - \( \cos(74^\circ) \approx 0.275 \) - \( \sin(74^\circ) \approx 0.961 \) #### **Moment equation:** \[ F_w \cdot 3.5 = 176.4 \cdot \left(\frac{4}{2} \cdot 0.275\right) + 637 \cdot (4 \cdot 0.275) \] \[ F_w \cdot 3.5 = 176.4 \cdot 0.55 + 637 \cdot 1.1 \] \[ F_w \cdot 3.5 = 97.02 + 700.7 = 797.72 \] \[ F_w = \frac{797.72}{3.5} = 227.92\, \text{N} \] So, **required friction force** \( f = F_w = 227.92\, \text{N} \) --- ## 5. **Maximum Static Friction** \[ f_{\text{max}} = \mu_s N = 0.29 \times 813.4 = 235.89\, \text{N} \] --- ## 6. **Final Answers** ### - **Friction force needed:** \( \boxed{228\, \text{N}} \) (rounded to 3 sig. figs.) ### - **Maximum friction supported:** \( \boxed{236\, \text{N}} \) (rounded to 3 sig. figs.) --- **Let me know if you need a free-body diagram or further breakdown!**