Let's solve the problem **step by step**: --- ## **Step 1: Understanding the Problem** You are asked to find the **equivalent force-couple system** at point **C** for the forces acting on the beam. ### **Given:** - Two downward forces: - 700 lb at 13 ft from C - 900 lb at 23 ft (13 + 10) from C - All forces act vertically downward. --- ## **Step 2: Resultant Force at C** Sum all vertical forces to get the **resultant force**: \[ F_{R} = 700\, \text{lb} + 900\, \text{lb} = 160\, \text{lb} \] - The resultant force acts **downward** at point C. --- ## **Step 3: Calculate the Resultant Moment about Point C** The **moment** (couple) about point C is the sum of the moments due to each force about C. ### **Moment from 700 lb force:** \[ M_{C,700} = 700\, \text{lb} \times 13\, \text{ft} = 910\, \text{lb}\cdot\text{ft} \] ### **Moment from 900 lb force:** \[ M_{C,900} = 900\, \text{lb} \times 23\, \text{ft} = 20700\, \text{lb}\cdot\text{ft} \] - Both moments cause **clockwise rotation**, so they are **negative** (if counterclockwise is positive). \[ M_C = -910\, \text{lb}\cdot\text{ft} - 20700\, \text{lb}\cdot\text{ft} = -29800\, \text{lb}\cdot\text{ft} \] --- ## **Step 4: Write the Final Equivalent Force-Couple System at C** \[ \boxed{ \begin{aligned} &\text{Equivalent Force at C:} \\ &\qquad \vec{F}_R = 160\, \text{lb} \quad \text{(downward)} \\[2ex] &\text{Equivalent Couple at C:} \\ &\qquad \vec{M}_C = -29800\, \text{lb}\cdot\text{ft} \quad \text{(clockwise)} \end{aligned} } \] --- ### **Summary Table** | Quantity | Value | Direction | |-----------------|------------------------|--------------| | Force at C | 160 lb | Downward | | Couple at C | 29,800 lb·ft | Clockwise | --- ## **Final Answer** > **The equivalent force-couple system at C is:** > > - A **160 lb downward force** > - A **29,800 lb·ft clockwise couple** (moment) Let me know if you'd like a free body diagram or further explanation!