Let's solve the problem **step by step**. We need to find the **magnitude of the resultant moment about point O** caused by all applied loads. --- ## **Step 1: List Given Data** - Couple moment at O: \( M = 104 \) kN·m (counterclockwise) - Distributed load: triangular, max intensity \( w = 5.7 \) kN/m, length = 12.0 m (7.5 + 4.5) - Point force \( F = 10.7 \) kN at the right end (12.0 m from O) --- ## **Step 2: Break Down the Loads** ### **1. Couple Moment at O** - Moment about O = **104 kN·m** (counterclockwise, by sign convention: positive) --- ### **2. Distributed Load (Triangular)** - **Total length = 12.0 m** - The load is triangular, starting from 0 at O to \( w = 5.7 \) kN/m at 12.0 m. #### **Resultant Force** (Area of triangle): \[ F_{\text{tri}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12.0 \times 5.7 = 34.2 \text{ kN} \] #### **Location of Resultant Force** - For a right triangle starting from O, the centroid is \( \frac{1}{3} \) from the high end (right), or \( \frac{2}{3} \) from O: \[ x_{F_{\text{tri}}} = \frac{2}{3} \times 12.0 = 8.0 \text{ m from O} \] #### **Moment about O (from this load)** \[ M_{\text{tri}} = F_{\text{tri}} \times x_{F_{\text{tri}}} = 34.2 \times 8.0 = 273.6 \text{ kN·m (clockwise)} \] - Clockwise moment is **negative** by convention: \[ M_{\text{tri}} = -273.6 \text{ kN·m} \] --- ### **3. Concentrated Force \( F \)** - Located at 12.0 m from O - Magnitude: \( F = 10.7 \) kN \[ M_F = F \times \text{distance from O} = 10.7 \times 12.0 = 128.4 \text{ kN·m (clockwise)} \] - Clockwise moment is **negative** by convention: \[ M_F = -128.4 \text{ kN·m} \] --- ## **Step 3: Sum Moments about O** \[ M_{\text{resultant}} = M + M_{\text{tri}} + M_F \] \[ M_{\text{resultant}} = 104 + (-273.6) + (-128.4) \] \[ M_{\text{resultant}} = 104 - 273.6 - 128.4 \] \[ M_{\text{resultant}} = 104 - 402.0 = -298.0 \text{ kN·m} \] --- ## **Step 4: Final Answer (Magnitude)** \[ \boxed{298.0 \text{ kN·m}} \] (The negative sign means the net moment is clockwise, but since the question asks for the **magnitude**, we report the positive value.) --- ## **Summary Table** | Load Type | Magnitude (kN·m) | Direction | |-------------------|------------------|-------------------| | Couple at O | +104.0 | Counterclockwise | | Triangular Load | -273.6 | Clockwise | | Point Force | -128.4 | Clockwise | | **Total** | **-298.0** | Clockwise | --- ### **Final Answer:** **The magnitude of the resultant moment about point O is** \[ \boxed{298.0\ \text{kN·m}} \]