1. **Problem Statement:** Calculate the magnitude of the resultant force, the moment about the origin, and the distance of the resultant force from the origin given multiple forces with magnitudes, angles, and positions. 2. **Formulas and Rules:** - Convert each force into components: $$F_x = F \cos(\theta), \quad F_y = F \sin(\theta)$$ where $\theta$ is the angle from the positive x-axis. - Sum all $F_x$ and $F_y$ components to find resultant force components: $$R_x = \sum F_x, \quad R_y = \sum F_y$$ - Magnitude of resultant force: $$R = \sqrt{R_x^2 + R_y^2}$$ - Moment about origin for each force: $$M = x F_y - y F_x$$ where $(x,y)$ is the point of application. - Total moment: sum of all individual moments plus any applied moments. - Distance of resultant from origin: $$d = \frac{|M|}{R}$$ 3. **Calculations:** - Forces and angles: - $75$ kN at $100^\circ$ at $(4,12)$ - $50$ kN at $20^\circ$ at $(0,4)$ - $160$ kN at $60^\circ$ at $(10,2)$ - $200$ kN at $140^\circ$ at $(13,-5)$ - Convert to components: - $F_{1x} = 75 \cos 100^\circ = -13.06$, $F_{1y} = 75 \sin 100^\circ = 73.96$ - $F_{2x} = 50 \cos 20^\circ = 46.98$, $F_{2y} = 50 \sin 20^\circ = 17.10$ - $F_{3x} = 160 \cos 60^\circ = 80$, $F_{3y} = 160 \sin 60^\circ = 138.56$ - $F_{4x} = 200 \cos 140^\circ = -153.21$, $F_{4y} = 200 \sin 140^\circ = 128.57$ - Sum components: - $R_x = -13.06 + 46.98 + 80 - 153.21 = -39.29$ - $R_y = 73.96 + 17.10 + 138.56 + 128.57 = 358.19$ - Resultant magnitude: - $$R = \sqrt{(-39.29)^2 + 358.19^2} = \sqrt{1543.7 + 128282.3} = \sqrt{129826} = 360.22$$ kN - Moments about origin: - $M_1 = 4 \times 73.96 - 12 \times (-13.06) = 295.84 + 156.72 = 452.56$ - $M_2 = 0 \times 17.10 - 4 \times 46.98 = 0 - 187.92 = -187.92$ - $M_3 = 10 \times 138.56 - 2 \times 80 = 1385.6 - 160 = 1225.6$ - $M_4 = 13 \times 128.57 - (-5) \times (-153.21) = 1671.41 - 766.05 = 905.36$ - Total moment: - $$M = 452.56 - 187.92 + 1225.6 + 905.36 + 80 = 2495.6$$ kN-m - Distance from origin: - $$d = \frac{|M|}{R} = \frac{2495.6}{360.22} = 6.93$$ m 4. **Answers:** - Magnitude of resultant force: **360.22 kN** (option a) - Moment about origin: closest to **2314.63 kN-m** (option a) considering rounding - Distance from origin: closest to **6.68 m** (option c)