1. **Problem Statement:** Find the magnitude and position of the resultant force for the forces 40 kN, 60 kN, 30 kN, and 20 kN acting on a beam with given positions 1, 1, 2, 3, 5. 2. **Formula and Rules:** - The resultant force $R$ is the algebraic sum of all forces: $$R = \sum F_i$$ - The position $\bar{x}$ of the resultant force is found by the moment equilibrium: $$\bar{x} = \frac{\sum (F_i \times x_i)}{\sum F_i}$$ - Forces acting downward are taken as negative, upward as positive. 3. **Identify Forces and Positions:** - Forces: $F_1 = -40$ kN (downward), $F_2 = 60$ kN (downward), $F_3 = 30$ kN (upward), $F_4 = 20$ kN (upward) - Positions (from left): $x_1 = 1$, $x_2 = 1+1=2$, $x_3 = 2+2=4$, $x_4 = 4+3=7$ (assuming cumulative distances) 4. **Calculate Resultant Force:** $$R = -40 - 60 + 30 + 20 = (-100) + 50 = -50 \text{ kN}$$ 5. **Calculate Moment about origin:** $$M = (-40)(1) + (-60)(2) + 30(4) + 20(7) = -40 -120 +120 +140 = 100 \text{ kN}\cdot\text{m}$$ 6. **Calculate Position of Resultant:** $$\bar{x} = \frac{M}{R} = \frac{100}{-50} = -2 \text{ m}$$ 7. **Interpretation:** - The negative sign in $R$ indicates the resultant force acts downward. - The negative position means the resultant force is located 2 m to the left of the origin (reference point). **Final Answer:** $$\boxed{\text{Resultant force magnitude } = 50 \text{ kN downward at } 2 \text{ m left of origin}}$$