1. **Problem Statement:** We are analyzing a hydraulic system where forces, areas, pressures, and distances relate through Pascal's law and the principle of moments. 2. **Key Formulas:** - Pressure: $$P = \frac{F}{A}$$ where $F$ is force and $A$ is area. - Area of a circle: $$A = \pi r^2$$ where $r$ is radius. - Moment balance: $$F_1 \cdot d_1 = F_2 \cdot d_2$$ where $d$ are distances from pivot. 3. **Calculate Areas:** - For radius $r_1 = 1$ m: $$A_1 = \pi (1)^2 = 3.14\, m^2$$ - For radius $r_2 = 10$ m: $$A_2 = \pi (10)^2 = 314.16\, m^2$$ 4. **Calculate Pressures:** - Left side: $$P_1 = \frac{F_1}{A_1} = \frac{50}{3.14} = 15.92\, Pa$$ - Right side: $$P_2 = \frac{F_2}{A_2} = \frac{5000}{314.16} = 15.92\, Pa$$ 5. **Verify Pascal's Law:** Pressures are equal ($P_1 = P_2$), confirming pressure transmits equally in the fluid. 6. **Moment Balance:** Given distances $d_1 = 5.68$ m and $d_2 = 0.06$ m, Calculate moments: $$F_1 \cdot d_1 = 50 \times 5.68 = 284\, Nm$$ $$F_2 \cdot d_2 = 5000 \times 0.06 = 300\, Nm$$ The moments are approximately equal, showing mechanical equilibrium. 7. **Summary:** - Pressure equality confirms Pascal's principle. - Moment balance shows forces and distances relate to maintain equilibrium. This explains how hydraulic systems amplify force and balance moments using fluid pressure and lever arms.