Let's break down each part of the problem step by step. --- ## a) **Compare the densities of the liquids** Given: - The three liquids are **not mixing** and are in **equilibrium** in a U-tube. - Heights and density notations: - Left arm: \( h_1, d_1 \) (blue liquid) - Right arm, top: \( h_3, d_3 \) (yellow liquid) - Right arm, bottom: \( h_2, d_2 \) (green liquid) **At equilibrium in a U-tube, the pressures at the same horizontal level (at the bottom of the tube) are equal.** Let's denote the bottom points: - Left side bottom: Point **A** - Right side bottom: Point **B** - Middle bottom: Point **C** For equilibrium: \[ \text{Pressure at A} = \text{Pressure at B} \] Pressure due to the column of liquid: \[ P = h \cdot d \cdot g \] where \( h \) is height, \( d \) is density, and \( g \) is gravitational acceleration. - At **A**: Only the blue liquid column is above. \[ P_A = h_1 \cdot d_1 \cdot g \] - At **B**: Both yellow (on top) and green (below) are above. \[ P_B = h_3 \cdot d_3 \cdot g + h_2 \cdot d_2 \cdot g \] Set these equal: \[ h_1 d_1 g = h_3 d_3 g + h_2 d_2 g \] \[ h_1 d_1 = h_3 d_3 + h_2 d_2 \] **Without numerical values, we cannot make a precise ranking, but:** - If the heights of the green and yellow are less than the blue, then their densities must be larger, or vice versa. --- ## b) **Compare the liquid pressures at points A and B on the bottom of the U-tube** As derived above, **at equilibrium**: \[ P_A = P_B \] So, **the pressures at A and B are equal**. --- ## c) **Compare the liquid pressure forces at point C** The force due to liquid pressure at point C: \[ F_C = P_C \cdot S \] where \( S \) is the area at point C. Since A and B are at the same level as C and pressures at A and B are the same, **the pressure at C is also the same:** \[ P_C = P_A = P_B \] Therefore, \[ F_C = P_C \cdot S = P_A \cdot S = P_B \cdot S \] So, **the liquid pressure forces at points A, B, and C are equal (assuming equal area at all points)**. --- # **Summary Table** | | Densities (comparison) | Pressures (A vs. B) | Forces (at C) | |------|------------------------|---------------------|-----------------------| | **a** | \(h_1 d_1 = h_3 d_3 + h_2 d_2\) | \(P_A = P_B\) | Same as at A and B | | **b** | | **Equal** | | | **c** | | | **Equal** | --- ### **Final Answers** **a)** Densities are related by: \( h_1 d_1 = h_3 d_3 + h_2 d_2 \). **b)** The liquid pressures at points A and B are **equal**. **c)** The liquid pressure force at point C is **equal** to the forces at A and B (assuming same area).