Let's solve the problem step by step: --- **Given:** - Density of cylinder, \( \rho_{cylinder} = 920 \, \text{kg/m}^3 \) - Density of water, \( \rho_{water} = 1000 \, \text{kg/m}^3 \) - Height of cylinder, \( h = 1.5 \, \text{m} \) - We need to find the **minimum diameter** \( d \) for stable floating equilibrium. --- ### **Step 1: Floating Condition (Buoyancy)** For the cylinder to float: - Weight of cylinder = Weight of water displaced Let \( V \) be the volume of the cylinder: \[ V = \pi \left(\frac{d}{2}\right)^2 h \] Let \( x \) be the submerged height: \[ \text{Weight of cylinder} = \rho_{cylinder} \cdot V \cdot g \] \[ \text{Weight of displaced water} = \rho_{water} \cdot V_{sub} \cdot g \] \[ V_{sub} = \pi \left(\frac{d}{2}\right)^2 x \] Equate: \[ \rho_{cylinder} \cdot \pi \left(\frac{d}{2}\right)^2 h = \rho_{water} \cdot \pi \left(\frac{d}{2}\right)^2 x \] \[ \rho_{cylinder} \cdot h = \rho_{water} \cdot x \] \[ x = \frac{\rho_{cylinder}}{\rho_{water}} h = \frac{920}{1000} \cdot 1.5 = 1.38 \, \text{m} \] --- ### **Step 2: Stability Condition** For stable equilibrium (metacentric height, \( GM > 0 \)): \[ GM = BM - BG > 0 \] Where: - \( BM = \frac{I}{V_{sub}} \) - \( I = \) second moment of area about the centerline \( = \frac{\pi}{64} d^4 \) - \( V_{sub} = \pi \left(\frac{d}{2}\right)^2 x \) - \( BG = \) distance between center of buoyancy and center of gravity \( = \frac{h}{2} - \frac{x}{2} \) So: \[ BM = \frac{\frac{\pi}{64} d^4}{\pi \frac{d^2}{4} x} = \frac{d^2}{16x} \] \[ BG = \frac{h - x}{2} \] \[ GM > 0 \implies \frac{d^2}{16x} > \frac{h - x}{2} \] \[ \frac{d^2}{16x} > \frac{h - x}{2} \] \[ d^2 > 8x(h - x) \] \[ d > \sqrt{8x(h - x)} \] --- ### **Step 3: Substitute Values** Recall: - \( x = 1.38 \, \text{m} \) - \( h = 1.5 \, \text{m} \) \[ d > \sqrt{8 \times 1.38 \times (1.5 - 1.38)} \] \[ d > \sqrt{8 \times 1.38 \times 0.12} \] \[ d > \sqrt{1.3248} \] \[ d > 1.151 \, \text{m} \] --- ### **Final Answer:** The **minimum diameter** should be **1.152 m** (rounded to 3 decimal places). #### **So, the correct answer is:** **1.152 m**