Floating Conditions for Thin Uniform Cylindrical Shell Partially Filled with Water

To determine the condition under which the cylindrical shell floats in a half-submerged state, we analyze the balance between the buoyant force and the total weight of the system.

1. Definitions and Parameters

• Let $V$ be the internal volume of the closed cylindrical shell.
• Let $V_m$ be the volume of the material composing the shell wall and end-caps.
• The total external volume of the shell is $V_{\text{total}} = V + V_m$.
• Let $V_w$ be the volume of water filled inside the shell.
• Let $\rho_w$ be the density of water.
• The relative density of the shell material with respect to water is $\rho_c$, so the density of the shell material is $\rho_m = \rho_c \rho_w$.

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2. Buoyant Force

The shell floats in a half-submerged state, meaning the volume of displaced water is equal to half of the total external volume of the shell: $V_{\text{submerged}} = \frac{1}{2} V_{\text{total}} = \frac{1}{2} (V + V_m)$

The buoyant force $F_B$ exerted by the surrounding water is given by Archimedes' principle: $F_B = V_{\text{submerged}} \rho_w g = \frac{1}{2} (V + V_m) \rho_w g$

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3. Total Weight of the System

The total mass of the floating system consists of two parts:

1. The mass of the shell material: $M_{\text{shell}} = V_m \rho_m = V_m \rho_c \rho_w$
2. The mass of the water inside the shell: $M_w = V_w \rho_w$

Thus, the total mass is: $M_{\text{total}} = M_{\text{shell}} + M_w = (V_m \rho_c + V_w) \rho_w$

The total downward gravitational force (weight) $W$ is: $W = M_{\text{total}} g = (V_w + \rho_c V_m) \rho_w g$

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4. Equilibrium Condition

For the shell to float in equilibrium, the upward buoyant force must equal the downward total weight: $F_B = W$

$\frac{1}{2} (V + V_m) \rho_w g = (V_w + \rho_c V_m) \rho_w g$

Dividing both sides by $\rho_w g$: $\frac{1}{2} V + \frac{1}{2} V_m = V_w + \rho_c V_m$

Rearranging the terms to solve for the internal water volume $V_w$: $V_w = \frac{1}{2} V + \left(\frac{1}{2} - \rho_c\right) V_m$

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5. Conclusion and Analysis

• Since $V_m > 0$ for any physical shell:
  • If $\rho_c < 0.5$, then $\left(\frac{1}{2} - \rho_c\right) > 0$, which implies $V_w > \frac{1}{2} V$. This means the shell is more than half-filled with water.
  • If $\rho_c > 0.5$, then $\left(\frac{1}{2} - \rho_c\right) < 0$, which implies $V_w < \frac{1}{2} V$. This means the shell is less than half-filled with water.
  • If $\rho_c = 0.5$, then $V_w = \frac{1}{2} V$, meaning the shell is exactly half-filled.

Hence, the correct assertion is that the shell is more than half-filled if $\rho_c$ is less than $0.5$.

Correct Option: A