A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If `rho_c` is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

Options

A. More than half-filled if ` rho _c ` is less than 0.5
B. More than half –filled if ` rho _c` is more than 1.0
C. Half-filled if ` rho_0` is more than 0.5
D. Less than half-filled if ` rho_c ` is less than 0.5

Explanation

To solve the problem, we need to analyze the conditions under which the thin uniform cylindrical shell, which is partially filled with water and floating vertically, is in equilibrium. We will derive the conditions based on the relative density of the shell material (`rho_c`) with respect to water.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a cylindrical shell that is partially filled with water and floating vertically in water.
- The shell is half-submerged, which means the height of the submerged part of the shell is \( \frac{H}{2} \), where \( H \) is the total height of the shell.

2. **Volume and Buoyant Force**:
- Let \( V \) be the volume of the shell (material) and the volume of water displaced is equal to the volume of the submerged part of the shell.
- Since the shell is half-submerged, the volume of water displaced is \( \frac{V}{2} \).
- The buoyant force \( F_b \) acting on the shell is given by:
\[ F_b = \rho_w \cdot g \cdot \text{Volume of water displaced} = \rho_w \cdot g \cdot \frac{V}{2} \]
- Here, \( \rho_w \) is the density of water and \( g \) is the acceleration due to gravity.

3. **Weight of the Shell**:
- The weight of the shell \( W_s \) can be expressed as:
\[ W_s = \rho_c \cdot V \cdot g \]
- Additionally, the weight of the water inside the shell (if filled to height \( h' \)) is:
\[ W_w = \rho_w \cdot A \cdot h' \cdot g \]
- Here, \( A \) is the cross-sectional area of the cylinder and \( h' \) is the height of the water inside the shell.

4. **Equilibrium Condition**:
- At equilibrium, the buoyant force equals the total weight of the shell plus the weight of the water inside it:
\[ \rho_w \cdot g \cdot \frac{V}{2} = \rho_c \cdot V \cdot g + \rho_w \cdot A \cdot h' \cdot g \]
- Canceling \( g \) from both sides gives:
\[ \rho_w \cdot \frac{V}{2} = \rho_c \cdot V + \rho_w \cdot A \cdot h' \]

5. **Solving for Height of Water**:
- Rearranging the equation:
\[ \frac{\rho_w}{2} = \rho_c + \frac{\rho_w \cdot A \cdot h'}{V} \]
- From this, we can express \( h' \) in terms of \( \rho_c \):
\[ h' = \frac{V}{A} \left( \frac{\rho_w}{2} - \rho_c \right) \]

6. **Analyzing Relative Density**:
- The relative density \( \rho_c \) determines how much of the shell is filled with water:
- If \( \rho_c < 0.5 \), then \( h' > \frac{H}{2} \) (more than half filled).
- If \( \rho_c > 1 \), then \( h' < \frac{H}{2} \) (less than half filled).
- If \( 0.5 < \rho_c < 1 \), then \( h' = \frac{H}{2} \) (half filled).

### Conclusion:
Based on the analysis, the correct statement regarding the state of the shell is:
- The shell is **more than half filled if \( \rho_c < 0.5 \)**.