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. (a) more than half-filled if `rho_c` is less than 0.5
B. (b) more than half-filled if `rho_c` is more than 1.0.
C. (c) half-filled if `rho_c` is more than 0.5.
D. (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 cylindrical shell floats in water and how its relative density affects the amount of water it contains. Here's a step-by-step solution:

### Step 1: Understand the System
We have a thin uniform cylindrical shell that is closed at both ends and partially filled with water. The shell is floating vertically in water, and it is half-submerged. The relative density of the shell material with respect to water is denoted as \( \rho_c \).

### Step 2: Define the Variables
- Let \( H \) be the total height of the cylindrical shell.
- The shell is half-submerged, so the submerged height is \( \frac{H}{2} \).
- Let \( H' \) be the height of water inside the shell.
- The cross-sectional area of the shell is \( A \).

### Step 3: Calculate the Volume Displaced
The volume of water displaced by the submerged part of the shell is equal to the volume of the shell that is submerged:
\[ V_{\text{displaced}} = A \cdot \frac{H}{2} \]

### Step 4: Calculate the Weight of the Shell
The weight of the shell can be expressed in terms of its relative density:
\[ W_{\text{shell}} = \rho_c \cdot V \cdot g \]
where \( V \) is the total volume of the shell.

### Step 5: Calculate the Weight of the Water Inside the Shell
The weight of the water inside the shell is given by:
\[ W_{\text{water}} = \rho_w \cdot A \cdot H' \cdot g \]
where \( \rho_w \) is the density of water.

### Step 6: Apply the Principle of Buoyancy
According to Archimedes' principle, the buoyant force acting on the shell is equal to the weight of the water displaced:
\[ F_b = W_{\text{displaced}} = \rho_w \cdot V_{\text{displaced}} \cdot g = \rho_w \cdot A \cdot \frac{H}{2} \cdot g \]

### Step 7: Set Up the Equilibrium Condition
For the shell to float in equilibrium, the total weight of the shell and the water inside must equal the buoyant force:
\[ W_{\text{shell}} + W_{\text{water}} = F_b \]
Substituting the expressions we derived:
\[ \rho_c \cdot V \cdot g + \rho_w \cdot A \cdot H' \cdot g = \rho_w \cdot A \cdot \frac{H}{2} \cdot g \]

### Step 8: Simplify the Equation
Cancel \( g \) from both sides:
\[ \rho_c \cdot V + \rho_w \cdot A \cdot H' = \rho_w \cdot A \cdot \frac{H}{2} \]

### Step 9: Analyze the Relative Density
From the equilibrium condition, we can derive the relationship between \( H' \) and \( \rho_c \):
- If \( \rho_c < 0.5 \), then \( H' > \frac{H}{2} \) (more than half filled).
- If \( \rho_c > 0.5 \), then \( H' < \frac{H}{2} \) (less than half filled).
- If \( \rho_c = 0.5 \), then \( H' = \frac{H}{2} \) (exactly half filled).

### Conclusion
The correct statement regarding the shell is:
- The shell is more than half filled if \( \rho_c < 0.5 \).
- The shell is less than half filled if \( \rho_c > 0.5 \).
- The shell is half filled if \( \rho_c = 0.5 \).

Thus, the correct answer is that the shell is more than half filled if \( \rho_c < 0.5 \).