A solid cylinder of length L is in equilibrium in two different liquid A and B as shown in the figure.The density of liquid A `(3ρ)/2` and liquid B is 3ρ.The density of cylinder is

Options

A. 7/5ρ
B. 13/5ρ
C. 5/2ρ
D. 2ρ

Answer

To solve the problem of finding the density of the solid cylinder in equilibrium in two different liquids, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have a solid cylinder of length \( L \) that is in equilibrium in two different liquids, A and B. The density of liquid A is \( \frac{3\rho}{2} \) and the density of liquid B is \( 3\rho \). We need to find the density of the cylinder, denoted as \( \rho_C \).

2. **Weight of the Cylinder**: The weight \( W \) of the cylinder can be calculated using the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the cylinder and \( g \) is the acceleration due to gravity. The mass \( m \) can be expressed in terms of density and volume:
\[ m = \rho_C \cdot V \]
Therefore, the weight can be rewritten as:
\[ W = \rho_C \cdot V \cdot g \]

3. **Buoyant Force**: The buoyant force \( B \) acting on the cylinder is the sum of the buoyant forces from both liquids A and B. The buoyant force can be calculated using the formula:
\[ B = \text{(Density of liquid A)} \cdot \text{(Volume submerged in liquid A)} \cdot g + \text{(Density of liquid B)} \cdot \text{(Volume submerged in liquid B)} \cdot g \]

4. **Volume Submerged**: Let's denote the volume of the cylinder as \( V \). According to the problem, the cylinder is submerged in liquid A for \( \frac{2}{3} \) of its volume and in liquid B for \( \frac{1}{3} \) of its volume:
- Volume submerged in liquid A: \( \frac{2V}{3} \)
- Volume submerged in liquid B: \( \frac{V}{3} \)

5. **Calculating Buoyant Force**:
- For liquid A:
\[ B_A = \left(\frac{3\rho}{2}\right) \cdot \left(\frac{2V}{3}\right) \cdot g = \frac{3\rho}{2} \cdot \frac{2Vg}{3} = \rho V g \]
- For liquid B:
\[ B_B = 3\rho \cdot \left(\frac{V}{3}\right) \cdot g = \rho V g \]
- Total buoyant force \( B \):
\[ B = B_A + B_B = \rho V g + \rho V g = 2\rho V g \]

6. **Setting Weight Equal to Buoyant Force**: For the cylinder to be in equilibrium, the weight must equal the buoyant force:
\[ \rho_C \cdot V \cdot g = 2\rho \cdot V \cdot g \]

7. **Cancelling Common Terms**: We can cancel \( V \) and \( g \) from both sides of the equation:
\[ \rho_C = 2\rho \]

### Final Answer:
The density of the cylinder \( \rho_C \) is:
\[ \rho_C = 2\rho \]