Two non-mixing liquids of densities `rho` and `nrho` `(n gt1)` are put in a container. The height of each liquid is `h`. A solid cylinder of length `L` and density `d` is put in this container. The cylinder floats with its axis vertical and length `pL(p lt 1)` in the denser liquid. The density `d` is equal to :

Options

A. `{2+(n+1)p}rho`
B. `{2+(n-1)p}rho`
C. `{1+(n-1)p}rho`
D. `{1+(n+1)p}rho`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the setup
We have two non-mixing liquids with densities \( \rho \) (upper liquid) and \( n\rho \) (lower liquid, where \( n > 1 \)). A solid cylinder of length \( L \) and density \( d \) is placed in this container. The cylinder floats vertically with a length \( pL \) (where \( p < 1 \)) submerged in the denser liquid.

### Step 2: Write the expression for the weight of the cylinder
The weight \( W \) of the cylinder can be expressed as:
\[ W = d \cdot V \cdot g \]
where \( V \) is the volume of the cylinder, which can be expressed as \( A \cdot L \) (where \( A \) is the cross-sectional area of the cylinder). Thus, we have:
\[ W = d \cdot A \cdot L \cdot g \]

### Step 3: Write the expression for the buoyant force
The buoyant force \( F_B \) acting on the cylinder is equal to the weight of the liquid displaced by the submerged part of the cylinder. The total buoyant force can be expressed as the sum of the buoyant forces from both liquids.

1. **Buoyant force from the upper liquid** (height \( L - pL = (1 - p)L \)):
\[ F_{B1} = \rho \cdot A \cdot (1 - p)L \cdot g \]

2. **Buoyant force from the lower liquid** (height \( pL \)):
\[ F_{B2} = n\rho \cdot A \cdot pL \cdot g \]

Thus, the total buoyant force \( F_B \) is:
\[ F_B = F_{B1} + F_{B2} = \rho \cdot A \cdot (1 - p)L \cdot g + n\rho \cdot A \cdot pL \cdot g \]

### Step 4: Set the weight equal to the buoyant force
Since the cylinder is floating, the weight of the cylinder is equal to the total buoyant force:
\[ d \cdot A \cdot L \cdot g = \rho \cdot A \cdot (1 - p)L \cdot g + n\rho \cdot A \cdot pL \cdot g \]

### Step 5: Cancel common terms
We can cancel \( A \), \( L \), and \( g \) from both sides:
\[ d = \rho(1 - p) + n\rho p \]

### Step 6: Simplify the equation
Now, we can simplify the equation:
\[ d = \rho(1 - p + np) \]
\[ d = \rho(1 + p(n - 1)) \]

### Step 7: Final expression for density \( d \)
Thus, the density \( d \) of the cylinder can be expressed as:
\[ d = \rho(1 + p(n - 1)) \]

### Conclusion
The density \( d \) of the cylinder is given by:
\[ d = \rho(1 + p(n - 1)) \]