Two non-mixing liquids of densities `rho` and `(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. `{1+(n+1)p}rho`
B. `{2+(n+1)p}rho`
C. `{2+(n-1)p}rho`
D. `{1+(n-1)p}rho`

Explanation

To solve the problem step by step, we need to analyze the situation involving the two non-mixing liquids and the floating cylinder.

### Step 1: Understand the setup
We have two liquids:
- The upper liquid has a density of \( \rho \).
- The lower liquid has a density of \( n\rho \) (where \( n > 1 \)).
- The height of each liquid is \( h \).
- A solid cylinder of length \( L \) and density \( d \) is placed in this container, floating vertically.

### Step 2: Determine the length of the cylinder submerged
The problem states that the cylinder floats with its axis vertical and that a length \( pL \) (where \( p < 1 \)) of the cylinder is submerged in the denser liquid (the one with density \( n\rho \)). Therefore, the length of the cylinder submerged in the upper liquid (density \( \rho \)) is:
\[ L_{upper} = L - pL = (1 - p)L \]

### Step 3: Calculate the buoyant force from each liquid
The buoyant force is equal to the weight of the liquid displaced by the submerged part of the cylinder.

1. **Buoyant force from the upper liquid** (density \( \rho \)):
- Volume displaced in the upper liquid:
\[ V_{upper} = A \cdot (1 - p)L \]
- Buoyant force from the upper liquid:
\[ U_1 = V_{upper} \cdot \rho \cdot g = A(1 - p)L \cdot \rho \cdot g \]

2. **Buoyant force from the lower liquid** (density \( n\rho \)):
- Volume displaced in the lower liquid:
\[ V_{lower} = A \cdot pL \]
- Buoyant force from the lower liquid:
\[ U_2 = V_{lower} \cdot (n\rho) \cdot g = A(pL) \cdot (n\rho) \cdot g \]

### Step 4: Set up the equilibrium condition
For the cylinder to float, the total buoyant force must equal the weight of the cylinder. Thus, we have:
\[ U_1 + U_2 = \text{Weight of the cylinder} \]
\[ A(1 - p)L \cdot \rho \cdot g + A(pL) \cdot (n\rho) \cdot g = A \cdot L \cdot d \cdot g \]

### Step 5: Simplify the equation
We can cancel \( A \) and \( g \) from all terms:
\[ (1 - p)L \cdot \rho + pL \cdot n\rho = L \cdot d \]
Dividing through by \( L \) (since \( L \neq 0 \)):
\[ (1 - p)\rho + p n\rho = d \]

### Step 6: Factor out \( \rho \)
\[ \rho(1 - p + pn) = d \]
This simplifies to:
\[ d = \rho(1 + p(n - 1)) \]

### Final Result
Thus, the density \( d \) of the cylinder is given by:
\[ d = \rho(1 + p(n - 1)) \]