A block of wood floats in water with `(4//5)th` of its volume submerged. If the same block just floats in a liquid, the density of liquid in `(kg m^(-3))` is

Options

A. 1250
B. 600
C. 400
D. 800

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We have a block of wood that floats in water with \( \frac{4}{5} \) of its volume submerged. We need to find the density of the liquid in which the same block just floats.

### Step 2: Define the variables
Let:
- \( V \) = Volume of the block
- \( \rho_b \) = Density of the block
- \( \rho_w \) = Density of water (approximately \( 1000 \, \text{kg/m}^3 \))
- \( \rho_l \) = Density of the liquid

### Step 3: Apply the principle of buoyancy in water
When the block is floating in water, the weight of the block is balanced by the buoyant force. The weight of the block can be expressed as:
\[ \text{Weight of block} = \rho_b \cdot V \cdot g \]
The buoyant force acting on the block when \( \frac{4}{5} \) of its volume is submerged is:
\[ \text{Buoyant force} = \rho_w \cdot \left(\frac{4}{5} V\right) \cdot g \]

### Step 4: Set up the equation for floating condition in water
Since the block is floating, we can equate the weight of the block to the buoyant force:
\[ \rho_b \cdot V \cdot g = \rho_w \cdot \left(\frac{4}{5} V\right) \cdot g \]
Cancelling \( V \) and \( g \) from both sides gives:
\[ \rho_b = \rho_w \cdot \frac{4}{5} \]

### Step 5: Substitute the density of water
Substituting \( \rho_w = 1000 \, \text{kg/m}^3 \):
\[ \rho_b = 1000 \cdot \frac{4}{5} = 800 \, \text{kg/m}^3 \]

### Step 6: Apply the principle of buoyancy in the liquid
When the block just floats in the liquid, the weight of the block is again balanced by the buoyant force:
\[ \rho_b \cdot V \cdot g = \rho_l \cdot V \cdot g \]
Cancelling \( V \) and \( g \) gives:
\[ \rho_b = \rho_l \]

### Step 7: Conclusion
Since we found that \( \rho_b = 800 \, \text{kg/m}^3 \), we can conclude that:
\[ \rho_l = 800 \, \text{kg/m}^3 \]

Thus, the density of the liquid is \( 800 \, \text{kg/m}^3 \).

### Final Answer
The density of the liquid is \( 800 \, \text{kg/m}^3 \).

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