Question

A block of wood floats in water with $ \frac{4}{5} $ th of its volume submerged. If the same block just floats in a liquid, the density of the liquid is (in $ \text{kg/m}^3 $)

• A. 1250
• B. 600
• C. 400
• D. 800

Hint:

The fraction of an object submerged in a fluid is inversely proportional to the density of the fluid. The denser the fluid, the less the object sinks.

Answer 1

The Correct Option is D

To determine the density of the liquid in which the block of wood just floats, we use the principle of flotation. According to this principle, the weight of the fluid displaced by the submerged part of the block is equal to the weight of the block.

Given that the block floats in water with $\frac{4}{5}$ of its volume submerged, the density of water is $\rho_w = 1000 \, \text{kg/m}^3$, and the following relationship holds:

$\rho_{block} \cdot V_{block} = \rho_{water} \cdot V_{submerged}$

where

• $\rho_{block}$ is the density of the block.
• $V_{block}$ is the volume of the block.
• $V_{submerged}$ is the submerged volume of the block, given by $ \frac{4}{5} V_{block} $.

Thus, the weight balance in water is:

$\rho_{block} = \frac{\rho_{water} \cdot \frac{4}{5} V_{block}}{V_{block}} = \frac{4}{5} \cdot \rho_w$

In this case,

$\rho_{block} = \frac{4}{5} \cdot 1000 = 800 \, \text{kg/m}^3$

Now, when the block floats in a different liquid, such that it just floats, the density of the liquid ($\rho_{liquid}$) equals the density of the block, which is:

$\rho_{liquid} = \rho_{block} = 800 \, \text{kg/m}^3$

Thus, the density of the liquid is 800 kg/m³.

Answer 2

When an object floats, the weight of the displaced liquid equals the weight of the object. For the block of wood floating in water: \[ \text{Weight of wood} = \text{Weight of displaced water} \] The fraction of the block submerged in water is \( \frac{4}{5} \). Using Archimedes' principle: \[ \text{Density of water} \times \text{Volume of displaced water} = \text{Density of wood} \times \text{Volume of wood} \] Given that the block is just floating in another liquid (with volume submerged), we use the same principle: Let the density of the liquid be \( \rho \). Then, the density of the wood can be expressed in terms of the density of water and the density of the new liquid: \[ \rho_{\text{liquid}} = 800 \, \text{kg/m}^3 \]
Thus, the density of the liquid is \( 800 \, \text{kg/m}^3 \). Therefore, the correct answer is: \[ \text{(4) } 800 \]