Question

A rectangular block with density of 720 kg/m$^3$ floats in fluid having the relative density of 0.8. What percentage of block will remain exposed?

• A. 12%
• B. 11%
• C. 10%
• D. 9%

Hint:

For any floating object, the ratio of its density to the fluid's density gives the fraction of the object that is submerged: \( \frac{V_{submerged}}{V_{total}} = \frac{\rho_{object}}{\rho_{fluid}} \). Always remember to convert relative density to actual density if the fluid is not water or if calculations require absolute density values.

Answer 1

The Correct Option is C

Step 1: Understand the principle of flotation.
For a floating object, the weight of the object is equal to the weight of the fluid displaced by the submerged part of the object. This is Archimedes' principle.
Mathematically, this can be expressed as:
Weight of block = Buoyant force
\( m_{block} \cdot g = m_{fluid\_displaced} \cdot g \)
\( \rho_{block} \cdot V_{block} \cdot g = \rho_{fluid} \cdot V_{submerged} \cdot g \)
This simplifies to:
\( \rho_{block} \cdot V_{block} = \rho_{fluid} \cdot V_{submerged} \)
Step 2: Identify the given densities.
Density of the block (\(\rho_{block}\)) = 720 kg/m\(^3\)
Relative density of the fluid = 0.8
Step 3: Calculate the density of the fluid.
Relative density (or specific gravity) is the ratio of the density of a substance to the density of a reference substance (usually water at 4°C, which has a density of 1000 kg/m\(^3\)). \[ \text{Relative density of fluid} = \frac{\rho_{fluid}}{\rho_{water}} \] So, \[ 0.8 = \frac{\rho_{fluid}}{1000 \, \text{kg/m}^3} \] \[ \rho_{fluid} = 0.8 \times 1000 \, \text{kg/m}^3 = 800 \, \text{kg/m}^3 \]
Step 4: Calculate the fraction of the block that is submerged.
From the principle of flotation (Step 1):
\( \rho_{block} \cdot V_{block} = \rho_{fluid} \cdot V_{submerged} \) Rearrange to find the ratio of submerged volume to total volume:
\[ \frac{V_{submerged}}{V_{block}} = \frac{\rho_{block}}{\rho_{fluid}} \] Substitute the density values: \[ \frac{V_{submerged}}{V_{block}} = \frac{720 \, \text{kg/m}^3}{800 \, \text{kg/m}^3} \] \[ \frac{V_{submerged}}{V_{block}} = \frac{72}{80} = \frac{9}{10} = 0.9 \] This means 90% of the block is submerged.
Step 5: Calculate the percentage of the block that remains exposed.
The percentage of the block exposed is the total volume percentage minus the submerged volume percentage: \[ \text{Percentage exposed} = 100% - \text{Percentage submerged} \] \[ \text{Percentage exposed} = 100% - (0.9 \times 100%) \] \[ \text{Percentage exposed} = 100% - 90% = 10% \] The final answer is \( \boxed{\text{10%}} \).