What fraction of the volume of solid piece of metal of sp gr 7.2 floats above the surface of a container of mercury of sp. gr. 13.6?

Fraction of Metal Floating Above Mercury

Problem Statement

A solid piece of metal with a specific gravity of 7.2 is placed in a container of mercury (specific gravity 13.6). Determine:

The fraction of the metal submerged in mercury.
The fraction of the metal floating above the mercury surface.

Solution

1. Define the Equilibrium Condition

The weight of the metal must equal the weight of the displaced mercury: \[ \gamma_{\text{body}} V = \gamma_{\text{mercury}} V’ \]

2. Substitute Values

\[ 7.2 \times 9810 \times V = 13.6 \times 9810 \times V’ \] Cancelling \( 9810 \) from both sides: \[ 7.2 V = 13.6 V’ \] \[ \frac{V’}{V} = \frac{7.2}{13.6} \] \[ \frac{V’}{V} = 0.53 \]

3. Find the Fraction Floating Above Mercury

\[ \text{Fraction of volume above mercury} = 1 – \frac{V’}{V} \] \[ = 1 – 0.53 \] \[ = 0.47 \]

Final Results:

Fraction of the metal submerged in mercury: 0.53
Fraction of the metal floating above mercury: 0.47

Explanation

1. Archimedes’ Principle:
When an object is placed in a liquid, it experiences an upward buoyant force equal to the weight of the liquid displaced. The object floats partially submerged if its density is less than that of the liquid.

2. Submersion Calculation:
The ratio of the submerged volume to the total volume is given by the ratio of specific gravities. Since the metal has a specific gravity of 7.2 and mercury has a specific gravity of 13.6, the metal will displace mercury equal to 53% of its volume.

3. Floating Portion Calculation:
The remaining 47% of the metal volume remains above the mercury surface. This fraction is calculated by subtracting the submerged fraction from 1.

4. Interpretation:
This means that the metal does not sink completely in mercury, as its density is lower than that of mercury but greater than that of water.

Physical Meaning

1. Application in Material Selection:
The concept of specific gravity is widely used in material science and engineering to determine whether materials will float or sink in different liquids.

2. Industrial and Laboratory Use:
This principle is applied in metallurgy, where molten metals of different densities are separated in a refining process.

3. Liquid Metal-Based Buoyancy Systems:
Mercury has a very high density, so even relatively dense materials like steel will float partially on mercury. This concept is used in certain specialized industrial applications.

4. Real-World Example:
A steel object, which sinks in water, will float partially when placed in mercury. This demonstrates how buoyancy is relative to the liquid’s density rather than an absolute property of the object.