A metallic body floats at the interface of mercury of specific gravity 13.6 and water in such a way that 30% of its volume is submerged in mercury and 70% in water. Find the density of the metallic body.
Problem Statement
A metallic body floats at the interface of mercury and water in such a way that:
- 30% of its volume is submerged in mercury.
- 70% of its volume is submerged in water.
- Specific gravity of mercury = 13.6
Determine the density of the metallic body.
Solution
1. Define the Equilibrium Condition
2. Substitute Given Values
3. Solve for \( \rho \)
- Density of the metallic body: 4780 kg/m³
Explanation
1. Floating Equilibrium:
The metallic body remains at rest at the interface of mercury and water because the total buoyant force acting on it is equal to its weight. The buoyant force is contributed by both mercury and water.
2. Archimedes’ Principle:
According to Archimedes’ principle, the buoyant force on an object is equal to the weight of the displaced fluid. Here, the metallic body displaces two fluids: mercury and water.
3. Calculation Approach:
– The fraction of the metallic body submerged in each fluid is known (30% in mercury and 70% in water).
– Using the density of mercury (\(13.6 \times 1000\) kg/m³) and water (1000 kg/m³), we calculate the total buoyant force.
– Equating the total buoyant force to the weight of the metallic body allows us to determine its density.
4. Importance of Density:
The obtained density of 4780 kg/m³ indicates that the metallic body is denser than water but much less dense than mercury. This is why it partially submerges in both fluids instead of floating completely on one.
Physical Meaning
1. Industrial Applications:
The concept of floating at the interface of two fluids is applied in density measurement, separation of materials, and metallurgical processes.
2. Ship Stability and Ballasting:
Ships and submarines use similar principles to control buoyancy, ensuring stability and proper flotation by balancing weights in different liquid layers.
3. Fluid Separation Techniques:
This principle is used in separating immiscible liquids of different densities, such as oil and water separation in refineries.
4. Floating and Sinking Behavior:
The specific gravity of a material relative to surrounding fluids determines whether it sinks, floats, or remains at an interface, as demonstrated in this problem.
