Determining the Relative Density of Oil Using Buoyancy Principles

Understanding the principles of buoyancy can help in solving intriguing problems related to fluid mechanics. In this article, we will walk through a problem where a body floats in water and oil, using buoyancy principles to find the relative density of the oil. Buoyancy is a fascinating concept, as it involves the relationship between a body's weight and the weight of the fluid it displaces.

Principle of Buoyancy

The principle of buoyancy is fundamental in fluid mechanics. It states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This force acts upward, counteracting the downward force due to gravity.

Problem Setup

Consider a body that partially floats in both water and oil. We need to determine the relative density of oil given the specific conditions of the body's floating in both mediums.

Step 1: Analyzing the Body in Water

When the body floats in water, 60% of its volume is submerged. This means that its density must be such that it displaces a volume of water equal to 60% of its own volume.

Volume Submerged in Water: ( V_s 0.6V )

The weight of the body is equal to the weight of the water displaced:

Weight of Water Displaced: ( W V_s cdot rho_w 0.6V cdot rho_w )

Step 2: Analyzing the Body in Oil

When the body is placed in oil, 40% of its volume is submerged.

Volume Submerged in Oil: ( V_s 0.4V )

The weight of the body is also equal to the weight of the oil displaced:

Weight of Oil Displaced: ( W V_s cdot rho_o 0.4V cdot rho_o )

Step 3: Setting Up the Equations

Equating the weight of the body in both scenarios, we have:

Water Scenario: ( W 0.6V cdot rho_w )

Oil Scenario: ( W 0.4V cdot rho_o )

Since both expressions are equal to the weight of the body, we can set them equal to each other:

0.6V cdot rho_w 0.4V cdot rho_o

Dividing both sides by V (assuming V ne; 0), we get:

0.6 cdot rho_w 0.4 cdot rho_o

Step 4: Solving for the Density of Oil

Rearranging the equation:

( rho_o frac{0.6}{0.4} cdot rho_w 1.5 cdot rho_w )

Given that the relative density of oil is ( frac{K}{2} ), we can write:

( frac{K}{2} 1.5 cdot rho_w )

Since the density of water (( rho_w )) is 1 g/cm3, we substitute and solve for K:

( frac{K}{2} 1.5 cdot 1 1.5 )

Multiplying both sides by 2:

( K 3 )

Hence, the value of K is 3.

Conclusion

By using the principles of buoyancy, we determined the relative density of the oil. This problem demonstrates the importance of understanding the relationship between the volume submerged in a fluid and the density of the body and fluid.

Key Takeaways:

The principle of buoyancy is critical for solving problems involving floating bodies. The volume of a body submerged in a fluid can give clues about its density. The relative density can be calculated based on the given conditions in different fluids.

Understanding these principles can be valuable for a variety of applications, including engineering, physics, and environmental science.