Understanding Buoyancy and Density: Calculating the Volume of a Floating Object

When an object floats in a fluid, it displaces a volume of the fluid equal to its own weight. This phenomenon, known as Archimedes' Principle, plays a crucial role in understanding the behavior of floating objects. In this article, we'll explore how to calculate the volume of an object given its density and the fraction of its volume that is immersed in a fluid.

Principles of Buoyancy

Buoyancy refers to the upward force exerted by a fluid on an object submerged or floating in it. This force is equal to the weight of the fluid displaced by the object. The principle of buoyancy can be mathematically represented as follows:

The buoyant force acting on an object is equal to the weight of the fluid displaced by the object.

The weight of the fluid displaced is equal to the weight of the object.

Key Concepts and Terminology

Density: Measured in mass per volume (e.g., kg/m3).

Buoyant Force: The upward force exerted by a fluid on a submerged or floating object.

Archimedes' Principle: The principle stating that the buoyant force on a submerged or floating object is equal to the weight of the fluid displaced by the object.

Calculating the Volume of a Floating Object

In the given scenario, an object has a density of 900 kg/m3 and 3/4 of its volume is immersed in a fluid. To find the volume of the object, we need to use the principle of buoyancy and the given information.

Step-by-Step Calculation

Denote the volume of the object as V.

Since 3/4 of the volume is immersed, the volume of the fluid displaced is 3/4 V.

The weight of the fluid displaced is equal to the weight of the object:

Weight of the fluid displaced Density of the fluid × Volume of the fluid displaced

Weight of the object Density of the object × Volume of the object

Equate the two:

Density of the fluid × 3/4 V Density of the object × V

Simplifying the equation to solve for V:

V 3/4 × Density of the fluid / Density of the object

Therefore, the volume of the object is given by the formula:

V 3/4 × Density of the fluid / Density of the object

Further Insights and Pitfalls

It's important to note a few key points:

The density provided in the question is given as kg/m2, which is incorrect. The correct unit for density should be kg/m3.

Given the information, we can determine the density of the surrounding fluid, not the volume of the object. For example, if an object of density 900 kg/m3 floats in water, it would be 75% immersed.

If the object were to float in a heavier fluid, such as mercury (density 13600 kg/m3), the fraction of the object's volume that would be immersed would be much smaller, approximately 6.6%.

Conclusion

In conclusion, by applying Archimedes' principle and understanding the relationship between density and buoyancy, we can accurately determine the volume of a floating object given its density and the fraction of its volume that is immersed. However, it is crucial to ensure that the units and provided data are correct to reach a valid and accurate conclusion.