The formula for the fraction of an object submerged in a fluid is given by the ratio of the object's density to the fluid's density.
Understanding the Formula
The exact formula provided in the reference is:
Fraction Submerged = ρ obj / ρ fl
Let's break down the components:
- ρ obj represents the density of the object being submerged. Density is a measure of mass per unit volume.
- ρ fl represents the density of the fluid in which the object is submerged.
This fundamental relationship arises from Archimedes' principle, which states that the buoyant force on an object equals the weight of the fluid displaced by the object. When an object floats, the buoyant force equals the object's weight.
How the Formula Works
When an object floats, a portion of its volume is submerged, displacing a certain volume of fluid. The weight of the object is equal to the buoyant force, which is the weight of the displaced fluid.
- Weight of object = (ρ obj) (Volume of object) g
- Weight of displaced fluid = (ρ fl) (Volume submerged) g
Since these weights are equal when floating:
(ρ obj) (Volume of object) g = (ρ fl) (Volume submerged) g
Cancelling 'g' (acceleration due to gravity) from both sides:
(ρ obj) (Volume of object) = (ρ fl) (Volume submerged)
Rearranging to find the fraction submerged (Volume submerged / Volume of object):
Volume submerged / Volume of object = ρ obj / ρ fl
This confirms the formula provided.
Applications and Practical Insights
The relationship fraction submerged = ρ obj / ρ fl is incredibly useful:
- Measuring Densities: As mentioned in the reference, this relationship can be used to measure densities. If you know the density of a fluid (like water) and can measure the fraction of a known object that is submerged, you can determine the object's density.
- Ship Design: The principle is critical in naval architecture. A ship floats because its average density (total mass divided by the total volume including the hull and air) is less than the density of water.
- Example: Figure 14.22 from the reference illustrates that "An unloaded ship (a) floats higher in the water than a loaded ship (b)." This is because adding cargo increases the ship's total mass, thus increasing its average density. To maintain equilibrium (buoyant force = weight), a larger volume of water must be displaced, meaning the ship sinks lower, and a larger fraction is submerged.
- Hydrometers: Devices used to measure the specific gravity (and thus density) of liquids work based on this principle. They float higher in denser liquids and lower in less dense liquids.
- Icebergs: Only about 10% of an iceberg is visible above the surface of seawater. This is because the density of ice is approximately 917 kg/m³, and the density of seawater is about 1025 kg/m³.
- Fraction submerged ≈ 917 / 1025 ≈ 0.895
- This means about 89.5% is submerged, leaving only about 10.5% above water.
Summary
The formula for the fraction of an object submerged in a fluid when it floats is the ratio of the object's density to the fluid's density: Fraction Submerged = ρ obj / ρ fl. This formula directly relates the densities to the proportion of the object that lies beneath the surface, providing a powerful tool for understanding buoyancy and measuring densities.
