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Archimedes' principle is a fundamental concept in fluid mechanics that describes the buoyancy experienced by objects submerged in a fluid. It states that an object submerged in a fluid is buoyed up by a force equal to the weight of the fluid that it displaces.
In our given problem, where a cylinder is floating in both water and oil, Archimedes' principle allows us to determine how much of the cylinder's volume will be submerged in each fluid. Since the total buoyant force is equal to the weight of the cylinder, this principle allows us to calculate the volume of water and oil displaced.
Understanding Archimedes' principle helps us see why objects float or sink depending on how their weight compares to the weight of the fluid they displace. In the case of the cylinder, the oil and water together provide enough buoyant force to keep it floating.

Fluid mechanics is the branch of physics concerned with the behavior of fluids (liquids and gases) and the forces on them. This discipline provides the tools needed to solve problems involving fluid motion and equilibrium.
When analyzing our floating cylinder, fluid mechanics principles are applied to understand the interaction between the cylinder, the water, and the oil. It allows us to understand how these liquid layers support the floating object through buoyancy.
Fluid mechanics involves concepts such as pressure, buoyancy, and viscosity, which are crucial for predicting how fluids behave under various conditions. By applying these principles, we can solve complex problems, such as calculating the portion of a cylinder submerged in displacing fluids.

Density is a property that measures how much mass is contained in a given volume of a substance, usually expressed as \( \rho = \frac{m}{V} \). This characteristic is pivotal in determining how much of an object will float or submerge in a fluid due to buoyancy.
In the context of the cylinder submersed in water and oil, density plays a key role in understanding the situation. The oil's density of 725 kg/m³ compared to the water's higher density of 1000 kg/m³ affects how much buoyant force each liquid exerts on the cylinder.
The differences in density explain why only a certain portion of the cylinder's height is submerged in oil, as water provides a stronger buoyant lift due to its higher density.

Displacement in fluid mechanics refers to the amount of fluid that is moved out of place when an object is submerged in a fluid. It directly ties into Archimedes' principle, where the displaced volume helps determine the buoyant force.
For the floating cylinder, the concept of displacement means the water and oil displaced by the cylinder's volume provide the necessary force to support its weight.
To find how much of the cylinder is immersed in oil, we first calculate how much water is displaced by using the \( V_{water} = \frac{m}{\rho_{water}} \). Once this is known, the remaining volume represents the displacement in the oil, which helps determine the height of the cylinder submerged in oil.