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Density is a measure of how much mass is contained in a given volume. It is an intrinsic property of matter, which means it does not change regardless of the size of the sample you have.
Density is usually expressed in units such as grams per cubic centimeter (g/cm extsuperscript{3}) or kilograms per cubic meter (kg/m extsuperscript{3}). \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] This formula helps us understand how much material is packed into a given space.

• For example, quartz has a density of 2.65 g/cm extsuperscript{3}. This means 1 cubic centimeter of quartz weighs 2.65 grams.
• On the other hand, mercury, which is denser, has a density of 13.6 g/cm extsuperscript{3}. This higher number signifies that mercury is much more tightly packed than quartz.

Recognizing the differences in density between substances helps us understand why some objects float while others sink. When an object of lower density is placed in a fluid of higher density, it floats because it displaces more fluid weight than its own weight.

The concept of the weight of displaced fluid is central to understanding why objects float, as explained by Archimedes’ principle. When an object is submerged in a fluid, it pushes or displaces some of that fluid out of the way. The weight of this displaced fluid provides a buoyant force, or an upward lift.
The weight of the displaced fluid can be calculated using the formula: \[ \text{Weight of Displaced Fluid} = \text{Submerged Volume} \times \text{Density of Fluid} \] This means, for the quartz example:

• When quartz is submerged in mercury, the displaced mercury's weight is given by the submerged quartz volume multiplied by the mercury's density (13.6 g/cm extsuperscript{3}).
• This weight of the displaced fluid must equal the weight of the object (quartz) for it to float, providing the necessary upward force to counter the gravitational force pulling the object downwards.

Practically, this explains why something might be able to "float" in mercury, despite mercury being a liquid metal because it is incredibly dense compared to the object.

The submerged volume fraction is the proportion of an object's volume that is beneath the surface when it is floating in a fluid. This concept is crucial in applications from shipbuilding to environmental science. In our example, the quartz piece is partially submerged in mercury, and we determine what fraction of its volume is underwater.Applying Archimedes' Principle, we established the relationship between displaced fluid's weight and the object's weight. Using equations, the submerged volume fraction is found by:\[ \frac{V_s}{V} = \frac{\text{Density of Quartz}}{\text{Density of Mercury}} = \frac{2.65}{13.6} \]Where:

• \( V_s \) is the submerged volume of quartz.
• \( V \) is the total volume of quartz.

Here, this results in roughly 19.49%, indicating that a smaller portion of the quartz is actually submerged. This happens because quartz is less dense than mercury.
This submerged volume fraction tells us how much of the quartz stays below the surface and is key for understanding buoyancy in different fluids.