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Density is a measure of how much mass is contained in a given volume. It is usually expressed in units such as grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). Density is an intrinsic property of a substance and influences many physical phenomena, including buoyancy.
Understanding the concept of density helps explain why certain objects float while others sink. A crucial point to remember is that if an object's density is less than that of the fluid it is placed in, it will float. If its density is higher, it will sink.
In the given exercise, the density of the plastic sphere is calculated using the fact that it sinks halfway in water. This is described by the equation:
\[ \rho_{sphere} = 0.50 \times \rho_{water} \]
Since water has a known density of 1 g/cm³, we conclude that the sphere's density is 0.50 g/cm³. Understanding density helps us describe and predict the behavior of materials in different environments.

Archimedes' Principle is a fundamental principle of fluid mechanics. It states that any object submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object. This is the principle that allows objects to float.
To visualize this, imagine placing an object in water. The object pushes some of the water out of the way. According to Archimedes' Principle, the upward buoyant force on an object is the same as the weight of the water displaced.
In the context of our exercise, this principle helps calculate the densities of both the sphere and glycerin. When the sphere floats in water with half of its volume submerged, the buoyant force equals the weight of the sphere. Here, Archimedes' Principle is used to establish that the weight of the displaced water is equal to the weight of the sphere.
By applying this principle, the buoyant force is also described when the sphere floats in glycerin, where only 40% of its volume submerges. It allows us to calculate the glycerin's density as well. Understanding Archimedes' Principle is crucial in solving problems involving floating or submerged objects.

Floating objects provide practical examples of the interplay between density and buoyancy. An object floats when its density is less than the density of the fluid it is in. The difference in densities results in an upward buoyant force that supports the object against gravity.
In our exercise, the plastic sphere floats in both water and glycerin, albeit with different proportions submerged. This occurs because their densities vary, thereby changing the degree of buoyancy experienced by the sphere.
When the sphere is in water, 50% of it is submerged because its density is 0.50 g/cm³. As the sphere's density is exactly half that of water, it displaces enough water to balance its own weight. On the other hand, when the sphere is in glycerin, only 40% is submerged. This indicates that glycerin has a higher density than the water but is lower than the solid sphere's average density.
Observing floating objects and understanding their dynamics with different liquids offers insight into larger natural and industrial processes, ranging from ships to hot air balloons. The patterns we observe when different objects float serve as practical demonstrations of fundamental principles in physics.