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Archimedes' Principle is a fundamental concept in physics that helps us understand how objects behave in fluids, such as water. The principle states that any object, wholly or partially submerged in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. This is why some objects float while others sink.
The principle can be mathematically expressed as:\[ F_b = V \cdot \rho_{\text{fluid}} \cdot g \]where:

• \( F_b \) is the buoyant force,
• \( V \) is the volume of the fluid displaced,
• \( \rho_{\text{fluid}} \) is the density of the fluid,
• \( g \) is the acceleration due to gravity.

Understanding this principle is crucial because it helps explain phenomena like why ships float, how submarines submerge, and many engineering applications.
Applying this to the exercise at hand, the paperweight displaces water equivalent to its own volume when submerged, which helps us calculate its volume using the change in apparent weight.

The concept of apparent weight comes into play when an object is submerged in a fluid. While an object has a true weight due to gravity, its apparent weight in a fluid is less due to the buoyant force acting upward. The apparent weight is the reading you would get from a scale when the object is in the fluid.
This phenomenon occurs because the fluid provides an upward force, which is the buoyant force, reducing the net force acting on the object compared to when it is in air.
Mathematically, apparent weight \( W' \) can be represented as the actual weight \( W \) minus the buoyant force \( F_b \):\[ W' = W - F_b \]In the given exercise, the paperweight's apparent weight in water is 4.3 N, compared to its weight in air of 6.9 N. The difference of 2.6 N corresponds to the buoyant force, helping us determine the paperweight's volume with the help of Archimedes' Principle.

The density of a fluid is an essential factor that affects buoyancy and is a key variable in solving problems related to submerged objects. For water, the standard density value is usually taken as \(1000 \, \text{kg/m}^3\), which is crucial for calculating how much water is displaced by an object.
Density is defined as mass per unit volume, and it helps determine the amount of fluid displaced by an object, which in turn helps calculate the buoyant force according to Archimedes' Principle. A crucial part of solving the paperweight problem was using water's density to find the volume of water displaced, which equals the volume of the paperweight itself.
Understanding the density of water is not only important for exercises like these but also for real-world applications such as designing ships, understanding aquatic life buoyancy, and in various engineering applications.