You have two identical silver spheres and two unknown fluids, \(A\) and \(B\). You place one sphere in fluid \(A\), and it sinks; you place the other sphere in fluid \(\mathrm{B}\), and it floats. What can you conclude about the buoyant force of fluid \(\mathrm{A}\) versus that of fluid \(\mathrm{B} ?\)

Understanding Archimedes' principle is crucial to grasp the concept of buoyancy. Simply put, this principle states that any object, when submerged in a fluid, is acted upon by an upward force known as the buoyant force. This force is equal to the weight of the fluid that the object displaces.
For example, imagine you're holding a basketball underwater. The ball wants to pop up to the surface because the water is pushing it up with a force equal to the weight of the water that would be in the space the ball is occupying. If the weight of the object is greater than the buoyant force, like a stone, it sinks. If the buoyant force is greater or equal to the weight, like a boat, it floats.
This principle is derived from experimentations by Archimedes of Syracuse, and it forms the foundation for understanding buoyancy and why objects float or sink in fluids.

Fluid mechanics is the branch of physics concerned with the behavior of liquids and gases. It can get quite complex with equations and concepts that describe how fluids flow and how they interact with their environment, but the key idea for our discussion is understanding how objects behave when they are in a fluid.
Fluids exert pressure in all directions, and when an object is submerged in a fluid, this pressure leads to the buoyant force we discussed earlier. The behavior of the fluid—how it moves and exerts force on objects—is governed by its own density, viscosity, and the gravitational force acting upon it.
In the case of the silver spheres in fluids A and B, fluid mechanics explains not just why they sink or float, but also provides the mathematical models to calculate the forces at play.

The concepts of density and buoyancy are intimately linked. Density is basically how much stuff is packed into a certain space - more technically, it's mass per unit volume. Objects with higher density than the fluid they're in tend to sink, while those with lower density float.
Buoyancy refers to the tendency of an object to float in a fluid. It is an upward force exerted by the fluid that opposes the weight of an immersed object. For the identical silver spheres mentioned in the exercise, density would be a deciding factor. Sphere in fluid A sinks because the fluid's buoyant force can't support the sphere's weight, signaling that the sphere is denser than fluid A. Conversely, the sphere in fluid B floats, indicating that fluid B has a greater density than the sphere, or at least, the densities are close enough, resulting in a buoyant force that can support or exceed the sphere's weight.

When comparing buoyant forces between two fluids, like A and B from the exercise, we look at the relative magnitudes of these forces on identical objects. Archimedes' principle allows us to understand that these forces are directly related to the weight of the fluid displaced.
This implies that if one sphere sinks in fluid A and another floats in fluid B, fluid B must be denser or similarly dense to support the sphere. Therefore, it exerts a greater buoyant force than fluid A. The sinking and floating behaviors are evidences we use to infer the relative buoyancies; they make it clear that the buoyant force in fluid B must be greater than or equal to the weight of the sphere, whereas the buoyant force in fluid A is insufficient to counteract the weight of the sphere, hence smaller in magnitude.