Question

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 \(B\), and it floats. What can you conclude about the buoyant force of fluid A versus that of fluid \(B\) ?

Short answer

Answer: Fluid B has a greater buoyant force.

Step-by-step solution

Preliminary Information

We can see from the given information that one sphere sinks in fluid A and the other sphere floats in fluid B. This will help us determine the buoyant force in each fluid. To do this, we'll use Archimedes' principle which states that: Buoyant force = weight of displaced fluid

Analyze Sphere in Fluid A

When the sphere is placed in fluid A, it sinks. This indicates that the buoyant force in fluid A is not sufficient enough to balance the weight of the silver sphere. Therefore, we can conclude that the buoyant force in fluid A is less than the weight of the sphere. Mathematically, this can be represented as: Buoyant force A < Weight of sphere

Analyze Sphere in Fluid B

When the sphere is placed in fluid B, it floats. This indicates that the buoyant force in fluid B is enough to balance the weight of the silver sphere or may even be greater. Therefore, we can conclude that the buoyant force in fluid B is greater than or equal to the weight of the sphere. Mathematically, this can be represented as: Buoyant force B ≥ Weight of sphere

Compare Buoyant Forces in Fluid A and Fluid B

Now, we have the buoyant forces in both fluids with respect to the weight of the silver sphere. Comparing them, we get: Buoyant force A < Weight of sphere ≤ Buoyant force B This inequality tells us that the buoyant force of fluid A is less than that of fluid B. In simpler terms, the buoyant force in fluid B is greater than the buoyant force in fluid A. #Conclusion# By observing the behavior of the identical silver spheres when placed in fluids A and B, and using Archimedes' principle, we conclude that the buoyant force of fluid A is less than that of fluid B.

Key concepts

Buoyant Force

When we talk about buoyant force, we're referring to the push that helps objects float in fluids, which include liquids and gases. According to Archimedes' principle, this force is equal to the weight of the fluid displaced by the object submerging in it.

Imagine submerging a basketball in a swimming pool. The ball pushes water out of the way — it displaces water. The pool water pushes back with a force equal to the weight of the water displaced. This force works upward against gravity, which helps the basketball float. If the buoyant force is less than the object's weight, the object sinks. Conversely, if the buoyant force is greater than or equal to the object's weight, the object floats or remains suspended.

In our exercise solution, the sphere in fluid A sinks because the buoyant force is less than the sphere's weight. In fluid B, the sphere floats, indicating the buoyant force is equal to or greater than its weight. This principle is critical in designing boats, ships, and even for understanding how hot air balloons rise.

Density and Buoyancy

The concepts of density and buoyancy are intimately connected. Density is mass per unit volume of a substance. An object will float if it's less dense than the fluid it's in. If it's denser, it will sink. That's because a fluid can only push back with a buoyant force equal to the weight of the fluid displaced.

Consider two blocks of equal volume, one made of cork and the other of iron. The iron block is denser and thus heavier than the cork. When placed in water, the iron block displaces a small amount of water before it's completely submerged, generating a small buoyant force. The cork displaces more water before it's fully submerged, resulting in a greater buoyant force. Therefore, the cork block floats while the iron sink.

In the case of our silver spheres, we infer that fluid B is denser than fluid A, because the sphere floats in B and sinks in A. This suggests that the sphere, which has a fixed density, displaces an amount of fluid B sufficient to generate a buoyant force to support its weight, ann indicative of a denser fluid.

Floating and Sinking Objects

Why do some objects float while others sink? The answer lies in the relative density of the objects and the fluid, along with the buoyant force we described earlier. If an object has a density lower than the fluid, it will float. If it's denser, it will sink. It's all about whether the object can displace enough fluid to create a counteracting buoyant force strong enough to support its weight.

Our silver spheres serve as great examples. Fluid B's ability to keep the sphere floating means that relative to the sphere, Fluid B must be dense enough to generate a sufficient buoyant force. Conversely, Fluid A's inability to keep the sphere afloat indicates that it is not as dense as Fluid B, and thus the sphere sinks.

Understanding why objects float or sink is essential in real-world applications. It allows us to design vessels that carry heavy loads across water and ensure that submarines can both surface and dive according to their operational needs. It's also the principle behind saline water in flotation therapies where higher salt density helps humans float effortlessly.