College Physics II – Mechanics, Sound, Oscillations, and Waves
Definition
$F_b = W_{displaced}$ is a fundamental relationship in the context of Archimedes' Principle and buoyancy. It states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
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The buoyant force, $F_b$, acts in the upward direction and is equal to the weight of the fluid displaced by the object, $W_{displaced}$.
The weight of the displaced fluid is determined by the volume of the object and the density of the fluid, according to the formula: $W_{displaced} = \rho_f \cdot V_{object}$.
Archimedes' Principle explains why some objects float while others sink, depending on their density relative to the density of the fluid they are placed in.
The buoyant force can be used to calculate the apparent weight of an object submerged in a fluid, which is less than its true weight in air.
Understanding the relationship between $F_b$ and $W_{displaced}$ is crucial for analyzing the behavior of objects in fluids, such as in the design of ships, submarines, and hot air balloons.
Review Questions
Explain how the relationship $F_b = W_{displaced}$ is derived from Archimedes' Principle.
Archimedes' Principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This means that the upward buoyant force, $F_b$, is directly proportional to the weight of the displaced fluid, $W_{displaced}$. The weight of the displaced fluid is determined by the volume of the object and the density of the fluid, according to the formula $W_{displaced} = \rho_f \cdot V_{object}$. Therefore, the relationship $F_b = W_{displaced}$ is a direct consequence of Archimedes' Principle.
Describe how the relationship $F_b = W_{displaced}$ can be used to determine the apparent weight of an object submerged in a fluid.
The relationship $F_b = W_{displaced}$ can be used to calculate the apparent weight of an object submerged in a fluid. The apparent weight of the object is its true weight in air minus the buoyant force acting on it. Since the buoyant force is equal to the weight of the displaced fluid, the apparent weight can be calculated as the object's true weight in air minus the weight of the displaced fluid, which is given by $W_{displaced} = \rho_f \cdot V_{object}$. This relationship is crucial for understanding the behavior of objects in fluids, such as the ability of objects to float or sink based on their density relative to the fluid.
Analyze how the relationship $F_b = W_{displaced}$ can be used to design and optimize the performance of objects that rely on buoyancy, such as ships, submarines, and hot air balloons.
The relationship $F_b = W_{displaced}$ is essential for the design and optimization of objects that rely on buoyancy, such as ships, submarines, and hot air balloons. By understanding this relationship, engineers can calculate the buoyant force acting on these objects and use it to determine their stability, load-carrying capacity, and overall performance. For example, in the design of ships, the relationship can be used to calculate the ship's displacement and the buoyant force acting on it, which is crucial for ensuring the ship's stability and seaworthiness. Similarly, in the design of submarines, the relationship can be used to calculate the buoyant force and adjust the vehicle's ballast to control its depth and maneuverability. For hot air balloons, the relationship can be used to determine the required volume of the balloon and the amount of lift generated by the heated air, which is essential for ensuring safe and efficient flight.
Archimedes' Principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
Buoyancy is the upward force exerted by a fluid on an object, which counteracts the object's weight and allows it to float.
Displaced Fluid: Displaced fluid refers to the volume of fluid that is pushed aside when an object is submerged in the fluid, and the weight of this displaced fluid is equal to the buoyant force acting on the object.