College Physics II – Mechanics, Sound, Oscillations, and Waves
Definition
$W_{object} = W_{displaced}$ is a key relationship that describes the buoyant force acting on an object submerged in a fluid. It states that the weight of the object is equal to the weight of the fluid displaced by the object, which is a fundamental principle in understanding Archimedes' Principle and buoyancy.
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The weight of the object ($W_{object}$) is equal to the weight of the fluid displaced by the object ($W_{displaced}$).
This relationship is a direct consequence of Archimedes' Principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object.
The weight of the fluid displaced is determined by the volume of the object and the density of the fluid, according to the formula: $W_{displaced} = \rho_{fluid} \cdot V_{object} \cdot g$.
The buoyant force on the object is directed upward and is equal in magnitude to the weight of the fluid displaced, which acts to support the weight of the object.
Understanding the relationship between $W_{object}$ and $W_{displaced}$ is crucial for analyzing the behavior of objects submerged in fluids, such as determining the apparent weight of an object in water or the force required to keep an object submerged.
Review Questions
Explain how the relationship $W_{object} = W_{displaced}$ is derived from Archimedes' Principle.
Archimedes' Principle states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object. The weight of the fluid displaced is determined by the volume of the object and the density of the fluid, according to the formula $W_{displaced} = \rho_{fluid} \cdot V_{object} \cdot g$. Since the buoyant force is equal to the weight of the fluid displaced, and the object's weight is balanced by the buoyant force, we can conclude that the weight of the object ($W_{object}$) is equal to the weight of the fluid displaced ($W_{displaced}$).
Describe how the relationship $W_{object} = W_{displaced}$ can be used to determine the apparent weight of an object submerged in a fluid.
When an object is submerged in a fluid, the apparent weight of the object is reduced due to the buoyant force acting on it. The relationship $W_{object} = W_{displaced}$ can be used to calculate the apparent weight of the object. Since the weight of the fluid displaced is equal to the weight of the object, the apparent weight of the object in the fluid is equal to the actual weight of the object minus the weight of the fluid displaced. This allows for the determination of the apparent weight, which is an important factor in understanding the behavior of objects submerged in fluids.
Analyze how the relationship $W_{object} = W_{displaced}$ can be used to predict the behavior of objects floating on the surface of a fluid.
When an object floats on the surface of a fluid, the weight of the object is equal to the weight of the fluid displaced by the object. This means that the buoyant force acting on the object is equal to the weight of the object. If the weight of the object is less than the weight of the fluid displaced, the object will float, and if the weight of the object is greater than the weight of the fluid displaced, the object will sink. By understanding the relationship $W_{object} = W_{displaced}$, one can predict the floating or sinking behavior of objects on the surface of a fluid, which is crucial in applications such as ship design and the study of flotation mechanisms in nature.