A fish maintains its depth in seawater by adjusting the air...

A fish maintains its depth in seawater by adjusting the air content of porous bone or air sacs to make its average density the same as that of the water. Suppose that with its air sacs collapsed a fish has a density of $1.08 \mathrm{~g} / \mathrm{cm}^{3} .$ To what fraction of its expanded body volume must the fish inflate the air sacs to reduce its average density to that of the water? Assume that the air density is $0.00121 \mathrm{~g} / \mathrm{cm}^{3}$.

Key Concepts

Density

Density is defined as mass per unit volume and is a key property determining whether an object will sink or float in a fluid. In the context of buoyancy, understanding how density relates to mass and volume is critical for predicting and controlling an object's behavior in a fluid medium.

Composite Density Calculation

When an object is made up of different components with varying densities, its overall or average density is determined by the weighted average of the densities of its components, based on their respective volume fractions. This concept is essential when substituting a portion of the object with a material of a contrasting density to achieve a desired average density.

Archimedes' Principle

Archimedes' principle explains that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body. This principle underpins the understanding of how adjustments in an object's density affect its buoyancy, ensuring it can float, sink, or remain neutrally buoyant.

Volume Fraction Adjustment

Volume fraction adjustment involves changing the proportion of different materials or components within an object to alter its overall density. By inflating air sacs, for example, a fish replaces a denser medium with a much lighter one, thereby reducing its average density to match that of the surrounding water and maintain neutral buoyancy.

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