Problem 1 - Fundamentals of Physics

Question

\(\cdot 1\) ILW A fish maintains its depth in fresh water 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 density to that of water?

Step-by-Step Solution

Answer

The fish must inflate the air sacs to replace about \(7.41\%\) of its volume with air.

Step 1: Understand the Problem

The problem states that a fish needs to adjust its density to match that of the water, which is approximately \(1.00 \ \mathrm{g/cm}^3\). The fish initially has a density of \(1.08 \ \mathrm{g/cm}^3\). We have to find the fraction of the fish's body volume that must be made up of air such that its overall density becomes \(1.00 \ \mathrm{g/cm}^3\).

Step 2: Establish Relationships

To solve this, we need to understand that the fish's density is the total mass divided by the total volume. Let \(V_f\) be the fraction of the total volume occupied by air at the expanded state, \(\rho_f\) be the final density (1.00 \(\mathrm{g/cm}^3\)), \(\rho_i\) be the initial density (1.08 \(\mathrm{g/cm}^3\)), and \(\rho_{air}\) be the density of air (negligible, or approximately \(0 \ \mathrm{g/cm}^3\)).

Key Concepts

Buoyancy in Fish

Fish adjust their buoyancy in the water by managing the air content in their bodies, specifically in structures like air sacs or swim bladders. This adjustment is crucial for maintaining the right depth in the water without exerting too much energy.
In simple terms, buoyancy refers to the ability of an object to float or remain submerged in a fluid. For fish, achieving the right buoyancy means balancing their body density with that of the surrounding water.

• When a fish wants to move up or remain at the current depth, it increases the air content in its air sacs.
• To descend, the fish decreases the air content, making their body denser compared to water.
• Ensuring a balanced density allows the fish to "hover" at a certain depth without actively swimming.

Thus, buoyancy control is a vital adaptative mechanism that helps fish conserve energy and respond to environmental changes swiftly.

Density Calculation

Understanding density is key to solving the problem of how much air a fish needs to maintain buoyancy. Density is defined as mass per unit volume, given by the formula \( \rho = \frac{m}{V} \), where \( \rho \) is density, \( m \) is mass, and \( V \) is volume.
For the fish, the initial density is 1.08 \, \mathrm{g/cm}^3, which is higher than the water density (1.00 \, \mathrm{g/cm}^3). This means the fish needs to reduce its density to match the water. This reduction is achieved by changing the volume of air within its body while the mass remains nearly constant.
Since air has negligible density compared to both the fish and water, increasing the air volume will significantly lower the overall average density of the fish's body. By formulating the problem, we express the density change with the equation:

• Desired density (final) \( \rho_f = 1.00 \, \mathrm{g/cm}^3 \)
• Initial density of fish \( \rho_i = 1.08 \, \mathrm{g/cm}^3 \)
• The fraction of volume made up of air \( V_f \) is adjusted to achieve this density change.

With this equation in place, we can find how much air is needed to reach the desired equilibrium at 1.00 \, \mathrm{g/cm}^3.

Volume and Density Relationship

The relationship between volume and density is pivotal in determining how much air fish should add to reach neutral buoyancy. Since density increases with mass and decreases with volume, altering one affects the others.
To successfully change their buoyancy, fish play with the volume part of the density equation while their mass largely stays the same.

• If the air volume within the fish increases (at near negligible mass), the overall volume of the fish body increases, thus decreasing its density.
• This adjustment happens within the fish's air sacs, slightly expanding the body volume without changing its total mass.

Mathematically, this realignment can be expressed as:
If a fish's body initially has a volume, \( V_0 \), it needs to adjust this through air expansion to achieve \( V_0 + V_f \) to balance its density to that of the water. In the context of the original exercise, this means calculating the necessary \( V_f \) that will let the fish float seamlessly at the desired depth, aligning exactly with the water's density.