Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 49

Chapter 14, Problem 49

The average density of the body of a fish is 1080 kg/m³ . To keep from sinking, a fish increases its volume by inflating an internal air bladder, known as a swim bladder, with air. By what percent must the fish increase its volume to be neutrally buoyant in fresh water? The density of air at 20°C is 119 kg/m³.

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Determine the condition for neutral buoyancy: For the fish to be neutrally buoyant, the average density of the fish (including the swim bladder) must equal the density of fresh water, which is approximately 1000 kg/m³.

Express the average density of the fish after inflating the swim bladder: Let the initial volume of the fish be \( V \) and its mass be \( m \). The initial density is \( \rho_{\text{fish}} = \frac{m}{V} \). After inflating the swim bladder, the new volume becomes \( V' = V + \Delta V \), where \( \Delta V \) is the increase in volume. The new average density is \( \rho_{\text{new}} = \frac{m}{V + \Delta V} \).

Set the new average density equal to the density of fresh water: \( \rho_{\text{new}} = \rho_{\text{water}} \). Substituting, \( \frac{m}{V + \Delta V} = 1000 \). Rearrange to solve for \( \Delta V \): \( \Delta V = \frac{m}{1000} - V \).

Relate the mass of the fish to its initial density and volume: Since \( m = \rho_{\text{fish}} \cdot V \), substitute this into the equation for \( \Delta V \): \( \Delta V = \frac{\rho_{\text{fish}} \cdot V}{1000} - V \). Factor out \( V \): \( \Delta V = V \left( \frac{\rho_{\text{fish}}}{1000} - 1 \right) \).

Calculate the percentage increase in volume: The percentage increase is given by \( \frac{\Delta V}{V} \times 100 \). Substituting \( \Delta V \), the percentage increase is \( \left( \frac{\rho_{\text{fish}}}{1000} - 1 \right) \times 100 \). Use \( \rho_{\text{fish}} = 1080 \) kg/m³ to find the final percentage.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is determined by the volume of fluid displaced by the object, which is governed by Archimedes' principle. For an object to be neutrally buoyant, the buoyant force must equal the object's weight, allowing it to neither sink nor float.

Density

Density is defined as mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It is a crucial factor in determining whether an object will float or sink in a fluid. In this scenario, the fish's density must be adjusted to match the density of fresh water (approximately 1000 kg/m³) to achieve neutral buoyancy.

Volume Increase for Neutral Buoyancy

To achieve neutral buoyancy, a fish must increase its volume so that its overall density matches that of the surrounding water. This involves inflating the swim bladder with air, which has a much lower density than water. The percentage increase in volume can be calculated by determining the required volume that results in a density equal to that of the water, considering the fish's original density.

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