Question

A Canada goose floats with \(25 \%\) of its volume below water. What is the average density of the goose?

Short answer

The average density of the goose is 250 kg/m³.

Step-by-step solution

Understand Buoyancy

According to the principle of buoyancy (Archimedes' principle), the fraction of an object's volume submerged is equal to the ratio of the object's average density to the density of the fluid. In this case, water.

Identify Given Values

We know that 25% of the goose's volume is submerged. Thus, the fraction of the volume submerged is 0.25. The density of water is approximately 1000 kg/m³.

Setup the Equation using Buoyancy Principle

According to the principle, the ratio of the goose's average density \( \rho_{goose} \) to the water's density \( \rho_{water} \) is equal to the submerged volume fraction. Thus, \( \rho_{goose} / \rho_{water} = 0.25 \).

Solve for Goose's Density

Since we know the density of water \( \rho_{water} \), which is 1000 kg/m³, we can solve for the density of the goose \( \rho_{goose} \). Multiply both sides by \( \rho_{water} \) to get: \( \rho_{goose} = 0.25 \times 1000 \).

Perform Calculation

Calculate \( \rho_{goose} = 250 \) kg/m³. Therefore, the average density of the goose is 250 kg/m³.

Key concepts

Archimedes' Principle

Archimedes' principle is an essential concept to understand when exploring buoyancy. It states that any object wholly or partially submerged in a fluid experiences an upward force equal to the weight of the fluid that is displaced by the object. This principle helps us to determine whether an object will float or sink, based on how much fluid it displaces when submerged. When an object is floating, the force of buoyancy is equal to the gravitational force acting on it.
This principle can be expressed with the equation \( F_b = \rho \cdot V \cdot g \), where \( F_b \) is the buoyant force, \( \rho \) is the fluid density, \( V \) is the volume of fluid displaced, and \( g \) is the acceleration due to gravity. Understanding this principle gives insight into how objects behave in different fluids and helps us solve problems in fluid mechanics.

Density

Density is a measure of how much mass is contained within a given volume. It's represented with the equation \( \rho = \frac{m}{V} \), where \( \rho \) is density, \( m \) is mass, and \( V \) is volume. Density is a critical factor in determining whether an object will float or sink, especially in fluid mechanics.
In the context of Archimedes' principle, the object's density compared to the density of the fluid determines the fraction of the object that will be submerged. If an object's density is less than the fluid's, it will float. If it is greater, it will sink.

• An object with a density equal to the fluid will neither sink nor float but will remain suspended.
• A floating object's density can be calculated using the fraction of its submerged volume, as illustrated with the floating goose example.

Floating Objects

Floating objects have to obey certain physical laws, among them Archimedes' principle and the concept of buoyancy. When considering objects floating in water, the object's density plays a significant role in determining how much of it will be submerged. For a floating object, the buoyant force is in equilibrium with the object's weight.
In the example of the Canada goose, it can float because its average density is less than the density of water. The fraction of its body submerged, which is \(25\%\), tells us the relationship between their densities. Such calculations are crucial in designing ships and other buoyant objects to ensure that they float as expected without sinking.

Fluid Mechanics

Fluid mechanics is the study of how fluids (liquids and gases) behave and interact with forces. It covers the concepts of buoyancy, pressure, and flow. One of the key aspects of fluid mechanics is understanding how fluids exert force on objects, and hence why some objects float while others sink.
The behavior of submersion and buoyancy can be analyzed through fluid properties like density and volume. Principles like those discovered by Archimedes help physicists and engineers predict and utilize these fluid behaviors in practical applications. From synchronizing water distribution systems to designing vessels that travel over and under water, understanding fluid mechanics is vital in various engineering fields.