Problem 38 - Physics

Question

A duck is floating on a lake with \(25 \%\) of its volume beneath the water. What is the average density of the duck?

Step-by-Step Solution

Answer

The average density of the duck is \(250 \text{ kg/m}^3\).

Step 1: Understanding the Problem

The duck is floating, which indicates it is in equilibrium. This means that the upward buoyant force equals the downward gravitational force. We need to find the average density of the duck given that 25% of its volume is submerged.

Step 2: Applying the Principle of Flotation

According to the principle of flotation, the buoyant force equals the weight of the water displaced by the submerged volume of the duck. This is expressed as:\[ V_{sub} \cdot \rho_{water} \cdot g = V_{total} \cdot \rho_{duck} \cdot g \]Where \( V_{sub} \) is the submerged volume of the duck, \( \rho_{water} \) is the density of water, \( V_{total} \) is the total volume of the duck, and \( \rho_{duck} \) is the duck's density.

Step 3: Relating Submerged Volume to Total Volume

Given that 25% of the duck's volume is submerged, we have:\[ V_{sub} = 0.25 \times V_{total} \]

Step 4: Solving for the Duck's Density

Substituting \( V_{sub} = 0.25 \times V_{total} \) into the flotation equation gives:\[ 0.25 \cdot V_{total} \cdot \rho_{water} \cdot g = V_{total} \cdot \rho_{duck} \cdot g \]Canceling \( V_{total} \) and \( g \) from both sides, we get:\[ 0.25 \cdot \rho_{water} = \rho_{duck} \]

Step 5: Calculating the Average Density of the Duck

The density of water is approximately \( 1000 \text{ kg/m}^3 \). Using this:\[ \rho_{duck} = 0.25 \times 1000 \]\[ \rho_{duck} = 250 \text{ kg/m}^3 \]

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume. It is crucial for understanding why objects float or sink. The formula for density is given by \[\rho = \frac{m}{V}\]where \(\rho\) is the density, \(m\) is the mass, and \(V\) is the volume. An object's density determines whether it will float or sink in a fluid.
If the density of the object is less than the fluid's density, it will float; otherwise, it will sink.
For our duck, the average density was calculated as \( 250 \text{ kg/m}^3 \), which is less than that of water, allowing it to float.

Flotation Principle

The flotation principle is fundamental in understanding buoyancy, the upward force that allows objects to float. According to this principle: - The buoyant force acting on a submerged object is equal to the weight of the fluid displaced by the object.
- For an object floating on a fluid, this buoyant force equals the gravitational force pulling the object downward.
The duck floats because the buoyant force from the water balances the gravitational force. This principle is expressed mathematically to find the density of the duck by calculating the weight of the displaced water, providing a key clue in understanding flotation dynamics.

Submerged Volume

In buoyancy problems, the submerged volume of an object is the volume below the surface of the fluid. For our floating duck, 25% of its total volume is submerged.
This part of the volume is crucial as it determines how much fluid the object displaces, which, in turn, affects the buoyant force. The submerged volume is connected to density through the equation:\[V_{sub} = 0.25 \times V_{total}\]This tells us that only a fraction of the duck's body displaces water, directly linking to its ability to float. The more an object is submerged, the higher its density relative to the fluid.

Equilibrium in Fluids

Equilibrium in fluids refers to a state where forces acting on an object are balanced. Two primary forces generally act on floating bodies: - Gravitational force pulling downward - Buoyant force pushing upward When these two forces are equal, the object is in equilibrium, neither sinking nor rising in the fluid.
For the floating duck, equilibrium is reached when the downward gravitational force matches the upward buoyant force. This balance is what keeps the duck afloat, and it's tied directly to the density and submerged volume.
Understanding equilibrium helps clarify why some objects, despite varying sizes and shapes, can float on water.