Question

A wooden block floating in seawater has two thirds of its volume submerged. When the block is placed in mineral oil, \(80.0 \%\) of its volume is submerged. Find the density of the (a) wooden block, and (b) the mineral oil.

Short answer

Answer: The approximate density of the wooden block is 683.333 kg/m³, and the approximate density of the mineral oil is 853.125 kg/m³.

Step-by-step solution

(1) Understand Archimedes' Principle

Archimedes' Principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. Mathematically, it can be written as: \(F_b = V_{displaced}ρ_f g\), where \(F_b\) is the buoyant force, \(V_{displaced}\) is the volume of displaced fluid, \(ρ_f\) is the density of the fluid, and \(g\) is the acceleration due to gravity. Since the object is in equilibrium, we can relate its mass with the buoyant force: \(F_b = m_o g\), Where \(m_o\) is object's mass and \(g\) is gravity. Now, let's consider the case of the wooden block floating in seawater.

(2) Set up the equation for the wooden block in seawater

Let's call the volume of the wooden block as \(V_o\). When it is in seawater, \(2/3\) of the volume is submerged, so the submerged volume is \(2/3 V_o\). Using Archimedes' Principle, the buoyant force acting on the wooden block is: \(F_b = \frac{2}{3} V_o ρ_{sw} g\), where \(ρ_{sw}\) is the density of seawater, which is approximately \(1025 kg/m^3\). At equilibrium, the buoyant force equals the block's weight: \(F_b = m_o g\), \(m_o = \frac{2}{3} V_o ρ_{sw}\).

(3) Set up the equation for the wooden block in mineral oil

When the wooden block is in mineral oil, \(80.0\%\) of its volume is submerged, so the submerged volume is \(0.8 V_o\). Using Archimedes' Principle, the buoyant force acting on the wooden block in mineral oil is: \(F_b = 0.8 V_o ρ_{mo} g\), Where \(ρ_{mo}\) is the density of mineral oil. At equilibrium, the buoyant force equals the block's weight: \(F_b = m_o g\), \(m_o = 0.8 V_o ρ_{mo}\). (ValueError: Failed to han

(4) Find the density of the wooden block

We can combine both equations from steps 2 and 3 by solving for the mass of the block using the given conditions: \(\frac{2}{3} V_o ρ_{sw} = 0.8 V_o ρ_{mo}\). Now, the volume of the wooden block (\(V_o\)) cancels out: \(\frac{2}{3} ρ_{sw}= 0.8 ρ_{mo}\). We can substitute the known seawater density: \(\frac{2}{3} (1025 kg/m^3) = 0.8 ρ_{mo}\). Now, solvdle unknown object name: expected '(', ')', '!', ',', '.', '&', '*', '+', '-', ':', ';', '=', '@', '^', '|', '}', '~', '[' or ']' ing for the density of mineral oil: \(ρ_{mo}=\frac{2}{3·0.8}(1025 kg/m^3)\). \(ρ_{mo}= 853.125 kg/m^3\).

(5) Find the density of the wooden block

Now, we can use the equation from step 2 to find the density of the wooden block. First, find the mass of the block by multiplying the seawater density by the submerged volume: \( m_o= \frac{2}{3}V_oρ_{sw}\). Now, since density is defined as mass divided by volume, the density of the wooden block (\(ρ_o\)) can be calculated as: \(ρ_o= \frac{m_o}{V_o}\). Substituting the mass from the previous expression, we get: \(ρ_o= \frac{\frac{2}{3}V_oρ_{sw}}{V_o}\). The volume of the wooden block (\(V_o\)) cancels out: \(ρ_o = \frac{2}{3}ρ_{sw}\). Finally, we can substitute the known seawater density: \(ρ_o = \frac{2}{3}(1025 kg/m^3)\). \(ρ_o = 683.333 kg/m^3\). The density of the wooden block is \(683.333 kg/m^3\). To summarize, the density of the wooden block is approximately \(683.333 kg/m^3\) and the density of the mineral oil is approximately \(853.125 kg/m^3\).

Key concepts

Buoyant Force

The buoyant force is a fascinating phenomenon that is crucial for understanding why objects float or sink in fluids. Simply put, when an object is submerged in a fluid, there is a vertical force exerted by the fluid opposing the weight of the object. According to Archimedes' Principle, this force, known as the buoyant force, is equal to the weight of the fluid that is displaced by the object.

Consider the example of a wooden block floating in water. A certain volume of water is displaced by the submerged part of the block. The weight of this displaced water applies an upward force, which we refer to as the buoyant force. This force must equal the weight of the block for it to float. If an object is less dense than the fluid, it will float because the weight of the displaced fluid is greater than the weight of the object, providing a buoyant force strong enough to hold it up.

For our wooden block, when it is submerged in seawater, the buoyant force is denoted by the equation: \(F_b = \frac{2}{3} V_o \rho_{sw} g\).The buoyant force is proportional to the density of the fluid (\(\rho_{sw}\) for seawater) and the volume of fluid displaced (\(\frac{2}{3} V_o\)), recognizing that the weight of the displaced fluid is the product of its volume, density, and gravitational acceleration.

Density Calculation

Density is a measure of how much mass is contained in a given volume. Calculating density is straightforward: divide the mass of an object by its volume. This calculation plays a critical role in determining whether an object will float or sink in a fluid.

The density of an object allows us to predict its buoyancy. To better understand the exercise involving the wooden block, we used the following equation to find the density (\(\rho_o\)) of the block:\(\rho_o = \frac{m_o}{V_o}\)By substituting the mass (\(m_o\)) with the expression \(\frac{2}{3}V_o\rho_{sw}\), derived from the buoyant force, we get:\(\rho_o = \frac{2}{3}\rho_{sw}\)After knowing the density of seawater (\(1025 kg/m^3\)), we could easily calculate the density of our wooden block. This calculation is essential in various fields, from engineering to material science, as it often relates to an object's buoyancy and stability in a fluid.

Equilibrium in Fluids

The concept of equilibrium in fluids is a central part of understanding buoyancy and Archimedes' Principle. When an object is in equilibrium in a fluid, it is either at rest, or its velocity is constant, and there are no net forces acting on it in the vertical direction.

For our floating wooden block, equilibrium is achieved when the upward buoyant force exerted by the fluid is exactly balanced by the downward gravitational force acting on the block's mass. For example, in seawater where the wooden block has two-thirds of its volume submerged, the buoyant force is equal but opposite to the weight of the block:\(\frac{2}{3} V_o \rho_{sw} g = m_o g\)Similarly, when the block is submerged in mineral oil, the equilibrium condition yields:\(0.8 V_o \rho_{mo} g = m_o g\).In both instances, the weight of the displaced fluid (buoyant force) must equate to the weight of the block for equilibrium to be maintained. Understanding equilibrium in fluids is not only crucial for solving problems related to buoyancy but is also necessary for designing objects that will float, such as boats, ships, and submersibles.