Buoyancy & Buoyant Force: Videos & Practice Problems

Buoyancy is the upward force that allows objects to float in a liquid, explained by Archimedes' principle, which states that the buoyant force equals the weight of the displaced liquid. The buoyant force (F_b) can be calculated using the equation: ${\rho }^{l}gV$, where ρ_l is the liquid's density, g is the acceleration due to gravity, and V is the submerged volume. An object sinks if its density is greater than the liquid's, and floats if it is less dense.

Intro to Buoyancy & Buoyant Force

Buoyancy is the phenomenon that allows objects to float or be pushed upward when immersed in a liquid. This occurs due to a force known as the buoyant force, denoted as \( F_b \). The buoyant force arises from a pressure difference between the top and bottom of an object submerged in a liquid. As depth increases, pressure also increases, resulting in a greater force acting on the bottom of the object compared to the top. This difference in pressure leads to a net upward force, which is the buoyant force.

One of the fundamental principles related to buoyancy is Archimedes' principle, which states that the magnitude of the buoyant force is equal to the weight of the liquid displaced by the object. To express this concept mathematically, the buoyant force can be calculated using the equation:

\[ F_b = \rho_{liquid} \cdot g \cdot V_{under} \]

In this equation, \( \rho_{liquid} \) represents the density of the liquid, \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) on Earth), and \( V_{under} \) is the volume of the object that is submerged in the liquid. It is crucial to note that the density used in this equation is that of the liquid, not the object itself. The volume considered is only the portion of the object that is underwater, which can differ from the object's total volume.

Understanding density is also essential, as it is defined as the mass of an object divided by its total volume:

\[ \text{Density} = \frac{mass_{total}}{volume_{total}} \]

In buoyancy problems, it is common to analyze forces using the equilibrium condition, where the sum of all forces acting on an object equals zero. This is often represented as:

\[ \sum F = 0 \]

In many cases, the object will be at rest, leading to a situation where the buoyant force equals the weight of the object:

\[ F_b = mg \]

By substituting the expression for buoyant force into this equation, we can derive relationships between the densities and volumes of the object and the liquid. For example, if an object is fully immersed and then reaches equilibrium with a portion above the water, the volume submerged can be expressed as a percentage of the total volume. If 30% of the object is above water, then 70% is submerged, which is critical for calculations.

In summary, buoyancy is a key concept in fluid mechanics, governed by the principles of pressure differences and Archimedes' principle. Understanding how to calculate buoyant force and the relationships between mass, volume, and density is essential for solving buoyancy-related problems effectively.

Comparing Buoyant Forces

In this buoyancy scenario, we analyze the behavior of two objects—wood and metal—both with the same volume when submerged in water. The key concept here is buoyant force, which is determined by the equation:

\( F_B = \rho \cdot g \cdot V \)

Where \( F_B \) is the buoyant force, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( V \) is the volume of the object submerged in the liquid. Since both objects are in the same water, the density of the liquid and the gravitational acceleration are constant for both, making them equal factors in the equation.

However, the crucial difference lies in the volume of each object that is submerged. While the wood floats, it has a portion of its volume above the water, meaning only part of it contributes to the buoyant force. In contrast, the metal, which sinks, has its entire volume submerged. This leads to the conclusion that the metal experiences a greater buoyant force because all of its volume is underwater, compared to the wood, which has some of its volume exposed.

It's a common misconception to assume that because the wood floats, it must have a greater buoyant force acting on it. In reality, the floating condition indicates that the weight of the wood is balanced by the buoyant force, but this does not mean the buoyant force is greater than that acting on the metal. Instead, the metal's greater submerged volume results in a stronger buoyant force, despite its sinking behavior.

In summary, when evaluating buoyancy, always refer to the equation and consider the volume submerged to accurately determine the buoyant forces acting on different objects. This approach helps clarify the often counterintuitive nature of buoyancy and floating versus sinking objects.

When an object of unknown mass and volume is fully immersed in large oil (800 kg/m³ ) container and released from rest, it stays at rest. Calculate the density of this object.

• A. 800 kg/m³
• B. 80.0 kg/m³
• C. 8000 kg/m³
• D. 8.00 kg/m³

Answer: (A)

Buoyancy / Three Common Cases

Buoyancy is a fundamental concept in fluid mechanics that describes how objects behave when placed in a fluid. The behavior of an object in a fluid is determined by its density relative to the density of the fluid. If an object is denser than the fluid, it will sink; if it is less dense, it will float. This relationship is crucial for understanding buoyant forces, which are the upward forces exerted by a fluid on an object submerged in it.

When an object floats, it displaces a volume of fluid equal to the weight of the object. This is described by the equilibrium condition where the buoyant force (F_b) acting upward equals the weight of the object (mg) acting downward. Mathematically, this can be expressed as:

F_b = mg

In the case of an object partially submerged in a fluid, the volume of fluid displaced is less than the total volume of the object. For example, if an object has a total volume of 100 units and 60 units are submerged, the buoyant force can be calculated based on the volume submerged and the density of the fluid. The density of the object can be inferred from the percentage of its volume that is submerged:

Density of the object = (Percent submerged) × (Density of the fluid)

In scenarios where an object is held underwater by a tension force, the equilibrium condition changes slightly. The forces acting on the object include the buoyant force, the weight of the object, and the tension in the cable. The equilibrium condition can be expressed as:

F_b = mg + T

When an object is fully submerged and remains at rest, it indicates that the density of the object is equal to the density of the fluid. This situation can be described as:

F_b = mg

In this case, the object neither sinks nor floats but remains suspended in the fluid.

For objects that sink, the weight of the object exceeds the buoyant force, leading to a downward acceleration. The forces can be expressed as:

mg > F_b

In this scenario, the object pushes against the bottom of the container, creating a normal force that also contributes to the overall force balance.

To calculate the apparent weight of an object submerged in a fluid, one can use the following relationship:

Normal force = mg - F_b

Where the buoyant force can be calculated using the density of the fluid and the volume of fluid displaced.

In practical applications, such as determining the reading on a scale when an object is submerged, the normal force can be calculated by considering the mass of the object, the density of the fluid, and the volume submerged. This understanding of buoyancy and the forces involved is essential for solving problems related to fluid mechanics.

A block floats with 40% of its volume above water. When you place it on an unknown liquid, it floats with 30% of its volume above. What is the density of the unknown liquid?

• A. 857 kg/m³
• B. 600 kg/m³
• C. 400 kg/m³
• D. 571 kg/m³

Answer: (A)

Is Crown Made of Gold? (Buoyancy)

To determine if a 100-gram crown is made of pure gold, we can use the concept of density. The density of gold is known to be 19.32 grams per cubic centimeter or 19,320 kilograms per cubic meter. To verify the material of the crown, we will calculate its density and compare it to the known density of gold.

The process begins by submerging the crown in water and measuring the tension in the string holding it, which is found to be 0.88 newtons. In this scenario, we can apply the principle of buoyancy, which states that the buoyant force acting on the crown plus the tension in the string equals the weight of the crown. This can be expressed mathematically as:

\[ F_b + T = mg \]

Where:

• F_b = buoyant force
• T = tension in the string (0.88 N)
• m = mass of the crown (0.1 kg, since 100 grams = 0.1 kg)
• g = acceleration due to gravity (approximately 9.8 m/s²)

Rearranging the equation gives us:

\[ F_b = mg - T \]

Substituting the known values, we can express the buoyant force in terms of the density of the liquid (water) and the volume of the crown:

\[ F_b = \rho_{liquid} \cdot g \cdot V \]

Where:

• ρ_liquid = density of water (approximately 1,000 kg/m³)
• V = volume of the crown

By equating the two expressions for buoyant force, we can derive the volume of the crown:

\[ \rho_{liquid} \cdot g \cdot V = mg - T \]

Substituting the values we have:

\[ 1,000 \cdot 9.8 \cdot V = 0.1 \cdot 9.8 - 0.88 \]

Solving for V gives us:

\[ V = \frac{0.1 \cdot 9.8 - 0.88}{1,000 \cdot 9.8} \]

Next, we can find the density of the crown using the formula:

\[ \rho_{object} = \frac{m}{V} \]

Substituting the mass of the crown and the calculated volume will yield the density of the crown. If this density is equal to or very close to 19,320 kg/m³, we can conclude that the crown is made of gold. However, if the calculated density is significantly lower, as in this case where it was found to be approximately 980 kg/m³, we can confidently state that the crown is not made of gold.

In summary, through the application of buoyancy principles and density calculations, we determined that the crown is not gold, as its density is far below that of pure gold.

An 8,000 cm³ block of wood is fully immersed in a deep water tank, then tied to the bottom. When the block is released and reaches equilibrium, you measure the tension on the string to be 12 N. What is the density of the wood?

• A. 1153 kg/m³
• B. 988 kg/m³
• C. 847 kg/m³
• D. 960 kg/m³

Answer: (C)

Maximum Load on Floating Board

Buoyancy is a fundamental concept in physics that describes how objects behave when placed in a fluid. In this example, we explore how to determine the maximum mass that can be placed on a wooden board floating on water without causing it to become submerged. This scenario is common in buoyancy problems, where the goal is to keep an object dry while maximizing the load it can support.

To analyze this situation, we start by recognizing that the board (denoted as m₁) and the additional mass placed on it (denoted as m₂) must be in equilibrium. This means that the sum of the forces acting on the board must equal zero, expressed mathematically as F = m a = 0. The forces acting on the board include its own weight (m₁g), the weight of the mass on top (m₂g), and the buoyant force (F_B) exerted by the water.

When the board is at its maximum load without getting wet, it is fully submerged, meaning the volume of the board underwater equals its total volume. The buoyant force can be calculated using the formula:

F_B = ρ_liquid g V_under

Here, ρ_liquid is the density of the liquid (in this case, saltwater, which has a density of approximately 1030 kg/m³), g is the acceleration due to gravity, and V_under is the volume of the submerged part of the board.

To find the maximum mass m₂ that can be added without submerging the board, we set the upward buoyant force equal to the total downward force:

F_B = m₁g + m₂g

Substituting the expression for buoyant force, we have:

ρ_liquid g V_under = m₁g + m₂g

Since gravity appears in all terms, it can be canceled out, simplifying our equation to:

ρ_liquid V_under = m₁ + m₂

Next, we need to express the mass of the board in terms of its density and volume:

m₁ = ρ₁ V₁

Substituting this into our equation gives:

ρ_liquid V_under = ρ₁ V₁ + m₂

In this case, the volume of the board is calculated as the area times the height. Given that the area is 1 m² and the height is 0.1 m, the total volume is:

V₁ = 1 m² × 0.1 m = 0.1 m³

Substituting the known values into the equation allows us to solve for m₂. After performing the calculations, we find that the maximum mass m₂ that can be placed on the board without it getting wet is 33 kg.

This example illustrates the importance of understanding buoyancy and equilibrium in fluid mechanics, as well as the application of fundamental physics principles to solve real-world problems.

You want to build a large storage container, with outer walls and an open top, as shown, so that you can load things into it, while it floats on fresh water, without any water getting inside. If the bottom face of the container measures 3.0 m by 8.0 m, how high should the side walls be, such that the combined mass of container and inside load is 100,000 kg?

• A. 5.0 m
• B. 2.9 m
• C. 9.1 m
• D. 4.2 m

Answer: (D)

Here’s what students ask on this topic:

What is buoyancy and how does it work?

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is explained by Archimedes' principle, which states that the buoyant force (F_b) on an object is equal to the weight of the fluid displaced by the object. The principle can be mathematically expressed as:

$F_b=\rho gV$

where $\rho$ is the density of the fluid, $g$ is the acceleration due to gravity, and $V$ is the volume of the fluid displaced. An object will float if its density is less than the fluid's density and sink if it is greater.

How do you calculate the buoyant force on an object?

The buoyant force (F_b) on an object can be calculated using the equation derived from Archimedes' principle:

$F_b=\rho gV$

where $\rho$ is the density of the liquid, $g$ is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and $V$ is the volume of the object submerged in the liquid. This equation helps determine the upward force exerted by the liquid on the object.

What is Archimedes' principle and why is it important?

Archimedes' principle states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object. This principle is crucial because it explains why objects float or sink in fluids. It can be mathematically expressed as:

$F_b=\rho gV$

where $\rho$ is the fluid's density, $g$ is the acceleration due to gravity, and $V$ is the volume of the displaced fluid. This principle is fundamental in fluid mechanics and has applications in engineering, shipbuilding, and various scientific fields.

How does the density of an object affect its buoyancy?

The density of an object determines whether it will float or sink in a fluid. If the object's density is less than the fluid's density, the object will float because the buoyant force is greater than the object's weight. Conversely, if the object's density is greater than the fluid's density, the object will sink because its weight exceeds the buoyant force. This relationship can be understood through Archimedes' principle, which states that the buoyant force is equal to the weight of the displaced fluid. Therefore, the relative densities of the object and the fluid are key factors in determining buoyancy.

What are the common cases of buoyancy and how do they differ?

There are three common cases of buoyancy:

1. Floating: The object is less dense than the fluid, so it floats with part of its volume above the fluid. The buoyant force equals the object's weight.

2. Neutral Buoyancy: The object's density is equal to the fluid's density, so it remains fully submerged but does not sink or rise. The buoyant force equals the object's weight.

3. Sinking: The object is denser than the fluid, so it sinks to the bottom. The buoyant force is less than the object's weight, and an additional normal force from the bottom surface balances the forces.

These cases illustrate how the density of the object relative to the fluid determines its buoyant behavior.