Problem 28 - University Physics with Modern Physics

Question

You are preparing some apparatus for a visit to a newly discovered planet Caasi having oceans of glycerine and a surface acceleration due to gravity of 4.15 \(\mathrm{m} / \mathrm{s}^{2} .\) If your appara- tus floats in the oceans on earth with 25.0\(\%\) of its volume submerged, what percentage will be submerged in the glycerine oceans of Caasi?

Step-by-Step Solution

Answer

47.1% of its volume will be submerged in the glycerine oceans of Caasi.

Step 1: Understand Archimedes' Principle

According to Archimedes' Principle, the buoyant force is equal to the weight of the fluid displaced by an object. On Earth, the buoyant force is determined by the volume of water displaced and the weight of the water displaced is equal to the weight of the apparatus submerged.

Step 2: Calculate the Weight of the Apparatus

Since 25% of the volume of the apparatus is submerged on Earth, the weight of the apparatus must be equal to the weight of 25% of its volume of water. Letting \( V \) be the total volume of the apparatus and \( \rho_{water} \) the density of water, the weight \( W \) of the apparatus is \( W = 0.25V \rho_{water} g_{earth} \).

Step 3: Calculate Buoyant Force on Caasi

On Caasi, the apparatus will float when the weight of the apparatus is equal to the weight of the glycerine displaced. This means \( W = V_s \rho_{glycerine} g_{caasi} \), where \( V_s \) is the submerged volume, \( \rho_{glycerine} \) is the density of glycerine, and \( g_{caasi} \) is the gravitational acceleration on the planet Caasi.

Step 4: Set the Equations Equal and Solve for Submerged Volume

Set the weight equations from Earth and Caasi equal: \( 0.25V \rho_{water} g_{earth} = V_s \rho_{glycerine} g_{caasi} \). Solve for \( V_s \), the submerged volume percentage in glycerine: \( V_s = 0.25V \frac{\rho_{water} g_{earth}}{\rho_{glycerine} g_{caasi}} \).

Step 5: Substitute Known Values and Calculate Submersion Percentage

The density of water \( \rho_{water} \) is approximately 1000 kg/m³, the density of glycerine \( \rho_{glycerine} \) is approximately 1260 kg/m³, and \( g_{earth} = 9.81 \mathrm{m/s}^2 \). Using these values: \( V_s = 0.25 \frac{1000 \times 9.81}{1260 \times 4.15} V \approx 0.471V \). This means approximately 47.1% of the apparatus’ volume will be submerged in Caasi.

Key Concepts

Buoyant Force

The buoyant force is a critical concept in understanding why objects float or sink in a fluid. It is based on Archimedes' Principle, which 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. This means that if an object is placed in water, it will experience an upward force that counteracts the downward pull of gravity. This upward force is what we call the buoyant force.

Consider an object floating in water. The portion of the object submerged displaces a certain volume of water. The weight of this displaced water determines the magnitude of the buoyant force. If the object's weight is equal to the buoyant force, the object will float.

• Buoyant Force Formula: The buoyant force can be expressed as \( F_b = \rho_{fluid} \cdot V_{displaced} \cdot g \), where \( \rho_{fluid} \) is the fluid's density, \( V_{displaced} \) is the volume of fluid displaced, and \( g \) is the gravitational acceleration.
• Balance for Floating: For an object to float in equilibrium, its weight must be equal to the buoyant force.

This principle helps us determine how much of an object will be submerged when it is floating on a fluid.

Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by an object due to the gravitational pull from a body like Earth or, in our problem, the planet Caasi. This force is what draws objects toward the center of the planet. Gravitational acceleration affects the weight of objects and, therefore, plays a crucial role in determining buoyancy.

On Earth, gravitational acceleration is approximately 9.81 m/s². However, on different planets, this value can change. For instance, in the problem with planet Caasi, gravitational acceleration is 4.15 m/s². This difference is important in calculating how much an object would weigh on a different planet and affects the buoyant force.

Key Concepts:

• Weight: The weight of an object is given by \( W = m \cdot g \), where \( m \) is the object's mass, and \( g \) is the gravitational acceleration.
• Effect on Buoyancy: Lower gravitational acceleration means the weight of the fluid displaced is less, impacting how much of an object is submerged.

Understanding gravitational acceleration helps us figure out how an object will behave in fluids on different planets.

Fluid Displacement

Fluid displacement is a fundamental concept in the study of buoyancy and is directly related to Archimedes' Principle. When an object is placed in a fluid, it pushes aside or "displaces" some of that fluid. The volume of the fluid displaced determines the buoyant force experienced by the object.

Simply put, the more fluid an object displaces, the greater the buoyant force acting on it. This displacement is why objects partially sink until the displaced fluid's weight equals the weight of the object.

• Volume of Displacement: This volume is equal to the submerged portion of the object.
• Equation Connection: In the equation \( F_b = \rho_{fluid} \cdot V_{displaced} \cdot g \), \( V_{displaced} \) is pivotal in calculating the buoyant force.

This concept explains why different parts of an object can be submerged depending on its density compared to the surrounding fluid. For instance, on Earth, only 25% of an apparatus might be submerged in water, while on another planet, like Caasi, this percentage might be higher or lower due to different gravitational accelerations and fluid densities.