Buoyancy Of Cork Calculating Immersed Fraction In Water

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In the realm of physics, the principle of buoyancy plays a pivotal role in understanding why objects float or sink in fluids. This fundamental concept, rooted in Archimedes' principle, governs the behavior of objects immersed in liquids or gases. In this comprehensive exploration, we delve into the intricacies of buoyancy by examining a classic scenario: a piece of cork floating in water. Cork, renowned for its low density, presents an intriguing case study for applying the principles of buoyancy. Our objective is to meticulously calculate the fraction of the cork that becomes submerged when it gracefully floats in water. This endeavor not only reinforces our grasp of buoyancy but also showcases the practical application of physics in everyday phenomena. Through a step-by-step analysis, we will unravel the interplay between the buoyant force, the weight of the cork, and the density of both the cork and water. Join us as we embark on this captivating journey into the world of fluid dynamics and discover the secrets behind the flotation of cork.

Buoyancy, a cornerstone of fluid mechanics, is the upward force exerted by a fluid that opposes the weight of an immersed object. This phenomenon arises from the pressure difference experienced by the object at varying depths within the fluid. According to Archimedes' principle, the buoyant force is precisely equal to the weight of the fluid displaced by the object. To fully grasp buoyancy, it's crucial to understand the interplay between several key factors: the object's weight, its volume, and the density of both the object and the fluid it's immersed in. The object's weight, a manifestation of gravity's pull, acts downward, while the buoyant force acts upward, counteracting the weight. The volume of the object determines the amount of fluid it displaces, directly influencing the magnitude of the buoyant force. Density, a measure of mass per unit volume, is a critical factor in determining whether an object floats or sinks. If the object's density is less than the fluid's density, the buoyant force will exceed the object's weight, causing it to float. Conversely, if the object's density is greater, it will sink. In the specific case of cork floating in water, the cork's lower density allows it to displace an amount of water whose weight equals the cork's weight, resulting in a state of equilibrium where the cork floats partially submerged. This balance between weight and buoyant force is the essence of buoyancy and the key to understanding why objects behave as they do in fluids. For a comprehensive understanding, we need to define buoyant force (\textbf{F_B}), weight of the object (W\textbf{W}), volume of the object (V\textbf{V}), density of the object (\textbf{\rho_o}), and density of the fluid (\textbf{\rho_f}). The buoyant force is given by $\textbf{F_B = $\rho_f$ V_d g}$, where $\textbf{V_d}$ is the volume of the fluid displaced and $\textbf{g}$ is the acceleration due to gravity. The weight of the object is $\textbf{W = $\rho_o$ V g}$. For an object to float, $\textbf{F_B = W}$.

Our focus now shifts to a specific problem that will allow us to apply the principles of buoyancy. We are presented with a piece of cork, a material known for its lightness and buoyancy. The cork has a density of 0.25 g/cm³, indicating that it is significantly less dense than water, which has a density of approximately 1 g/cm³. The cork's mass is given as 20 g, a crucial piece of information that will enable us to determine its volume. The core question we aim to answer is: what fraction of the cork's volume will be immersed when it floats in water? This question delves into the heart of buoyancy, requiring us to calculate the equilibrium point where the buoyant force acting on the submerged portion of the cork precisely balances the weight of the entire cork. To solve this problem, we will need to employ Archimedes' principle and the relationships between density, volume, and mass. By carefully analyzing the forces at play and utilizing the given data, we can quantitatively determine the fraction of the cork that will reside below the water's surface. This exercise not only enhances our understanding of buoyancy but also demonstrates how physics can be used to predict and explain real-world phenomena. The problem essentially asks us to find the ratio of the volume of the cork submerged in water to the total volume of the cork.

To solve this problem, we will follow a step-by-step approach, utilizing the principles of buoyancy and the given information. First, we need to determine the volume of the cork. We know the density (ρ\rho) and the mass (m) of the cork, and we can use the relationship:

\textbf{Density (\rho) = Mass (m) / Volume (V)}$

Rearranging the formula to solve for volume, we get:

\textbf{Volume (V) = Mass (m) / Density (\rho)}$

Substituting the given values:

V = 20 g / 0.25 g/cm³ = 80 cm³\textbf{V = 20 g / 0.25 g/cm³ = 80 cm³}$

Now we know the total volume of the cork is 80 cm³. Next, we consider the buoyant force. When the cork floats, the buoyant force (\textbf{F_B}) acting on it must equal its weight (W\textbf{W}). The buoyant force is the weight of the water displaced by the submerged part of the cork. Let V_immersed be the volume of the cork immersed in water. The density of water (ρwater\rho_water) is approximately 1 g/cm³. The weight of the displaced water, which is the buoyant force, is given by:

$\textbf{F_B = V_immersed × ρwater\rho_water × g}$

Where g is the acceleration due to gravity, but since we're equating forces, we can work in terms of mass and not weight (which includes g). The weight of the cork is its mass (20 g). So, for the cork to float:

Weight of cork = Buoyant force\textbf{Weight of cork = Buoyant force}$

\textbf{20 g = V_immersed × 1 g/cm³}$

Solving for V_immersed:

\textbf{V_immersed = 20 cm³}$

Now, to find the fraction of the cork that is immersed, we divide the immersed volume by the total volume:

\textbf{Fraction immersed = V_immersed / V}$

Fraction immersed = 20 cm³ / 80 cm³ = 0.25\textbf{Fraction immersed = 20 cm³ / 80 cm³ = 0.25}$

Therefore, 0.25 or 1/4 of the cork is immersed when it floats in water.

In conclusion, our analysis reveals that 0.25, or one-quarter, of the cork's volume is immersed when it floats in water. This result elegantly demonstrates the principles of buoyancy at work. The cork, being less dense than water, displaces a volume of water whose weight equals the cork's own weight. This equilibrium is what allows the cork to float, and the fraction immersed is a direct consequence of the density difference between the cork and the water. This exercise not only reinforces our understanding of buoyancy but also highlights the power of physics in explaining everyday phenomena. The step-by-step calculation, from determining the cork's volume to equating the buoyant force with its weight, showcases the practical application of physical principles in solving real-world problems. The example of cork floating in water serves as a clear and concise illustration of buoyancy, a fundamental concept in fluid mechanics that governs the behavior of objects in fluids. Understanding buoyancy is essential in various fields, from naval architecture to meteorology, and this analysis provides a solid foundation for further exploration of fluid dynamics. By calculating the immersed fraction of the cork, we've gained a deeper appreciation for the delicate balance of forces that dictates whether an object floats or sinks. This understanding can be extended to a wide range of scenarios, making the principles of buoyancy a valuable tool in scientific inquiry and engineering applications.