Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 12: Fluid Mechanics

Problem 27

Chapter 12, Problem 27

A 950-kg cylindrical can buoy floats vertically in sea-water. The diameter of the buoy is 0.900 m. Calculate the additional distance the buoy will sink when an 80.0-kg man stands on top of it.

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First, determine the volume of the cylindrical buoy that is submerged when the buoy is floating without the man. Use the formula for the volume of a cylinder: \( V = \pi r^2 h \), where \( r \) is the radius of the buoy and \( h \) is the height submerged.

Calculate the buoyant force acting on the buoy using Archimedes' principle, which states that the buoyant force is equal to the weight of the displaced water. The buoyant force can be expressed as \( F_b = \rho_{water} V_{submerged} g \), where \( \rho_{water} \) is the density of sea-water, \( V_{submerged} \) is the submerged volume, and \( g \) is the acceleration due to gravity.

Determine the total weight acting on the buoy when the man stands on it. This is the sum of the weight of the buoy and the weight of the man: \( W_{total} = m_{buoy} g + m_{man} g \), where \( m_{buoy} \) is the mass of the buoy and \( m_{man} \) is the mass of the man.

Set the total weight equal to the new buoyant force to find the new submerged volume: \( W_{total} = \rho_{water} V_{new} g \). Solve for \( V_{new} \), the new submerged volume.

Calculate the additional distance the buoy will sink by finding the difference in the submerged height before and after the man stands on it. Use the relationship between the submerged volume and height: \( \Delta h = \frac{V_{new} - V_{submerged}}{\pi r^2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object submerged in it, counteracting the weight of the object. According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced by the object. Understanding buoyancy is crucial for determining how much the buoy will sink when additional weight is added.

Archimedes' Principle

Archimedes' Principle states that any object submerged in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. This principle is essential for calculating the change in the buoy's submerged volume when the man stands on it, as it allows us to equate the additional weight to the weight of the displaced water.

Volume and Density

The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height. Density, defined as mass per unit volume, is crucial for determining the weight of the displaced seawater. These concepts help in calculating the additional submerged volume of the buoy when the man's weight is added.

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