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 28

Chapter 12, Problem 28

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 65.0-kg woman to be able to stand on it without getting her feet wet?

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First, understand that the problem involves buoyancy, which is the ability of an object to float in a fluid. The slab of ice must displace a volume of water equal to the weight of the woman plus the weight of the ice itself.

Use Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. The buoyant force must equal the total weight of the woman and the ice for the slab to float without submerging.

Calculate the weight of the woman using the formula: \( \text{Weight} = m \cdot g \), where \( m = 65.0 \text{ kg} \) is the mass of the woman and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity.

Determine the density of freshwater, which is approximately \( 1000 \text{ kg/m}^3 \). The volume of water displaced by the ice slab must be equal to the total weight divided by the density of water.

Set up the equation for the volume of the ice slab: \( V = \frac{\text{Weight of woman} + \text{Weight of ice}}{\text{Density of water}} \). Solve for \( V \) to find the minimum volume of the ice slab required to support the woman without submerging.

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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 object's weight. For an object to float, the buoyant force must equal the object's weight. In this scenario, the ice slab must displace a volume of water equal to the combined weight of the slab and the woman to keep her feet dry.

Density

Density is the mass per unit volume of a substance, often expressed in kilograms per cubic meter (kg/m³). The density of ice is less than that of water, which is why ice floats. Understanding the density difference between ice and water is crucial to calculating the volume of ice needed to support additional weight without submerging.

Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This principle helps determine the minimum volume of the ice slab required to support the woman's weight by ensuring the displaced water's weight equals the total weight of the woman and the ice.

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Textbook Question

A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m³ and the tension in the cord is 1120 N. The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

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Textbook Question

A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m³ and the tension in the cord is 1120 N. (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere?

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