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Ch 12: Fluid Mechanics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 12: Fluid MechanicsProblem 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.
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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.
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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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Related Practice
Textbook Question

Exploring Venus. The surface pressure on Venus is 92 atm, and the acceleration due to gravity there is 0.894g. In a future exploratory mission, an upright cylindrical tank of benzene is sealed at the top but still pressurized at 92 atm just above the benzene. The tank has a diameter of 1.72 m, and the benzene column is 11.50 m tall. Ignore any effects due to the very high temperature on Venus. What force does the Venusian atmosphere exert on the outside surface of the bottom of the tank?

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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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Textbook Question

An ore sample weighs 17.50 N in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 11.20 N. Find the total volume and the density of the sample.

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Textbook Question

A rock has mass 1.80 kg. When the rock is suspended from the lower end of a string and totally immersed in water, the tension in the string is 12.8 N. What is the smallest density of a liquid in which the rock will float?

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Textbook Question

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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