Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 12: Fluid Mechanics

Problem 29

Chapter 12, Problem 29

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.

Verified step by step guidance

First, understand that the weight of the ore sample in air (17.50 N) is the gravitational force acting on it, which can be expressed as \( F_{\text{gravity}} = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity (approximately 9.81 m/s²).

Next, recognize that when the sample is immersed in water, it experiences a buoyant force. The tension in the cord (11.20 N) is the apparent weight of the sample in water. The buoyant force can be calculated using the equation \( F_{\text{buoyant}} = F_{\text{gravity}} - F_{\text{tension}} \).

The buoyant force is also equal to the weight of the water displaced by the sample, which can be expressed as \( F_{\text{buoyant}} = \rho_{\text{water}} V g \), where \( \rho_{\text{water}} \) is the density of water (approximately 1000 kg/m³) and \( V \) is the volume of the sample. Use this equation to solve for the volume \( V \).

Once the volume \( V \) is found, calculate the mass of the sample using its weight in air: \( m = \frac{F_{\text{gravity}}}{g} \).

Finally, determine the density of the sample using the formula \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume of the sample.

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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. This force is equal to the weight of the fluid displaced by the object. In this problem, the difference in weight of the ore sample in air and in water is due to the buoyant force acting on it, which helps in determining the volume of the sample.

Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid that the object displaces. This principle is crucial for calculating the volume of the ore sample, as the difference in tension when the sample is submerged provides the weight of the displaced water.

Density

Density is defined as mass per unit volume and is a measure of how much matter is packed into a given space. To find the density of the ore sample, we need to calculate its mass from the weight in air and then divide it by the volume obtained using the buoyant force. This will give us the sample's density, which is a key property in material identification.

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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

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