Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 27

Chapter 17, Problem 27

An aluminum bar has the desired length when at 12°C. How much stress is required to keep it at this length if the temperature increases to 38°C? [See Table 12–1.]

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Identify the key concepts involved: This problem involves thermal expansion and stress. Thermal expansion describes how materials expand or contract with temperature changes, and stress is the force per unit area required to counteract this expansion or contraction.

Write the formula for linear thermal expansion: ΔL = αL₀ΔT, where ΔL is the change in length, α is the coefficient of linear expansion for aluminum (from Table 12-1), L₀ is the original length, and ΔT is the temperature change (T_final - T_initial).

Recognize that the bar is being constrained to its original length, so ΔL = 0. This means the thermal expansion is being counteracted by stress. The relationship between stress (σ), strain (ε), and Young's modulus (Y) is given by σ = Yε.

Relate strain to thermal expansion: Strain is defined as ε = ΔL / L₀. Since ΔL = αL₀ΔT, substitute this into the strain equation to get ε = αΔT.

Combine the equations: Substitute ε = αΔT into σ = Yε to get σ = YαΔT. Use the given temperature change (ΔT = 38°C - 12°C) and the values for Y (Young's modulus for aluminum) and α (coefficient of linear expansion for aluminum) from Table 12-1 to calculate the stress required to keep the bar at its original length.

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Key Concepts

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

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in size or volume in response to changes in temperature. For solids, this is typically characterized by a linear expansion coefficient, which quantifies how much a material expands per degree of temperature increase. In the context of the aluminum bar, understanding thermal expansion is crucial to determine how much the bar would naturally expand when the temperature rises from 12°C to 38°C.

Stress and Strain

Stress is defined as the force applied per unit area within materials, while strain is the deformation experienced by the material in response to that stress. In this scenario, the stress required to maintain the aluminum bar's length at a higher temperature must counteract the natural expansion due to thermal effects. Understanding the relationship between stress and strain is essential for calculating the necessary force to keep the bar at its desired length.

Young's Modulus

Young's modulus is a measure of the stiffness of a material, defined as the ratio of stress to strain in the elastic region of the material's deformation. It provides insight into how much a material will deform under a given stress. In this problem, knowing Young's modulus for aluminum allows us to calculate the stress needed to prevent the bar from expanding beyond its desired length when subjected to increased temperature.

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Determine a formula for the change in surface area of a uniform solid sphere of radius r if its coefficient of linear expansion is α (assumed constant) and its temperature is changed by ∆T.

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If 12.50 mol of helium gas is at 10.0°C and a gauge pressure of 0.350 atm, calculate

(a) the volume of the helium gas under these conditions and

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