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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: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 17 - Temperature, Thermal Expansion, and the Ideal Gas LawProblema 16a
Capitolo 17, Problema 16a

A uniform rectangular plate of length ℓ and width ω has a coefficient of linear expansion α. Show that, if we neglect very small quantities, the change in area of the plate due to a temperature change ∆T is ∆A = 2αℓω ∆T. See Fig. 17–21.
Diagram of a rectangular plate labeled with length ℓ, width w, and changes in dimensions Δℓ and Δw due to thermal expansion.

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Start by recalling the formula for linear expansion: the change in length of an object due to a temperature change is given by Δℓ = αℓΔT, where α is the coefficient of linear expansion, ℓ is the original length, and ΔT is the temperature change.
For a rectangular plate, the area A is given by A = ℓω, where ℓ is the length and ω is the width. When the temperature changes, both the length and width expand linearly.
The new length of the plate after expansion is ℓ + Δℓ = ℓ + αℓΔT, and the new width is ω + Δω = ω + αωΔT. The new area of the plate is therefore (ℓ + αℓΔT)(ω + αωΔT).
Expand the expression for the new area: (ℓ + αℓΔT)(ω + αωΔT) = ℓω + ℓ(αωΔT) + ω(αℓΔT) + (αℓΔT)(αωΔT). The term (αℓΔT)(αωΔT) is very small compared to the other terms and can be neglected.
Simplify the expression: The change in area ΔA = (new area) - (original area) = [ℓω + ℓ(αωΔT) + ω(αℓΔT)] - ℓω = 2αℓωΔT. This shows that the change in area is proportional to the coefficient of linear expansion, the original area, and the temperature change.

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Coefficient of Linear Expansion

The coefficient of linear expansion (α) quantifies how much a material expands per unit length for each degree of temperature increase. It is a crucial property in understanding how solids respond to temperature changes, as it directly influences the change in dimensions of an object, such as length and width.
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Linear Thermal Expansion

Area Expansion

Area expansion refers to the increase in surface area of a two-dimensional object when subjected to a temperature change. For a rectangular plate, the change in area (∆A) can be derived from the linear expansions of its length and width, leading to the formula ∆A = 2αℓω∆T, which accounts for the contributions from both dimensions.
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Volume Thermal Expansion

Temperature Change (∆T)

Temperature change (∆T) is the difference in temperature that causes materials to expand or contract. In the context of thermal expansion, it is essential to determine how much a material will change in size, as the extent of expansion is directly proportional to this temperature difference, influencing the overall change in area of the plate.
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Specific Heat & Temperature Changes
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