Question

When the temperature of a bar of initial length \(L_{0}\) (Fig. \(2-11\) ) rises by an amount \Deltat, the bar will elongate to a new length \(L\), where \(L\) is given by $$L=L_{0}(1+\alpha \Delta t)$$ Here \(\alpha\) is the temperature coefficient of expansion for the material from which the bar is made. Simplify this expression by multiplying out.

Short answer

The simplified expression for the new length \(L\) after multiplying out is \(L = L_{0} + L_{0}\alpha \Delta t\).

Step-by-step solution

Identify Given Values

The problem provides the initial length of the bar as \(L_{0}\), the temperature change as \(\Delta t\), and the temperature coefficient of expansion as \(\alpha\). The new length \(L\) after the temperature change is expressed by the formula \(L = L_{0}(1 + \alpha \Delta t)\).

Apply Distributive Property

To simplify the expression for the new length \(L\), apply the distributive property (multiplying out), which states that for any real numbers a, b, and c, the equation \(a(b + c) = ab + ac\) holds. Using this property on the given formula yields \(L = L_{0}\times 1 + L_{0}\times\alpha \Delta t\).

Simplify Expression

Simplify the expression by equating it to \(L_{0} + L_{0}\alpha \Delta t\), which is the result of distributing \(L_{0}\) to both terms in the parenthesis.

Key concepts

Temperature Coefficient of Expansion

Understanding the concept of the temperature coefficient of expansion is crucial when analyzing the behavior of solids in response to temperature changes. In essence, this coefficient, often denoted by the symbol \(\alpha\), is a measure of how much a material's length, area, or volume will change with a one-degree change in temperature.

Materials tend to expand when heated and contract when cooled, and the temperature coefficient provides a proportional value that can help predict this change. Different materials have different coefficients. Metals, for instance, typically have higher coefficients compared to materials like glass or plastic.

When considering the linear expansion of a solid, the change in length \(\Delta L\) can be calculated using \(\Delta L = \alpha L_{0}\Delta t\), where \(L_{0}\) is the original length and \(\Delta t\) is the temperature change. This relation helps in designing structures or components by accommodating the thermal expansion to prevent structural failure or damage.

Linear Expansion Formula

Expanding on the topic of thermal expansion, the linear expansion formula is a practical mathematical expression that allows us to calculate the new length of an object after it has been subjected to a temperature change. This formula integrates the temperature coefficient of expansion and is given by: \[L = L_{0}(1 + \alpha \Delta t)\].

In this equation, \(L_{0}\) represents the original length before any heat was applied, and \(\Delta t\) indicates the magnitude of the temperature change in degrees. The term \(1 + \alpha \Delta t\) signifies how much the original length has been scaled as a result of the temperature variation. While the linear expansion formula is relatively straightforward, it plays a critical role in fields such as materials science, construction, and engineering, where precise control over the dimensions of materials is necessary.

Distributive Property of Multiplication

Delving into the mathematical principles, the distributive property of multiplication is a fundamental rule that guides the simplification of expressions and the solving of equations. This property states that the product of a number and the sum of two or more numbers is equal to the sum of the individual products of the number and each addend. Mathematically, this is interpreted as \(a(b + c) = ab + ac\), where \(a\), \(b\), and \(c\) are any real numbers or algebraic expressions.

Applying the distributive property can be particularly useful when working with expressions involving variables, such as the linear expansion formula. As seen in the textbook exercise, the property was applied to multiply the original length \(L_{0}\) by each term within the parentheses, which simplified the equation to its expanded form \(L = L_{0} + L_{0}\alpha \Delta t\). Grasping this concept is crucial for students as it simplifies complex problems and is widely used in various mathematical and scientific calculations.