Problem 34 - Technical Mathematics with Calculus

Question

When a bar of length \(L_{0}\) having a coefficient of linear thermal expansion \(\alpha\) is increased in temperature by an amount \(\Delta t,\) it will expand to a new length \(L\) where $$L=L_{0}(1+\alpha \Delta t)$$ Solve this equation for the distance measured, \(L.\)

Step-by-Step Solution

Answer

The distance measured after expansion is \(L = L_{0}(1 + \alpha \Delta t)\), which is the new length of the bar.

Step 1: Identify Given Variables

Identify and list the given variables. In this case, we have the original length of the bar, which is denoted as \(L_{0}\), the coefficient of linear thermal expansion \(\alpha\), and the change in temperature denoted as \(\Delta t\).

Step 2: Write Down the Expansion Formula

Write the formula for the final length after thermal expansion, which is \[L = L_{0}(1 + \alpha \Delta t)\].

Step 3: Insert the Known Values

Insert the known values into the formula. Since this step is about understanding the relationship, actual numerical values aren't required unless provided.

Step 4: Calculate the New Length

Perform the calculation to solve for \(L\). This involves multiplying \(L_{0}\) by the quantity \(1 + \alpha \Delta t\) to find the new length. Without specific values for \(L_{0}\), \(\alpha\), and \(\Delta t\), the expression remains as the formula itself.

Key Concepts

Thermal Expansion

Imagine a metal rod lying in the sun on a chilly morning. As the day progresses and the sun heats the rod, it begins to grow longer. This phenomenon is known as thermal expansion, which occurs because the particles within the material move more vigorously as they absorb heat, causing the material to expand. Thermal expansion is not just a trivial fact; it's a crucial consideration in fields like construction, manufacturing, and engineering. For example, gaps are often left between sections of bridges to allow for the expansion and contraction that occurs due to temperature changes.

In essence, thermal expansion helps us understand why things don't stay the exact same size when they heat up or cool down.

Coefficient of Linear Expansion

Each material has its unique response to temperature changes known as the coefficient of linear expansion, represented by the Greek letter \(\alpha\). This coefficient is a numerical value that represents how much a length of material will expand (or contract) per degree change in temperature. To put it simply: the higher the coefficient of linear expansion, the more the material will expand for each degree of temperature increase. Metals typically have higher coefficients compared to non-metals, meaning they expand more noticeably when heated. Understanding the coefficient of linear expansion is critical in designing anything that must maintain its size and shape over a range of temperatures.

Temperature Change in Solids

When discussing temperature change in solids, it's essential to concentrate on how the solid's dimensions alter as the temperature varies. Most solids expand when heated and contract when cooled - a basic physics principle often observed in everyday life. To predict the exact amount of expansion or contraction, we apply the formula for linear thermal expansion. This takes into consideration the initial length of the solid, the temperature change it undergoes, and its material's specific coefficient of linear expansion. Carefully accommodating temperature-induced size changes is vital in engineering to avoid structural failures like buckling in rails or cracking in roadways during temperature fluctuations.

Mathematical Problem Solving

Approaching a mathematical problem like thermal expansion includes specific steps that aid in understanding and solving the problem. Begin by identifying the known variables, then apply the relevant formula - in the case of thermal expansion, this is \(L = L_{0}(1 + \alpha \Delta t)\). Insert any provided numerical values you may have, and through algebraic manipulation, solve for the unknown. If specific values aren't given, as with some textbook problems, the exercise becomes more about grasping the concept than producing a numerical answer. These step-by-step methods aren't exclusive to physics or engineering; they're applicable to a broad range of mathematical endeavors, fostering a systematic approach to problem-solving in various fields.