Question

A steel bar and a brass bar are both at a temperature of \(31.03^{\circ} \mathrm{C}\) The brass bar is \(272.47 \mathrm{~cm}\) long. At a temperature of \(227.27{ }^{\circ} \mathrm{C},\) the two bars have the same length. What is the length of the steel bar at \(31.03^{\circ} \mathrm{C} ?\) Take the linear expansion coefficient of steel to be \(13.00 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\) and the linear expansion coefficient of brass to be \(19.00 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\).

Short answer

Answer: The length of the steel bar at 31.03°C is approximately 259.61 cm.

Step-by-step solution

Understand the linear expansion formula

Linear expansion formula relates initial length, final length, change in temperature, and coefficient of linear expansion of a material. The formula is given by: \(L = L_0 [ 1 + \alpha(T - T_0)]\) where L is the final length, \(L_0\) is the initial length, \(\alpha\) is the coefficient of linear expansion, T is the final temperature, and \(T_0\) is the initial temperature.

Set up two equations for Brass and Steel Bars

At the given temperature, the lengths of both steel and brass bars are equal. We can set up two equations using the linear expansion formula for both bars. For Brass bar: \(L_{B} = 272.47 [ 1 + 19.00 \cdot 10^{-6}(227.27 - 31.03)]\) Simplify the expression to find the final length of the Brass bar. For Steel bar: Let \(L_0\) be the length of the steel bar at \(31.03^{\circ} \mathrm{C}\) \(L_{S} = L_0 [ 1 + 13.00 \cdot 10^{-6}(227.27 - 31.03)]\) Since both bars have the same length at \(227.27^{\circ} \mathrm{C}\), we can say: \(L_{B} = L_{S}\)

Solve the equation

Substitute the expressions for \(L_{B}\) and \(L_{S}\) from the previous step and set them equal: \(272.47 [ 1 + 19.00 \cdot 10^{-6}(227.27 - 31.03)] = L_0 [ 1 + 13.00 \cdot 10^{-6}(227.27 - 31.03)]\) Now, solve for \(L_0\): \(L_0 = \frac{272.47 [ 1 + 19.00 \cdot 10^{-6}(227.27 - 31.03)]}{[ 1 + 13.00 \cdot 10^{-6}(227.27 - 31.03)]}\) Calculate the value of \(L_0\) to find the length of the steel bar at \(31.03^{\circ} \mathrm{C}\). \(L_0 \approx 259.61 \mathrm{~cm}\) The length of the steel bar at \(31.03^{\circ} \mathrm{C}\) is approximately \(259.61 \mathrm{~cm}\).

Key concepts

Thermal Expansion

Imagine a hot summer day causing the pavement to crack or a metal bridge to expand — these are examples of thermal expansion, a natural phenomenon where the increase in temperature leads to an increase in the volume of a material. Think of it like this: when materials heat up, their particles move faster and tend to take up more space, causing the material to expand.

Ample everyday examples include railway tracks expanding and contracting with the temperature changes, power lines sagging on a hot day, or even the simple act of a metal lid on a glass jar becoming looser when run under hot water. Thermal expansion is not just a fact of life; it is a crucial factor to consider in engineering, construction, and design, ensuring that structures can withstand temperature changes without breaking.

Coefficient of Linear Expansion

The coefficient of linear expansion is a fundamental aspect of thermal physics that quantifies how much a material will expand along its length per degree of temperature increase. The coefficient is unique for each material, symbolized by the Greek letter \(\alpha\) and typically expressed in units of inverse degrees Celsius \(\text{°C}^{-1}\).

Now, back to the example in the exercise: steel and brass have different coefficients of expansion, meaning they expand at different rates when heated. This property is crucial when these materials are used in structures that experience temperature variation. It aids in predicting the behavior of materials, preventing potential damage or failure due to temperature changes. So, in the given exercise, knowing the coefficients of steel and brass, along with their initial lengths and temperatures, allows us to calculate and predict the lengths at different temperatures.

Thermal Physics

The overarching field studying these phenomena is thermal physics, a branch of physics that deals with heat and temperature and their relation to energy and work. It encompasses the study of thermodynamics, kinetic theory, and statistical mechanics.

Thermal physics explains not only why materials expand or contract with temperature changes but also other phenomena such as phase changes (like ice melting), heat transfer, and even the behavior of gases under different thermal conditions. Understanding these concepts is essential for developments in technology and the advancement of various sciences, including meteorology, engineering, and environmental science. This field provides the theoretical framework that lets us understand the practical problem solved in the exercise from the textbook, connecting the dots between abstract principles and real-world applications.