Question

A steel bar and a brass bar are both at a temperature of \(28.73^{\circ} \mathrm{C}\). The steel bar is \(270.73 \mathrm{~cm}\) long. At a temperature of \(214.07^{\circ} \mathrm{C}\), the two bars have the same length. What is the length of the brass bar at \(28.73^{\circ} \mathrm{C} ?\) 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: To find the initial length of the brass bar, we followed these steps: 1. We understood the formula for linear expansion of solids: ΔL = L_0 * α * ΔT. 2. We found the change in length of the steel bar: ΔL_steel = 270.73 * (13.00 *10^{-6}) * (214.07 - 28.73). 3. We found the length of the steel bar at the final temperature: L_steel,final = L_steel,0 + ΔL_steel. 4. We found the change in length of the brass bar: L_brass,final - L_brass,0 = L_brass,0 * α_brass * (T_final - T_initial). 5. Finally, we solved for the initial length of the brass bar: L_brass,0 = L_steel,final / (1 + (19.00 * 10^{-6} * (214.07 - 28.73))). By calculating these values, we found the initial length of the brass bar at 28.73 °C.

Step-by-step solution

Understand the formula for linear expansion of solids

The formula for linear expansion of a solid is given by: ΔL = L_0 * α * ΔT, where ΔL is the change in length, L_0 is the initial length, α is the linear expansion coefficient, and ΔT is the change in temperature.

Find the change in length of the steel bar

Since we know the initial length and temperature of the steel bar and the final temperature, we can find the ΔL for the steel bar. The expression for the change in length of the steel bar can be written as: ΔL_steel = L_steel,0 * α_steel * (T_final - T_initial). Plug in the given values: L_steel,0 = 270.73 cm, α_steel = 13.00 * 10^{-6} °C^{-1}, T_initial = 28.73 °C, and T_final = 214.07 °C. ΔL_steel = 270.73 * (13.00 *10^{-6}) * (214.07 - 28.73) Compute the value of ΔL_steel.

Find the length of the steel bar at the final temperature

Now, we need to find the length of the steel bar at the final temperature 214.07 °C. This can be done by adding the change in length (ΔL_steel) to the initial length (L_steel,0). L_steel,final = L_steel,0 + ΔL_steel Compute the value of L_steel,final.

Find the change in length of the brass bar

At the final temperature, the lengths of both steel and brass bars are equal. So, L_brass,final = L_steel,final. Now, let's find the change in length of the brass bar by using the expression: ΔL_brass = L_brass,0 * α_brass * (T_final - T_initial). We know α_brass = 19.00 * 10^{-6} °C^{-1}, and we can rearrange the expression for ΔL_brass as follows: L_brass,final - L_brass,0 = L_brass,0 * α_brass * (T_final - T_initial)

Solve for the initial length of the brass bar

Now, we can plug in the values and solve for L_brass,0. We know that L_brass,final = L_steel,final and α_brass = 19.00 * 10^{-6} °C^{-1}. L_steel,final - L_brass,0 = L_brass,0 * (19.00 * 10^{-6}) * (214.07 - 28.73) Rearrange the equation to solve for L_brass,0: L_brass,0 = L_steel,final / (1 + (19.00 * 10^{-6} * (214.07 - 28.73))) Compute the value of L_brass,0, which represents the initial length of the brass bar at 28.73 °C.

Key concepts

Understanding Thermal Expansion

Thermal expansion is a physical property of matter to change in size or volume as a result of a change in temperature. When solids are heated, their molecules begin to vibrate more vigorously due to an increase in energy, leading to an expansion of the material.

In the context of the exercise, both the steel and brass bars expand when their temperature is raised from a common initial temperature to a higher final temperature. This behavior is not unique to these metals; it's a general property of solids, and it is crucial for engineers and architects to take into account when designing structures to ensure they can withstand temperature changes without damage.

Linear Expansion Coefficient

The linear expansion coefficient is a measure of how much a material expands per unit length for each degree of temperature increase. This coefficient, usually denoted as \(\alpha\), is unique to each material and is essential in calculating the amount of thermal expansion a solid will undergo when heated or cooled.

Understanding how to apply the linear expansion coefficient is central to solving problems involving thermal expansion. For instance, in the given exercise, the coefficients for steel and brass are provided, and they are used in conjunction with the initial lengths and temperature change to calculate the respective expansions of the bars.

Initial and Final Length

The terms 'initial length' and 'final length' refer to the length of an object before and after thermal expansion, respectively. These terms are represented in the formula \(\Delta L = L_0 \cdot \alpha \cdot \Delta T\), where \(\Delta L\) is the change in length, \(L_0\) is the initial length, \(\alpha\) is the coefficient of linear expansion, and \(\Delta T\) is the change in temperature.

To find the final length after expansion, the initial length is added to the change in length. In our exercise, both bars are initially at the same temperature. However, when they are heated to the final temperature, they expand to different lengths due to having different linear expansion coefficients. Yet, due to the nature of the problem given, which states that both bars have the same final length at the higher temperature, we can use this information to determine the unknown initial length of the brass bar.