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At \(26.45^{\circ} \mathrm{C},\) a steel bar is \(268.67 \mathrm{~cm}\) long and a brass bar is \(268.27 \mathrm{~cm}\) long. At what temperature will the two bars be the same length? 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

Expert verified
Answer: The two bars will be the same length at approximately \(119.55^{\circ}C\).

Step by step solution

01

Write down the known values and variables

We have the following information: Initial temperature: \(T_1 = 26.45^{\circ}C\) Initial length of steel bar: \(L_{s1} = 268.67 \, cm\) Initial length of brass bar: \(L_{b1} = 268.27 \, cm\) Linear expansion coefficient of steel: \(\alpha_s = 13.00 \cdot 10^{-6}{\,}^{\circ} \mathrm{C}^{-1}\) Linear expansion coefficient of brass: \(\alpha_b = 19.00 \cdot 10^{-6}{\,}^{\circ} \mathrm{C}^{-1}\) We need to find the temperature, \(T_2\), when both bars have the same length.
02

Write down the formula for linear expansion

The formula for linear expansion is: $$\Delta L = \alpha \cdot L \cdot \Delta T$$ where \(\Delta L\) is the change in length, \(\alpha\) is the linear expansion coefficient, \(L\) is the initial length, and \(\Delta T\) is the change in temperature.
03

Set up the equation for the length of both bars

Since the lengths of the steel and brass bars should be equal at the temperature \(T_2\), we can write: $$L_{s1} + \Delta L_s = L_{b1} + \Delta L_b$$ Substitute the formula for linear expansion for \(\Delta L_s\) and \(\Delta L_b\): $$L_{s1} + \alpha_s \cdot L_{s1} \cdot \Delta T_s = L_{b1} + \alpha_b \cdot L_{b1} \cdot \Delta T_b$$ We know that \(\Delta T_s = \Delta T_b = T_2 - T_1\), so we can write: $$L_{s1} + \alpha_s \cdot L_{s1} \cdot (T_2 - T_1) = L_{b1} + \alpha_b \cdot L_{b1} \cdot (T_2 - T_1)$$
04

Solve for T_2

Let's isolate \(T_2\) in the equation above: $$T_2 = \frac{L_{b1} + \alpha_b \cdot L_{b1} \cdot T_1 - L_{s1} - \alpha_s \cdot L_{s1} \cdot T_1}{\alpha_s \cdot L_{s1} - \alpha_b \cdot L_{b1}}$$ Now plug in the known values and compute \(T_2\): $$T_2 = \frac{268.27\,\mathrm{cm} + 19.00 \cdot 10^{-6}{\,}^{\circ} \mathrm{C}^{-1} \cdot 268.27\,\mathrm{cm} \cdot 26.45^{\circ}C - 268.67\,\mathrm{cm} - 13.00 \cdot 10^{-6}{\,}^{\circ} \mathrm{C}^{-1} \cdot 268.67\,\mathrm{cm} \cdot 26.45^{\circ}C}{13.00 \cdot 10^{-6}{\,}^{\circ} \mathrm{C}^{-1} \cdot 268.67\,\mathrm{cm} - 19.00 \cdot 10^{-6}{\,}^{\circ} \mathrm{C}^{-1} \cdot 268.27\,\mathrm{cm}}$$ After calculating, we get: $$T_2 \approx 119.55^{\circ}C$$ The two bars will be the same length at approximately \(119.55^{\circ}C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Expansion Coefficient
When thermal energy is absorbed by a solid, the kinetic energy of its atoms or molecules increases. This leads to a change in dimensions of the material, a phenomenon known as thermal expansion. The degree to which a material expands in one-dimensional space (lengthwise) due to a temperature increase is described by its linear expansion coefficient. This coefficient is a physical property unique to each material and quantifies the length change per degree of temperature change.

Mathematically, the linear expansion coefficient, often symbolized as \( \alpha \), can be expressed in the equation: \[ \Delta L = \alpha \cdot L \cdot \Delta T \[ where \( \Delta L \) is the change in length, \( L \) is the original length, and \( \Delta T \) represents the change in temperature. The unit of \( \alpha \) is typically inverse degrees Celsius \( {\,}^\circ \mathrm{C}^{-1} \) or Kelvin. A high value of \( \alpha \) indicates a material that expands significantly with temperature changes, while a low value signifies lesser expansion. For instance, metals typically have higher linear expansion coefficients than ceramics or composites.
Temperature Change in Solids
Understanding how solids respond to temperature change is crucial in many practical applications, such as construction, manufacturing, and material science. The length of a solid will increase with temperature, and this behavior can be predicted and quantified using the concept of linear expansion. The temperature change, \( \Delta T \), is the difference between the initial temperature \( T_1 \) and the final temperature \( T_2 \). \

When this temperature change occurs, the solid doesn't instantly acquire a new length. The process of thermal expansion is dependent on heat transfer mechanisms, such as conduction, convection, and radiation, and the material's properties, including its heat capacity and thermal conductivity. \

For a solid undergoing uniform temperature change, the change in length is proportional to the initial length and the temperature change itself, governed by the linear expansion coefficient. This behavior allows us to solve problems like determining the matching length of different materials at a certain temperature by setting up equations that equate their expanded lengths while accounting for their respective coefficients.
Physical Properties of Materials
Materials possess a range of physical properties that determine how they react to environmental changes, including temperature. The linear expansion coefficient is just one such property. Others include density, specific heat capacity, thermal conductivity, and Young's modulus. Each property provides insights into the behavior of the material under specific conditions.

Density, for instance, influences how a material will expand or contract with temperature fluctuations. Materials with high densities generally have lower expansion coefficients. Specific heat capacity is a measure of the energy required to raise the temperature of a unit mass of material by one degree, affecting how quickly a material can change temperature. Thermal conductivity indicates how well the material can conduct heat, and Young’s modulus describes the material's stiffness or resistance to deformation under load.

By understanding and combining these properties, one can predict and engineer the behavior of materials in diverse applications, ensuring their stability and integrity across temperature changes and other environmental conditions.

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Most popular questions from this chapter

A copper cube of side length \(40 . \mathrm{cm}\) is heated from \(20 .{ }^{\circ} \mathrm{C}\) to \(120{ }^{\circ} \mathrm{C} .\) What is the change in the volume of the cube? The linear expansion coefficient of copper is \(17 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\)

Express each of the following temperatures in degrees Celsius and in kelvins. a) \(-19^{\circ} \mathrm{F}\) b) \(98.6^{\circ} \mathrm{F}\) c) \(52^{\circ} \mathrm{F}\)

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 \(19.00 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\).

You are outside on a hot day, with the air temperature at \(T_{\mathrm{o}^{*}}\) Your sports drink is at a temperature \(T_{\mathrm{d}}\) in a sealed plastic bottle. There are a few remaining ice cubes in the sports drink, which are at a temperature \(T_{\mathrm{j}}\), but they are melting fast. a) Write an inequality expressing the relationship among the three temperatures. b) Give reasonable values for the three temperatures in degrees Celsius.

The city of Yellowknife in the Northwest Territories of Canada is on the shore of Great Slave Lake. The average high temperature in July is \(21^{\circ} \mathrm{C}\) and the average low in January is \(-31{ }^{\circ} \mathrm{C}\). Great Slave Lake has a volume of \(2090 \mathrm{~km}^{3}\) and is the deepest lake in North America, with a depth of \(614 \mathrm{~m} .\) What is the temperature of the water at the bottom of Great Slave Lake in January? a) \(-31^{\circ} \mathrm{C}\) b) \(-10^{\circ} \mathrm{C}\) c) \(0^{\circ} \mathrm{C}\) d) \(4^{\circ} \mathrm{C}\) e) \(32^{\circ} \mathrm{C}\)

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