Thermal Expansion Calculation Of Steel Railroad Track Length Change With Temperature

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When exploring the fascinating world of physics, thermal expansion stands out as a fundamental concept that governs how materials respond to temperature changes. This phenomenon is particularly evident in large structures like steel railroad tracks, where even slight expansions can have significant implications. Let's delve into a practical scenario to understand this better: Imagine a steel railroad track measuring 20 meters in length on a cool, rainy day with a temperature of 20°C. Now, what happens to its length when the weather shifts to a hot, dry day, and the temperature soars to 40°C? This seemingly simple question opens the door to a detailed exploration of linear thermal expansion, the properties of steel, and the calculations required to predict changes in length.

The Physics Behind Thermal Expansion

Thermal expansion is the tendency of matter to change in volume in response to temperature changes. When a substance is heated, its particles move more, thus maintaining a greater average separation. Because thermometers are calibrated against these effects, thermal expansion was among the earliest studied macroscopic indicators of temperature. A solid material will expand in all directions. Thermal expansion is a physical property of materials that describes how much their size changes in response to changes in temperature. At the microscopic level, the atoms in a solid are constantly vibrating. The amplitude of these vibrations increases with temperature. As the temperature rises, the atoms vibrate more vigorously, pushing each other further apart. This increased separation between atoms manifests as an overall expansion of the material. The amount of expansion depends on several factors, including the material itself, the initial temperature, and the change in temperature.

For most solids, thermal expansion is linear, meaning the change in length is directly proportional to the change in temperature. This relationship is expressed by the following formula:

ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T

Where:

  • \Delta L$ is the change in length

  • \alpha$ is the coefficient of linear expansion (a material property)

  • L_0$ is the original length

  • \Delta T$ is the change in temperature

Steel and Its Coefficient of Linear Expansion

Steel is a common material used in the construction of railroad tracks due to its high tensile strength and durability. However, like all materials, steel is subject to thermal expansion. The coefficient of linear expansion for steel is approximately $12 \times 10^{-6} , \text{°C}^{-1}$. This value represents the fractional change in length per degree Celsius change in temperature. In simpler terms, for every degree Celsius increase in temperature, a steel object will expand by 12 millionths of its original length.

This seemingly small number has a significant impact on large structures. For instance, in our railroad track scenario, even a 20°C temperature change can lead to a noticeable expansion. This expansion needs to be accounted for in the design and construction of railroad tracks to prevent buckling or other structural issues. Expansion joints are often incorporated into the tracks to allow for this thermal expansion and contraction.

Calculating the Expansion of the Railroad Track

Now, let's apply the formula for linear thermal expansion to our specific problem. We have:

  • Original length, $L_0 = 20 , \text{m}$
  • Coefficient of linear expansion for steel, $\alpha = 12 \times 10^{-6} , \text{°C}^{-1}$
  • Initial temperature, $T_1 = 20 , \text{°C}$
  • Final temperature, $T_2 = 40 , \text{°C}$
  • Change in temperature, $\Delta T = T_2 - T_1 = 40 , \text{°C} - 20 , \text{°C} = 20 , \text{°C}$

Using the formula, we can calculate the change in length:

ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T

ΔL=(12×10−6 °C−1)(20 m)(20 °C)\Delta L = (12 \times 10^{-6} \, \text{°C}^{-1}) (20 \, \text{m}) (20 \, \text{°C})

ΔL=0.0048 m\Delta L = 0.0048 \, \text{m}

This means the steel railroad track will expand by 0.0048 meters, or 4.8 millimeters, when the temperature increases from 20°C to 40°C. While this may seem like a small amount, it is crucial to consider in engineering applications, especially over long distances.

The Final Length of the Railroad Track

To find the final length of the railroad track, we simply add the change in length to the original length:

Lfinal=L0+ΔLL_{\text{final}} = L_0 + \Delta L

Lfinal=20 m+0.0048 mL_{\text{final}} = 20 \, \text{m} + 0.0048 \, \text{m}

Lfinal=20.0048 mL_{\text{final}} = 20.0048 \, \text{m}

Therefore, the length of the steel railroad track on a hot, dry day when the temperature is 40°C will be 20.0048 meters. This calculation demonstrates the practical application of the concept of thermal expansion and its importance in real-world engineering scenarios.

Practical Implications and Considerations

The expansion of materials due to temperature changes has significant implications in various engineering fields. In the case of railroad tracks, engineers must design the tracks with expansion joints to accommodate the changes in length due to temperature variations. These joints allow the tracks to expand and contract without causing stress or deformation that could lead to accidents.

Bridges are another example where thermal expansion is a critical consideration. The expansion and contraction of bridge materials can create substantial forces, which must be accounted for in the design to ensure the bridge's structural integrity. Similarly, in the construction of buildings, expansion joints are used to prevent cracks and structural damage caused by thermal expansion.

In summary, understanding thermal expansion is crucial for engineers and designers to create safe and durable structures. The principles of thermal expansion are not only applicable to large-scale projects but also play a role in everyday applications, such as the design of thermostats and bimetallic strips used in various devices.

Thermal Stress

Thermal stress is stress created in a material due to changes in temperature. Thermal stress can lead to fractures or plastic yielding depending on the nature of the materials. This can happen in several situations, such as when two materials with different coefficients of thermal expansion are used to make an object. Another case is when an object is held fixed only at its ends, and experiences a change in temperature. In this scenario, thermal stress will appear in the body.

Expansion Joints

As mentioned before, expansion joints are incorporated into structures like bridges and railroad tracks to accommodate the thermal expansion of materials. These joints are gaps built into the structure that allow the materials to expand and contract without putting undue stress on the structure itself. Expansion joints are often filled with flexible materials that can compress and expand with the movement of the structure.

The design and placement of expansion joints are critical to the long-term durability of structures. Engineers carefully calculate the expected thermal expansion and contraction to determine the appropriate size and spacing of the joints. Proper implementation of expansion joints can prevent costly repairs and ensure the safety of the structure.

Other Applications of Thermal Expansion

Besides railroad tracks and bridges, thermal expansion plays a role in various other applications:

  • Bimetallic Strips: These strips are made of two different metals with different coefficients of thermal expansion. When heated, one metal expands more than the other, causing the strip to bend. Bimetallic strips are used in thermostats, circuit breakers, and other temperature-sensitive devices.
  • Thermometers: Traditional liquid-in-glass thermometers rely on the thermal expansion of a liquid, such as mercury or alcohol, to measure temperature. The liquid expands and rises in a glass tube as the temperature increases.
  • Hot Air Balloons: The principle of thermal expansion is also used in hot air balloons. Heating the air inside the balloon causes it to expand, making it less dense than the surrounding air. This difference in density creates buoyancy, allowing the balloon to float.

Conclusion

In conclusion, the phenomenon of thermal expansion is a fundamental concept in physics with significant practical implications. By understanding how materials respond to temperature changes, engineers can design structures that are safe, durable, and able to withstand the effects of thermal stress. In the case of our steel railroad track, calculating the expansion due to a temperature change allows us to appreciate the importance of considering thermal expansion in engineering design. The final length of the track, 20.0048 meters, underscores the need for expansion joints and careful planning in the construction of railways and other large structures. From bridges and buildings to thermostats and hot air balloons, thermal expansion is a ubiquitous principle that shapes the world around us.

By examining this specific scenario, we've highlighted the practical application of thermal expansion and its relevance in the field of engineering. The ability to calculate and predict these changes is vital for ensuring the safety and longevity of various structures and devices. This understanding underscores the importance of physics in everyday life and its role in shaping the world we inhabit.