Question

A steel antenna for television broadcasting is \(501.9 \mathrm{~m}\) tall on a summer day, when the outside temperature is \(28.09^{\circ} \mathrm{C} .\) The steel from which the antenna was constructed has a linear expansion coefficient of \(13.89 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1} .\) On a cold winter day, the temperature is \(-15.91{ }^{\circ} \mathrm{C} .\) What is the change in the height of the antenna?

Short answer

Answer: The change in height of the steel antenna during the transition from a hot summer day to a cold winter day is approximately 0.306 meters.

Step-by-step solution

Identify the given values

We are given the following values: - Initial height of the antenna, \(H_1 = 501.9 \mathrm{~m}\) - Summer temperature, \(T_1 = 28.09^{\circ} \mathrm{C}\) - Winter temperature, \(T_2 = -15.91^{\circ} \mathrm{C}\) - Linear expansion coefficient of steel, \(\alpha = 13.89 \cdot 10^{-6} {}^{\circ}\mathrm{C}^{-1}\)

Calculate the change in temperature

To find the change in temperature during the transition from summer to winter, subtract the winter temperature from the summer temperature: \(\Delta T = T_1 - T_2 = 28.09^{\circ} \mathrm{C} - (-15.91^{\circ} \mathrm{C}) = 44^{\circ} \mathrm{C}\)

Apply the formula for linear expansion

The formula for linear expansion is: \(\Delta L = L_1 \cdot \alpha \cdot \Delta T\) Where: - \(\Delta L\) is the change in length (or height in this case), - \(L_1\) is the initial length (or height), - \(\alpha\) is the linear expansion coefficient, and - \(\Delta T\) is the change in temperature.

Calculate the change in height

Plugging in the given values into the above formula, we have: \(\Delta L = 501.9 \mathrm{~m} \cdot (13.89 \cdot 10^{-6} {})^{\circ} \mathrm{C}^{-1} \cdot 44^{\circ} \mathrm{C}\) \(\Delta L = 501.9 \mathrm{~m} \cdot (13.89 \cdot 10^{-6} {})^{\circ} \mathrm{C}^{-1} \cdot 44^{\circ} \mathrm{C} \approx 0.306 \mathrm{~m}\)

State the change in height

The change in height of the steel antenna during the transition from a hot summer day to a cold winter day is approximately \(0.306 \mathrm{~m}\).

Key concepts

Linear Expansion Coefficient

When materials are heated, they usually expand, and when they are cooled, they contract. The linear expansion coefficient, often symbolized as \( \alpha \), is a property that quantifies how much a material's length changes with temperature. Specifically, the linear expansion coefficient is defined as the fractional change in length per degree change in temperature. To put it in the context of the solved problem, the steel used for the antenna has a linear expansion coefficient of \( 13.89 \times 10^{-6} { }^{\circ} \mathrm{C}^{-1} \).

If you're imagining the antenna, just know that if the material heats up or cools down, every little segment of the antenna nominally 'slides' a tiny bit farther from or closer to its immediate neighbor — depending on whether it's getting warmer or colder. The linear expansion coefficient of steel means that for every degree increase in temperature, a one-meter length of steel would expand by \( 13.89 \times 10^{-6} { }^{\circ} \mathrm{C}^{-1} \) meters. Whether you are working on a building structure, or something as tall as a broadcasting antenna, the knowledge of \( \alpha \) is crucial to ensure safety and functionality as temperatures change.

Temperature Change

Temperature change, denoted as \( \Delta T \) in physics, is the difference between the final and initial temperatures of an object or system when heat is transferred. In thermal physics, it's important to know that temperatures are usually measured in degrees Celsius (\( { }^{\circ} \mathrm{C} \)) or Kelvin (K).

In the textbook example, the winter temperature is lower than the summer's, leading to \( \Delta T = 28.09{ }^{\circ} \mathrm{C} - (-15.91{ }^{\circ} \mathrm{C}) = 44{ }^{\circ} \mathrm{C} \). Since the steel antenna contracts when the temperature drops, we need to understand this change to calculate the new height of the antenna.

It's also essential to grasp that temperature change isn't just about how the numbers shift; it's fundamentally about energy. Warmer temperatures mean the steel's particles are vibrating more eagerly, taking up more space. Conversely, the lack of heat makes them 'calm down' and huddle together a bit - that’s why materials expand and contract with temperature changes.

Length Contraction and Expansion

The concepts of length contraction and length expansion are direct consequences of the linear expansion coefficient and temperature change discussed above. Length contraction happens when a material decreases in size due to a decrease in temperature, while length expansion refers to an increase in size with an increase in temperature.

In the example, the steel antenna undergoes length contraction because the temperature drops from a warm 28.09{ }^{\circ} \mathrm{C} to a frigid -15.91{ }^{\circ} \mathrm{C}. This decrease in temperature leads to a contraction of the antenna's steel, shortening its height by about \( 0.306 \mathrm{~m} \).

Practical Implications

Engineers must account for these changes in the design process of structures to prevent damage. In construction, expansion joints are incorporated to safely allow the expansion and contraction of materials. Thus, observing these thermal properties is not just academic; it ensures the longevity and safety of real-life structures such as bridges, railways, and yes – tall antennas that reach for the sky in all seasons.