Problem 7 - College Physics

Question

\(\cdot\) The Eiffel Tower in Paris is 984 ft tall and is made mostly of steel. If this is its height in winter when its temperature is \(-8.00^{\circ} \mathrm{C},\) how much additional vertical distance must you cover if you decide to climb it during a summer heat wave when its temperature is \(40.0^{\circ} \mathrm{C}\) ? (b) Express the coefficient of linear expansion of steel in terms of Fahrenheit degrees.

Step-by-Step Solution

Answer

The additional distance to cover is 0.520 ft; coefficient in Fahrenheit is \(6.11 \times 10^{-6} ^{\circ}\text{F}^{-1}\).

Step 1: Understand Thermal Expansion

The problem involves the concept of thermal expansion where materials expand or contract with temperature changes. We will be using the formula for linear expansion: \[ \Delta L = L_0 \alpha \Delta T \]where \( \Delta L \) is the change in length, \( L_0 \) is the original length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

Step 2: Calculate Change in Temperature

First, calculate the temperature change from winter to summer: \[ \Delta T = 40.0^{\circ} \text{C} - (-8.00^{\circ} \text{C}) = 48.0^{\circ} \text{C} \]

Step 3: Find Coefficient of Linear Expansion for Steel

The coefficient of linear expansion for steel is generally \( \alpha = 11 \times 10^{-6} \underline{\phantom{xxx}} ^{\circ}\text{C}^{-1} \). This will be used to calculate the expansion of the Eiffel Tower.

Step 4: Apply the Linear Expansion Formula

Using the formula: \[ \Delta L = 984 \text{ ft} \times 11 \times 10^{-6} \underline{\phantom{xxx}} ^{\circ}\text{C}^{-1} \times 48.0^{\circ} \text{C} \]Calculating gives:\[ \Delta L \approx 0.520 \text{ ft} \]

Step 5: Coefficient of Linear Expansion in Fahrenheit

The relationship between Celsius and Fahrenheit is \( \Delta T_{\text{Fahrenheit}} = \Delta T_{\text{Celsius}} \times \frac{9}{5} \). Thus, the coefficient of linear expansion in terms of Fahrenheit is calculated by adjusting \( \alpha \) to:\[ \alpha_{\text{F}} = \frac{\alpha}{\frac{9}{5}} = 11 \times 10^{-6} ^{\circ}\text{C}^{-1} \times \frac{5}{9} = 6.11 \times 10^{-6} ^{\circ}\text{F}^{-1} \]

Key Concepts

Coefficient of Linear Expansion

The coefficient of linear expansion, often denoted by the Greek letter alpha (\( \alpha \)), is a crucial property of materials that defines how much a material expands per unit length with a change in temperature. For steel, which is the primary material used in the Eiffel Tower, the coefficient is typically \( 11 \times 10^{-6} \, ^{\circ}\text{C}^{-1} \). This means that for each degree Celsius increase in temperature, every meter of steel will expand by 11 millionths of a meter.
It's important to note that this coefficient is specific to materials and defines how reactive they are to temperature changes. Different materials have different coefficients, which is why some materials may expand or contract more than others when exposed to the same temperature changes.

Linear Expansion Formula

When a material's temperature changes, so does its size. This change in size, specifically length, can be calculated using the linear expansion formula:\[ \Delta L = L_0 \alpha \Delta T \]

• \( \Delta L \) is the change in length.
• \( L_0 \) is the original length of the material before temperature change.
• \( \alpha \) is the coefficient of linear expansion of the material.
• \( \Delta T \) is the temperature change the material undergoes.

Let's consider an example involving the Eiffel Tower. Given its original height in winter (\( L_0 = 984 \, \text{ft} \)), if it experiences a temperature change of \( \Delta T = 48.0^{\circ} \text{C} \) (from \(-8.00^{\circ} \text{C} \) to \( 40.0^{\circ} \text{C} \)), and using steel’s coefficient of linear expansion, you can calculate how much taller the tower becomes. This application of the formula shows how engineering and material science consider thermal effects in design and construction.

Temperature Change

Temperature change, represented as \( \Delta T \), is a straightforward concept but essential in thermal expansion calculations. It is simply the difference between the final temperature and the initial temperature:\[\Delta T = T_{\text{final}} - T_{\text{initial}}\]Using the Eiffel Tower example, the initial winter temperature is \(-8.00^{\circ} \text{C} \)and the final summer temperature is \(40.0^{\circ} \text{C} \). Thus, the temperature change \( \Delta T \) is found by:\[\Delta T = 40.0^{\circ} \text{C} - (-8.00^{\circ} \text{C}) = 48.0^{\circ} \text{C}\]Understanding \( \Delta T \) helps in determining how much a material will expand or contract when subjected to various thermal environments. It is crucial in architectural and engineering designs that account for temperature fluctuations.

Celsius to Fahrenheit Conversion

Sometimes, temperature change calculations require conversions between different units. In particular, converting temperature change from Celsius to Fahrenheit can be crucial when working with global data.The conversion formula is:\[\Delta T_{\text{Fahrenheit}} = \Delta T_{\text{Celsius}} \times \frac{9}{5}\]This formula says that to convert a temperature change from Celsius to Fahrenheit, you multiply by \(\frac{9}{5}\).Using this, we can find that the same \( 48.0^{\circ} \text{C} \)temperature change for the Eiffel Tower becomes:\[\Delta T_{\text{Fahrenheit}} = 48.0^{\circ} \text{C} \times \frac{9}{5} = 86.4^{\circ} \text{F}\]Conversions like this are essential in international projects where materials and temperatures might be measured and reported using different systems.