Problem 96 - Physics

Question

Concept Questions (a) When an object, such as a rod, is heated, what factors determine how much its length increases? (b) Suppose two rods are made from materials that have different coefficients of linear expansion, but their lengths change by the same amount when their temperatures change by the same amount. Are their initial lengths the same or different? Explain. Problem One rod is made from lead and another from quartz. They are heated and experience the same change in temperature. If the initial length of the lead rod is \(0.10 \mathrm{~m}\), what is the initial length of the quartz rod? Be sure that your answer is consistent with your answer to the Concept Questions.

Step-by-Step Solution

Answer

The initial length of the quartz rod is 5.8 meters.

Step 1: Understanding Linear Expansion

When an object is heated, its length increases. The increase in length depends on three main factors: the original length of the object, the change in temperature, and the material's coefficient of linear expansion, which is a measure of how much the material expands for a given temperature increase.

Step 2: Comparing Two Rods

If two rods experience the same change in temperature and their lengths increase by the same amount, it indicates a relationship between their initial lengths and their coefficients of linear expansion. The rods must have different initial lengths to compensate for differing coefficients of linear expansion, assuming both rods underwent identical temperature changes.

Step 3: Coefficient of Linear Expansion Formula

The formula for linear expansion is:\[ \Delta L = \alpha L_0 \Delta T \]where \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of linear expansion, \( L_0 \) is the initial length, and \( \Delta T \) is the change in temperature.

Step 4: Given Data and Problem Setup

We know the initial length of the lead rod (L_{lead,0} = 0.10 ext{ m}), and we want to find the initial length of the quartz rod. The expansion for both rods is equal, and they experience the same temperature change. The coefficient of linear expansion for lead and quartz, respectively, might be given, e.g., \( \alpha_{lead} = 29 \times 10^{-6} \underline{\phantom{xxx}} ^{\circ} ext{C}^{-1} \) and \( \alpha_{quartz} = 0.5 \times 10^{-6} \underline{\phantom{xxx}} ^{\circ} ext{C}^{-1} \).

Step 5: Setting Up the Mathematical Equality

Use the equation given in the previous step. Equating the expansion of both rods, we get:\[ \alpha_{lead} \times L_{lead,0} \times \Delta T = \alpha_{quartz} \times L_{quartz,0} \times \Delta T \]After cancelling \( \Delta T \) from both sides, we have:\[ \alpha_{lead} \times 0.10 = \alpha_{quartz} \times L_{quartz,0} \]

Step 6: Solving for Quartz Initial Length

Substitute the given values for \( \alpha_{lead} \) and \( \alpha_{quartz} \) into the equation:\[ 29 \times 10^{-6} \times 0.10 = 0.5 \times 10^{-6} \times L_{quartz,0} \]Solve for \( L_{quartz,0} \):\[ L_{quartz,0} = \frac{29 \times 10^{-6} \times 0.10}{0.5 \times 10^{-6}} = 5.8 \text{ m} \]

Key Concepts

Coefficient of Linear Expansion

The coefficient of linear expansion is a value that characterizes how much a material expands per degree change in temperature. It is usually denoted by the Greek letter \( \alpha \). This coefficient varies with different materials and is crucial in determining how significantly an object will increase in length when heated.
For example:

• Lead has a relatively high coefficient of linear expansion compared to quartz.
• Quartz, on the other hand, has a very low coefficient of linear expansion.

This means that under the same temperature change, lead will expand more than quartz. Knowing the coefficients is important in applications where precise measurements are necessary and stability is crucial. It's akin to a material's thermal footprint, telling us how sensitive it is to temperature changes.
You can calculate the actual change in length by multiplying this coefficient by the initial length and the temperature change, as shown in the formula:\[ \Delta L = \alpha L_0 \Delta T \]where \( \Delta L \) is the change in length, \( L_0 \) is the initial length, and \( \Delta T \) is the change in temperature. This equation lets us predict how different materials will react under thermal conditions, allowing us to engineer constructions that account for these changes.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change in shape, area, and volume in response to a change in temperature. For solid materials, this typically manifests as an increase in length, known as linear expansion.
Understanding thermal expansion is vital:

• It informs the design and engineering of structures, from bridges to everyday items like thermostats.
• Different materials react to temperature changes in unique ways, necessitating careful material selection in engineering and construction.
• Thermal expansion calculations are crucial for preventing material stress and structural failure.

With the formula \( \Delta L = \alpha L_0 \Delta T \), thermal expansion becomes a predictable variable rather than an unpredictable factor.
For instance, engineers design gaps in bridges to allow for expansion and contraction with temperature changes. Without understanding thermal expansion, these critical accommodations would go unmade, likely leading to damage.

Initial Length

Initial length, represented as \( L_0 \) in equations, is the original measurement of a material before any temperature change. It is a crucial component in calculating how much a material will expand or contract with temperature variations.
Why is the initial length important?

• It acts as a base measurement from which changes are evaluated, ensuring precision in mathematical predictions and real-world applications.
• In equality situations, initial lengths can provide balance across materials with different coefficients of linear expansion, allowing for accurate thermal expansion comparisons.

In our problem scenario, the lead rod and the quartz rod start with different initial lengths. This difference compensates for their varying thermal expansion properties, ensuring they expand by the same amount under identical temperature changes. By solving:\[ \alpha_{lead} \times 0.10 = \alpha_{quartz} \times L_{quartz,0} \]we see that calculating the initial length of the quartz rod involves accounting for its significantly lower coefficient of linear expansion while still producing the same expansion increase as the lead rod.