Problem 14 - Physics

Question

When the temperature of a coin is raised by \(75 \mathrm{C}^{\circ},\) the coin's diameter increases by \(2.3 \times 10^{-5} \mathrm{~m}\). If the original diameter of the coin is \(1.8 \times 10^{-2} \mathrm{~m}\), find the coefficient of linear expansion.

Step-by-Step Solution

Answer

The coefficient of linear expansion is \( 1.7 \times 10^{-6} \mathrm{m/}\mathrm{m}\cdot\mathrm{C}^{\circ} \).

Step 1: Understanding the Coefficient of Linear Expansion

The coefficient of linear expansion, denoted as \( \alpha \), measures how much a material's length changes per degree change in temperature. The formula to calculate it is given by: \( \Delta L = \alpha L_0 \Delta T \), where \( \Delta L \) is the change in length, \( L_0 \) is the original length, and \( \Delta T \) is the temperature change.

Step 2: Identifying Given Values

From the problem, we have: \( \Delta L = 2.3 \times 10^{-5} \) m, \( L_0 = 1.8 \times 10^{-2} \) m, and \( \Delta T = 75 \mathrm{C}^{\circ} \). We need to use these values to find the coefficient of linear expansion \( \alpha \).

Step 3: Rearranging the Formula

Rearrange the formula for the coefficient of linear expansion to solve for \( \alpha \):\[\alpha = \frac{\Delta L}{L_0 \Delta T}\] Substituting the given values will allow us to find \( \alpha \).

Step 4: Substituting Values and Calculating

Substitute the known values into the equation:\[\alpha = \frac{2.3 \times 10^{-5}}{1.8 \times 10^{-2} \times 75}\]Calculate the coefficient: \[\alpha = \frac{2.3 \times 10^{-5}}{1.35} = 1.7 \times 10^{-6} \mathrm{~m/}\mathrm{m}\cdot\mathrm{C}^{\circ}\]

Step 5: Final Result

The coefficient of linear expansion for the coin is \( 1.7 \times 10^{-6} \mathrm{m/}\mathrm{m}\cdot\mathrm{C}^{\circ} \). This means that for every degree Celsius change, the coin's diameter changes by this coefficient relative to its original size.

Key Concepts

Thermal Expansion

Thermal expansion is an important concept in physics and engineering. It describes how a material expands when its temperature increases. When a substance is heated, its particles begin to move more vigorously. This causes the substance to take up more space and results in the material expanding.
There are three types of thermal expansion: linear expansion, area expansion, and volume expansion.

• Linear expansion is focused on changes in length.
• Area expansion applies to changes in surface area.
• Volume expansion considers changes in overall volume.

In the context of the original problem, we are dealing with linear expansion. This is because we are interested in how the diameter of the coin changes with temperature. The degree of expansion depends on the material's coefficient of linear expansion. Understanding this property helps in designing objects that will undergo temperature changes, like bridges and thermostats.

Linear Expansion Formula

The linear expansion formula is an essential tool for calculating how much a material will increase in size with a temperature change. It's given by:\[\Delta L = \alpha L_0 \Delta T\]where:

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

In our exercise, \( \Delta L \) is the increase in the coin's diameter, \( L_0 \) is the original diameter of the coin, and \( \Delta T \) is the temperature change. Rearranging this formula allows one to solve for any of the variables, provided the other values are known. This flexibility makes it incredibly useful in practical applications, such as determining necessary material clearances and tolerances in manufacturing.

Temperature Change Effect

The temperature change effect is the principle that the size of materials will change when exposed to different temperatures. When a material is heated, its particles gain energy and move more, causes it to expand. Conversely, when cooled, it contracts as particle movement decreases.

In the example of the coin, heating by \(75 \mathrm{C}^{\circ}\) leads to an increase in the diameter. The extent of this effect depends on the coefficient of linear expansion, \(\alpha\). A higher \(\alpha\) implies a larger change for the same temperature change.

• Materials with tightly packed atomic structures have lower expansion coefficients.
• Materials like metals usually have higher coefficients, making them more susceptible to noticeable size changes with temperature fluctuations.

Understanding the temperature change effect and having accurate \(\alpha\) values is crucial in construction, as buildings must accommodate thermal expansion without compromising structural integrity.