Problem 13 - University Physics with Modern Physics

Question

A U.S. penny has a diameter of 1.9000 \(\mathrm{cm}\) at \(20.0^{\circ} \mathrm{C} .\) The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is \(2.6 \times 10^{-5} \mathrm{K}^{-1} .\) What would its diameter be on a hot day in Death Valley \(\left(48.0^{\circ} \mathrm{C}\right) ?\) On a cold night in the mountains of Greenland \(\left(-53^{\circ} \mathrm{C}\right) ?\)

Step-by-Step Solution

Answer

1.9014 cm in Death Valley; 1.8964 cm in Greenland.

Step 1: Identify the Given Values

The initial diameter of the penny, \(d_0\) is \(1.9000\, \mathrm{cm}\). The coefficient of linear expansion, \(\alpha\), is \(2.6 \times 10^{-5} \, \mathrm{K}^{-1}\). The initial temperature, \(T_0\), is \(20.0^{\circ} \mathrm{C}\). The final temperatures are \(48.0^{\circ} \mathrm{C}\) for Death Valley and \(-53^{\circ} \mathrm{C}\) for Greenland.

Step 2: Apply the Linear Expansion Formula

The formula for linear expansion is: \[ d = d_0 (1 + \alpha \Delta T) \] where \(\Delta T\) is the change in temperature and \(d\) is the final diameter.

Step 3: Calculate Diameter for Death Valley

For Death Valley, \(\Delta T = 48.0^{\circ} \mathrm{C} - 20.0^{\circ} \mathrm{C} = 28.0^{\circ} \mathrm{C}\). Substitute values into the formula: \[ d = 1.9000 \times (1 + 2.6 \times 10^{-5} \times 28) \]Calculate the final diameter, \(d\): \[ d \approx 1.9000 \times 1.000728 \approx 1.9014\, \mathrm{cm} \]

Step 4: Calculate Diameter for Greenland

For Greenland, \(\Delta T = -53.0^{\circ} \mathrm{C} - 20.0^{\circ} \mathrm{C} = -73.0^{\circ} \mathrm{C}\). Substitute values into the formula: \[ d = 1.9000 \times (1 + 2.6 \times 10^{-5} \times -73) \]Calculate the final diameter, \(d\): \[ d \approx 1.9000 \times 0.998102 \approx 1.8964\, \mathrm{cm} \]

Step 5: Conclusion

During a hot day in Death Valley, the diameter of the penny expands to approximately \(1.9014\, \mathrm{cm}\), while on a cold night in Greenland, it contracts to approximately \(1.8964\, \mathrm{cm}\).

Key Concepts

Linear Expansion

Linear expansion refers to the change in length, width, or diameter of a material when it's subjected to a change in temperature. This happens because the particles within the material gain energy and move more vigorously, causing the entire structure to expand. This effect is linear, meaning it scales proportionally with the original dimension and the temperature change.
When something undergoes linear expansion, only one dimension changes. For a long, thin object like a wire or rod, this might be its length. But for other objects, like the diameter of a penny, it could be the width that expands.
Understanding this concept helps us foresee how materials behave in different temperatures. Engineers and designers use knowledge of linear expansion to ensure that structures remain safe and functional over a range of temperatures.

Temperature Change

The change in temperature (\(\Delta T\)) is a critical factor in thermal expansion. It is simply the difference between the initial and final temperatures.
An increase in temperature generally causes materials to expand, whereas a decrease causes them to contract.

• For Death Valley: The temperature increased from \(20^{\circ} \mathrm{C}\) to \(48^{\circ} \mathrm{C}\), so the temperature change \(\Delta T\) is \(28^{\circ} \mathrm{C}\).
• For Greenland: The temperature dropped from \(20^{\circ} \mathrm{C}\) to \(-53^{\circ} \mathrm{C}\), resulting in a temperature change of \(-73^{\circ} \mathrm{C}\).

Accurately determining \(\Delta T\) is essential for applying the linear expansion formula effectively.

Coefficient of Linear Expansion

The coefficient of linear expansion, denoted as \(\alpha\), is a constant unique to each material that describes how much it expands or contracts per degree change in temperature per unit length. This coefficient thus provides a measure of how sensitive a material is to temperature changes.
For the metal alloy used in the penny, the coefficient of linear expansion is \(2.6 \times 10^{-5} \mathrm{K}^{-1}\). This relatively small number indicates that the penny will not change drastically in size with temperature. However, over large temperature ranges like those considered here, the change in size becomes noticeable.
Using the coefficient of linear expansion is crucial when calculating how an object's size will change with temperature, as it directly influences the degree of expansion or contraction the object will experience.

Diameter Calculation

Calculating the change in diameter of an object subjected to thermal expansion involves using the linear expansion formula: \[d = d_0 (1 + \alpha \Delta T)\]In this formula:

• \(d\) is the final diameter or size of the object.
• \(d_0\) is the initial diameter.
• \(\alpha\) is the coefficient of linear expansion.
• \(\Delta T\) is the change in temperature.

Applying this to the penny for different temperature scenarios:

• **Death Valley:** For a temperature increase to \(48^{\circ}\mathrm{C}\), the diameter increases to approximately \(1.9014\, \mathrm{cm}\).
• **Greenland:** For a temperature decrease to \(-53^{\circ}\mathrm{C}\), the diameter decreases to approximately \(1.8964\, \mathrm{cm}\).

Understanding this calculation helps us predict how objects will behave in practice under varying thermal conditions.