A Swiss 1 -franc coin has a diameter of 23.200 mm at 20.0^∘ C. The coin is made of a copper-nic

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A Swiss 1 -franc coin has a diameter of 23.200 mm at 20.0^∘...

A Swiss 1 -franc coin has a diameter of $23.200 \mathrm{~mm}$ at $20.0^{\circ} \mathrm{C}$. The coin is made of a copper-nickel alloy (75\% copper and $25 \%$ nickel) for which the coefficient of linear expansion is $16.3 \times 10^{-6} \mathrm{~K}^{-1}$. What would its diameter be on a hot day in Dubai $\left(45^{\circ} \mathrm{C}\right)$ ? On a cold night in the mountains of Greenland $\left(-53^{\circ} \mathrm{C}\right)$ ?

Key Concepts

Linear Thermal Expansion Formula

The linear thermal expansion formula, ?L = ? L? ?T, describes the relationship between the change in a material's length (or any linear dimension) and the temperature change. In this formula, ?L is the change in length, L? is the original length, ?T is the temperature change, and ? is the coefficient of linear expansion. This formula is essential for calculating the dimensional variations of materials in varying thermal conditions.

Temperature Change (?T)

Temperature change, denoted as ?T, is the difference between the final temperature and the initial temperature of a system or material. In the context of thermal expansion, ?T determines whether a material will expand (with an increase in temperature) or contract (with a decrease in temperature), and by how much. This concept is critical for accurately applying the thermal expansion formula in real-world scenarios.

Thermal Expansion

Thermal expansion is the phenomenon where materials change their size (length, area, or volume) in response to a change in temperature. As the temperature increases, the kinetic energy of the particles in the material increases, causing them to occupy more space and thus the material expands. Conversely, when the temperature decreases, the material contracts. This concept is fundamental in physics and engineering as it impacts the design and functionality of various structures and devices.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific parameter that quantifies the extent to which a material's linear dimensions change per degree change in temperature. Represented by the symbol ?, it is typically expressed in units of K?¹. This coefficient enables engineers and scientists to predict how much an object will expand or contract when subjected to temperature changes, making it crucial for applications where precise dimensions are important.

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