Consider a wire that undergoes an infinitesimal change from an initial equilibrium state to a final equilibrium state. (a) Show that the change of tension is equal to $$ d \mathcal{F}=-\alpha A Y d T+\frac{A Y}{L} d L $$ (b) A nickel wire of cross-sectional area \(0.0085 \mathrm{~cm}^{2}\) under a tension of \(20 \mathrm{~N}\) and a temperature of \(20^{\circ} \mathrm{C}\) is stretched between two rigid supports \(1 \mathrm{~m}\) apart. If the temperature is reduced to \(8^{\circ} \mathrm{C}\), what is the final tension? (Note: Assume that \(\alpha\) and \(Y\) remain constant at the values of \(1.33 \times 10^{-5} \mathrm{~K}^{-1}\) and \(2.1 \times 10^{6} \mathrm{~Pa}\), respectively.)

When a material such as a wire is heated or cooled, its length changes. This phenomenon is known as linear thermal expansion.
The amount of change in length \( \Delta L \) of a material is directly proportional to the initial length \( L_0 \) and the temperature change \( \Delta T \).
The formula that describes this relationship is:
\[ \Delta L = \alpha L_0 \Delta T \]
Here, \( \alpha \) represents the coefficient of linear thermal expansion, a material-specific constant.

For example, if you heat a metal wire, it expands because the heat increases the motion of the atoms, pushing them farther apart. Conversely, if you cool the wire, it contracts because the motion of the atoms decreases.
Understanding linear thermal expansion is crucial for various engineering and physics applications, such as designing bridges, buildings, and electrical components that must endure temperature changes without failing.

Young's modulus, often represented by \( Y \), is a measure of the stiffness of a solid material. It is a fundamental concept in materials science and mechanical engineering.
It quantifies the relationship between stress (force per unit area) and strain (proportional deformation) in a material.
The formula for Young's modulus is:
\[ Y = \frac{Stress}{Strain} = \frac{F/A}{\Delta L/L} \]
Where:
- \( F \) is the force applied
- \( A \) is the cross-sectional area
- \( \Delta L \) is the change in length
- \( L \) is the original length

A higher Young's modulus indicates a stiffer material, meaning it requires more force to deform it.
For example, steel, with a high Young's modulus, is much stiffer compared to rubber, which has a low Young's modulus.
This property is essential for predicting how materials will behave under different types of loads, ensuring they remain safe and effective in their applications.

Thermodynamic equilibrium is a state where a system's properties are unchanging in time because the system is in perfect balance.
Consider a wire in thermodynamic equilibrium. The temperature is uniform throughout the wire, and there is no net flow of energy or matter in or out of the system.
In this state, all the forces and energy exchanges within the wire are balanced, leading to a stable situation.
If the wire's equilibrium state is disturbed, for example, by changing the temperature, the system will eventually return to a new equilibrium state.
For example, cooling the wire will initially cause it to contract due to linear thermal expansion properties. The wire will adjust its tension due to Young's modulus properties until it reaches a new equilibrium. Understanding thermodynamic equilibrium is vital in physics and engineering because it helps predict how systems respond to changes, ensuring they function correctly under various conditions.