91Ó°ÊÓ

Young’s Modulus, denoted by the symbol \( Y \), is a fundamental concept in physics and engineering. It measures a material's ability to withstand length changes under tensile (stretching) or compressive (squeezing) forces. In simpler terms, it tells us how "stiff" or "flexible" a material is. For example, steel with a high Young’s Modulus is very stiff.
Young's Modulus is expressed mathematically as:

• \( Y = \frac{\text{stress}}{\text{strain}} \)

Here, stress is the force per unit area (\( \sigma = \frac{F}{A} \)) and strain is the fractional change in length (\( \epsilon = \frac{\Delta L}{L} \)).
This dimensionless ratio highlights how much force a material can endure before it deforms.
In our exercise, Young’s Modulus is given as \( 10^{11} \text{ N/m}^2 \), indicating the wire’s strong resistance to deformation under stress.

The Coefficient of Linear Expansion, often represented by \( \alpha \), describes how the size of an object changes with temperature. Specifically, it measures the fractional change in length per degree change in temperature. Materials expand when heated and contract when cooled, and \( \alpha \) quantifies this behavior.
The formula for calculating the change in length is:

• \( \Delta L = \alpha \cdot L \cdot \Delta T \)

Here, \( L \) is the original length, and \( \Delta T \) is the temperature change.
In simpler terms, \( \alpha \) tells us how much longer or shorter an object becomes when the temperature changes. For example, materials with a high \( \alpha \) would expand significantly with a small temperature rise.
In the given exercise, \( \alpha \) is \( 10^{-5} /^{\circ}\text{C} \), indicating that the wire expands slightly for each degree Celsius increase in temperature.

Temperature changes have a substantial impact on the physical dimensions of solids, which can cause stress if the expansion or contraction is restricted. When solids are heated, their atoms vibrate more vigorously, resulting in expansion and an increase in length.
In our scenario, to prevent the wire from expanding when heated from \( 0^{\circ} \text{C} \) to \( 100^{\circ} \text{C} \), an internal force is required. This results in thermal stress:

• Calculating this force involves the formula: \( \sigma = Y \alpha \Delta T \)
• Where \( \Delta T \) is the change in temperature, \( Y \) and \( \alpha \) are material properties.

This thermal stress is, in essence, the stress developed when the material doesn't expand due to external constraints. It's crucial in engineering, particularly in designing structures that sustain temperature fluctuation, to prevent damage or failure.
Understanding the effect of temperature changes on solids assists in predicting behavior and planning for material selection and design processes.