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Thermal expansion is a fascinating concept explaining how materials change size with temperature. When a material, like metal or plastic, is heated, its molecules vibrate more and take up more space. This is why most substances grow larger when they get hotter. Conversely, when they cool down, they shrink. The physical property that describes this behavior is called the **coefficient of linear expansion**.

For example, in the case of the ruler from the problem, the initial temperature is 25°C, well within the normal room temperature range. When the temperature drops to -14°C, the ruler's length decreases. This can be calculated using the formula for linear expansion:

• **\( \Delta L = \alpha L_0 \Delta T \)**

Here,

• **\( \alpha \)** is the coefficient of linear expansion**
• **\( L_0 \)** is the original length**
• **\( \Delta T \)** is the change in temperature.**

In this scenario, as we substitute the given values, we can compute how much the ruler shrinks due to this change in temperature.

Hooke's law is a principle of physics that states the force needed to extend or compress a spring is proportional to the distance you stretch it. This concept is not just limited to springs but applies to most elastic materials to some extent. The law is usually formulated as:

• **\( F = k \Delta x \)**

Where

• **\( F \)** is the force applied**
• **\( \Delta x \)** is the extension (or compression)**
• **\( k \)** is the spring constant or stiffness**

However, for our ruler example, Hooke's law is adapted for stretching using Young's modulus, which characterizes how a material deforms under stress. The adapted formula becomes:

• **\( F = Y \frac{\Delta L}{L_0} A \)**

Where

• **\( Y \)** is Young's modulus which measures material stiffness**
• **\( \Delta L \)** is the change in length**
• **\( L_0 \)** is the original length**
• **\( A \)** is the cross-sectional area**

By knowing how much force (\( 1.2 \times 10^3 \) N) is applied to counteract the shrinking, we can solve for Young's modulus to understand the elastic nature of the material.

Linear expansion occurs when the length of an object changes due to temperature shifts. While thermal expansion can happen three-dimensionally— affecting length, width, and height—linear expansion simplifies the concept to just one dimension along the length.

To better understand, think about how a railroad track or a metal bridge expands in the heat. Shouldn't they crack in warmer weather? Luckily, engineers design them with tiny gaps allowing room to expand and contract. The concept involves relying on the coefficient of linear expansion which differs between materials. It determines how much a material expands per degree change in temperature.

• **Formula: \( \Delta L = \alpha L_0 \Delta T \)**

Using our previous parameters:

• \( \alpha \)** is the coefficient of linear expansion measured in per °C.**
• \( \Delta T \)** stands for temperature change**
• \( L_0 \)** is the initial length**

In the exercise, the problem tasks us with calculating the length change of a ruler exposed to different temperatures. The value is pivotal as it provides insight into how much the material contracts or expands, impacting the calculation of Young's modulus and ensuring the correct measurement from a structural perspective.