Problem 33 - Master Resource Book in JEE Main Physics

Question

A bar of iron is \(10 \mathrm{~cm}\) at \(20^{\circ} \mathrm{C}\). At \(19^{\circ} \mathrm{C}\) it will be \((\alpha\) of iron \(\left.=11 \times 10^{-6} /{ }^{\circ} \mathrm{C}\right)\) (a) \(11 \times 10^{-6} \mathrm{~cm}\) longer (b) \(11 \times 10^{-6} \mathrm{~cm}\) shorter (c) \(11 \times 10^{-5} \mathrm{~cm}\) shorter (d) \(11 \times 10^{-5} \mathrm{~cm}\) longer

Step-by-Step Solution

Answer

(b) The iron bar will be 11 \times 10^{-6} cm shorter.

Step 1: Understand the Concept of Linear Expansion

Linear expansion is the change in length of an object due to a change in temperature. It is given by the formula: \( \Delta L = \alpha \cdot L_0 \cdot \Delta T \), where \(\Delta L\) is the change in length, \(\alpha\) is the coefficient of linear expansion, \(L_0\) is the initial length, and \(\Delta T\) is the change in temperature.

Step 2: Identify Known Values

From the problem statement, we know the following: \( \alpha = 11 \times 10^{-6} /{\degree}C\), \(L_0 = 10 \text{ cm}\), and the change in temperature \(\Delta T = 19^{\circ}C - 20^{\circ}C = -1^{\circ}C\).

Step 3: Calculate the Change in Length

Substitute the known values into the linear expansion formula: \( \Delta L = 11 \times 10^{-6} \times 10 \times (-1) \). This calculates to \( \Delta L = -11 \times 10^{-6} \text{ cm} \).

Step 4: Determine the Nature of Change

Since \(\Delta L\) is negative, it indicates a reduction in length. Therefore, the bar becomes shorter by \(11 \times 10^{-6} \text{ cm}\).

Step 5: Select the Correct Answer

Looking at the options provided, option (b) \(11 \times 10^{-6} \text{ cm}\) shorter matches our calculated result.

Key Concepts

Coefficient of Linear Expansion

The coefficient of linear expansion, often symbolized as \(\alpha\), is a material-specific constant that describes how much a material's length changes with a change in temperature. It's a crucial factor for predicting how different materials behave when their temperatures vary.
This coefficient is given in units of reciprocal temperature, typically in \(\text{°C}^{-1}\). For instance, the coefficient of linear expansion for iron is \(11 \times 10^{-6} \, /\text{°C}\).
This means, for every degree Celsius change in temperature, each centimeter of iron will change in length by \(11 \times 10^{-6}\) times its original length.

• Material-specific: Each material has its own unique coefficient.
• Predictive: It helps in calculating precise length changes.

Understanding \(\alpha\) allows engineers to design structures and systems that can safely withstand temperature changes.

Temperature Change Effect

The effect of temperature change on an object's dimensions is a fundamental concept in physics. When a material undergoes a temperature change, its particles either spread out or come closer together, leading to expansion or contraction.
Here's the essential bit:
The total change in an object's length due to temperature is calculated using \(\Delta L = \alpha \cdot L_0 \cdot \Delta T\), where:

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

For example, if the temperature drops from \(20^{\circ} \text{C}\) to \(19^{\circ} \text{C}\), the particles of the iron bar in the exercise move closer, causing it to contract. Understanding temperature change effects ensures safe and efficient design of everyday applications, such as bridges or railways, which expand and contract with the weather.

Thermal Contraction Calculation

Thermal contraction is the decrease in physical dimensions of a material when it is cooled. This results in the particles of the substance coming closer as the temperature decreases. In the provided exercise, the bar contracts when cooled from \(20^{\circ} \text{C}\) to \(19^{\circ} \text{C}\).
To calculate how much the iron bar shortens, we use the linear expansion formula:
- Substitute the values: \(\Delta L = 11 \times 10^{-6} \cdot 10 \cdot (-1) = -11 \times 10^{-6} \text{ cm}\).
This negative change indicates a reduction in length, not an increase.
Understanding thermal contraction is vital in fields such as engineering and construction, as failure to account for it can lead to structural failures. Recognizing contraction helps in determining correct material tolerances in various temperature conditions.