Problem 13 - Arihant AIEEE Physics

Question

At \(20^{\circ} \mathrm{C}\), a steel ruler of \(20 \mathrm{~cm}\) long is graduated to give correct reading, but when it is used at a temperature of \(40^{\circ} \mathrm{C}\), what will be the actual length of the steel ruler? \(\left[\alpha_{\text {stèel }}=1.2 \times 10^{-5}\left({ }^{\circ} \mathrm{C}\right)^{-1}\right]\) (a) \(22.02 \mathrm{~cm}\) (b) \(19.6 \mathrm{~cm}\) (c) \(20.0048 \mathrm{~cm}\) (d) \(18.0002 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The actual length of the ruler at 40°C is 20.0048 cm, matching option (c).

Step 1: Identify the Initial Parameters

Identify the necessary initial parameters for the problem. We have the initial length of the ruler, \( L_0 = 20 \text{ cm} \), initial temperature \( T_0 = 20^{\circ} C \), the final temperature \( T = 40^{\circ} C \), and the coefficient of linear expansion for steel \( \alpha = 1.2 \times 10^{-5} \left({ }^{\circ}\text{C}\right)^{-1} \).

Step 2: Use the Linear Expansion Formula

Apply the linear expansion formula, which is \( \Delta L = L_0 \alpha \Delta T \). \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

Step 3: Calculate Change in Temperature

Calculate the change in temperature: \( \Delta T = T - T_0 = 40^{\circ} C - 20^{\circ} C = 20^{\circ} C \).

Step 4: Calculate Change in Length

Substitute the values into the linear expansion formula: \( \Delta L = 20 \times 1.2 \times 10^{-5} \times 20 = 4.8 \times 10^{-3} \text{ cm} \).

Step 5: Calculate the Actual Length of Ruler

Add the change in length to the initial length to find the new length: \( L = L_0 + \Delta L = 20 \text{ cm} + 0.0048 \text{ cm} = 20.0048 \text{ cm} \).

Step 6: Select the Correct Answer

Compare the calculated actual length with the given options. The actual length of 20.0048 cm matches option (c).

Key Concepts

Coefficient of Linear Expansion

The coefficient of linear expansion, often denoted by the symbol \( \alpha \), is a fundamental concept in understanding how materials respond to changes in temperature. It measures how much a material expands per unit length per degree change in temperature. Each material has its own unique coefficient of linear expansion, often influenced by its atomic structure and bonding.

For instance, in the original exercise, we considered a steel ruler that has a coefficient of linear expansion \( \alpha = 1.2 \times 10^{-5} \left({ }^{\circ}\text{C}\right)^{-1} \). This specific value for steel means that for every degree Celsius increase in temperature, each centimeter of the steel will expand by \( 1.2 \times 10^{-5} \) cm.

Understanding this coefficient is crucial because it indicates how much an object will expand or contract when exposed to temperature variations. It's particularly important in engineering and manufacturing, where materials are often subjected to temperature changes.

Linear Expansion Formula

The linear expansion formula is key to solving problems involving temperature-induced changes in the dimensions of solids. The formula is expressed as:
\[ \Delta L = L_0 \alpha \Delta T \]
where:

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

This formula enables us to predict how much an object's length will alter in response to a temperature change.

In the exercise on the steel ruler, we were tasked to calculate the change in length when the ruler was subjected to a 20°C increase in temperature. By substituting the given values into the formula, we determined that the change in the ruler's length was \( 4.8 \times 10^{-3} \) cm. This simple calculation used the properties of the material and the extent of temperature change to reach an accurate result.

Temperature Change Calculation

Temperature change calculations are a straightforward yet essential step in problems involving thermal expansion. To compute the change in temperature, you simply subtract the initial temperature from the final temperature:

\[ \Delta T = T - T_0 \]
where:

• \( T \) is the final temperature.
• \( T_0 \) is the initial temperature.

In our exercise, the temperature change was calculated by subtracting the initial temperature of \( 20^{\circ} \text{C} \) from the final temperature of \( 40^{\circ} \text{C} \), yielding a \( 20^{\circ} \text{C} \) increase.

Recognizing this temperature difference allows you to apply the linear expansion formula correctly and solve the problem at hand. While this step seems simple, it is fundamental in ensuring accurate and reliable outcomes in thermal expansion calculations.