Problem 1065 - JEE Physics

Question

An iron bar of length \(10 \mathrm{~m}\) is heated from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\). If the coefficient of linear thermal expansion of iron is \(\left[\left\\{10 \times 10^{-6}\right\\} / \mathrm{C}\right]\) the increase in the length of bar is (A) \(0.5 \mathrm{~cm}\) (B) \(1.0 \mathrm{~cm}\) (C) \(1.5 \mathrm{~cm}\) (D) \(2.0 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The increase in the length of the iron bar when heated from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) is (B) \(1.0 \mathrm{~cm}\).

Step 1: Identify the given values

The initial length of the iron bar, \(L_0\), is \(10 \mathrm{~m}\). The coefficient of linear thermal expansion, \(α\), is \(\left[\left\\{10 \times 10^{-6}\right\\} / \mathrm{C}\right]\). The initial temperature, \(T_1\), is \(0^{\circ} \mathrm{C}\) and the final temperature, \(T_2\), is \(100^{\circ} \mathrm{C}\).

Step 2: Calculate the change in temperature ΔT

To find the change in temperature, subtract the initial temperature from the final temperature: \(ΔT = T_2 - T_1\). \(ΔT = 100^{\circ} \mathrm{C} - 0^{\circ} \mathrm{C} = 100^{\circ} \mathrm{C}\)

Step 3: Apply the formula for linear expansion

Now, use the formula for the change in length due to thermal expansion: \(ΔL = L_0 × α × ΔT\). Plug in the given values: \(ΔL = 10 \mathrm{~m} \times \left[\left\\{10 \times 10^{-6}\right\\} / \mathrm{C}\right] \times 100^{\circ} \mathrm{C}\).

Step 4: Calculate the increase in length ΔL

By performing the calculation, we find the change in length: \(ΔL = 10 \mathrm{~m} \times \left[\left\\{10 \times 10^{-6}\right\\} / \mathrm{C}\right] \times 100^{\circ} \mathrm{C} = 0.01 \mathrm{~m}\) To express the increase in length in centimeters, we can convert the change in length from meters to centimeters by multiplying by \(100\): \(ΔL = 0.01 \mathrm{~m} \times 100 = 1 \mathrm{~cm}\)

Step 5: Identify the answer in the given options

Since the increase in length is \(1 \mathrm{~cm}\), the correct answer is (B) \(1.0 \mathrm{~cm}\).

Key Concepts

Coefficient of Linear Expansion

The **coefficient of linear expansion** is a critical property when studying how materials expand or contract with temperature changes. It indicates how much a material's length increases per degree Celsius of temperature change. Typically, it is denoted by the symbol \(\alpha\). In the given problem involving an iron bar, the coefficient is given as \([10 \times 10^{-6} / ^\circ\mathrm{C}]\). This value reveals that for every degree increase in temperature, each meter of iron expands by \(10 \mu\mathrm{m}\).
Understanding the coefficient of linear expansion is important for applications where temperature changes can significantly affect the structural integrity or dimensions of materials. For engineers and designers, knowing these values helps in planning and managing potential length variations. For iron, having a relatively small coefficient means changes in length due to temperature variations are minimal compared to materials with higher coefficients.

Temperature Change

In thermodynamics, **temperature change** is an essential concept used to compute how materials respond physically to heat differences. The exercise specifies a temperature change through heating from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\). The change in temperature, represented as \(\Delta T\), is calculated by the difference between the final and initial temperatures: \(\Delta T = T_2 - T_1\).
In the case of the iron bar, the initial temperature \(T_1\) is \(0^{\circ} \mathrm{C}\) and the final temperature \(T_2\) is \(100^{\circ} \mathrm{C}\), making \(\Delta T = 100^{\circ} \mathrm{C}\).
Understanding how to determine the temperature change is critical since it directly affects the calculations of expansion or contraction in materials when applying the formula for linear expansion. In broader applications, this allows prediction of how environmental conditions can alter the size and shape of objects and structures.

Length Increase Calculation

The **length increase calculation** helps determine how much an object expands when subjected to thermal changes. This can be calculated using the formula for linear thermal expansion: \(\Delta L = L_0 \times \alpha \times \Delta T\). Here, \(\Delta L\) represents the change in length, \(L_0\) is the original length, \(\alpha\) is the coefficient of linear thermal expansion, and \(\Delta T\) is the temperature change.
For the iron bar problem, plugging in the values gives \(\Delta L = 10 \mathrm{~m} \times [10 \times 10^{-6} / ^\circ\mathrm{C}] \times 100^{\circ} \mathrm{C} = 0.01 \mathrm{~m}\). To convert this value from meters to centimeters for easier comprehension, multiply by 100, resulting in a change in length of \(1\, \mathrm{cm}\).
This length increase is essential for practical purposes like construction and manufacturing, where precise dimensions are needed despite temperature fluctuations. It shows the importance of considering how materials behave with temperature changes to avoid structural issues.