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Chapter 7: Problem 1065

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) \(2.0 \mathrm{~cm}\)

Short Answer

Expert verified
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 by step solution

01

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 \(100^{\circ} \mathrm{C}\).
02

Calculate the change in temperature ΔT

To find the change in temperature, subtract the initial temperature from the final temperature: \(ΔT = T_2 - T_1\). \(ΔT = 100^{\circ} \mathrm{C} - 0^{\circ} \mathrm{C} = 100^{\circ} \mathrm{C}\)
03

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: \(ΔL = 10 \mathrm{~m} \times \left[\left\\{10 \times 10^{-6}\right\\} / \mathrm{C}\right] \times 100^{\circ} \mathrm{C}\).
04

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 \(100\): \(ΔL = 0.01 \mathrm{~m} \times 100 = 1 \mathrm{~cm}\)
05

Identify the answer in the given options

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

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