Question

At $\mathbf{20}\mathbf{°}\mathbf{\text{C}}$, a rod is exactly$\mathbf{20.05}\mathbf{\text{\hspace{0.17em}cm}}$long on a steel ruler. Both the rod and the ruler are placed in an oven at$\mathbf{270}\mathbf{°}\mathbf{\text{C}}$, where the rod now measures$\mathbf{20.11}\mathbf{\text{\hspace{0.17em}cm}}\mathbf{}$on the same ruler. What is the coefficient of linear expansion for the material of which the rod is made?

Short answer

Coefficient of linear expansion for the given material is$23\times {10}^{-6}/°\text{\hspace{0.17em}C}$

Step-by-step solution

The given data

1. Initial temperature before expansion,${\mathrm{T}}_1=20°\text{C}$
2. Final temperature after expansion,${\mathrm{T}}_2=270°\text{C}$
3. The linear coefficient of expansion,$\mathrm{\alpha }=11\times {10}^{-6}/°\text{\hspace{0.17em}C}$
4. Initial length before the expansion,${\mathrm{L}}_1=20.05\text{\hspace{0.17em}cm}$
5. Final length after the expansion,${\mathrm{L}}_2=20.11\text{\hspace{0.17em}cm}$

Understanding the concept of thermal expansion

When an object's temperature changes, it expands and grows larger, a process known as thermal expansion. If the temperature of a metal rod of length L is raised by an amount$\mathbf{\Delta }\mathbf{\text{T}}$, its length is found to be increased by the amount$\mathit{\Delta }\mathbf{\text{L}}\mathbf{=}\mathbf{\text{L}}\mathit{\alpha }\mathit{\Delta }\mathbf{\text{T}}$where,$\mathit{\alpha }$is the coefficient of linear expansion of the material.

Formula:

The linear expansion of a body, $\mathbf{\Delta L}\mathbf{=}\mathbf{L\alpha }\mathbf{}\mathbf{\Delta T}\mathbf{}$ …(i)

where, $\mathrm{\alpha }$is the coefficient of linear expansion of body.

Calculation of coefficient of linear expansion

The change in length for the section of the steel ruler between its$\text{20.05cm}$mark and$\text{20.11cm}$ mark is given using equation (i) as:

$\begin{array}{c}\mathrm{\Delta L}=\left(20.11\text{\hspace{0.17em}cm}\right)(11\times {10}^{-6}/°\text{\hspace{0.17em}C})(270°\text{C}-20°\text{C})\\ =0.0551\text{\hspace{0.17em}cm}\end{array}$

Thus, the actual change in length for the rod is given as:

$\begin{array}{c}\mathrm{\Delta }\text{L}=(20.11-20.05)\text{cm}+0.055\text{\hspace{0.17em}cm}\\ =0.115\text{\hspace{0.17em}cm}\end{array}$

The coefficient of thermal expansion for the material of which the rod is made using equation (i) is given as:

$\begin{array}{c}\mathrm{\alpha }=\frac{\mathrm{\Delta L}}{\mathrm{L}\cdot \mathrm{\Delta T}}\\ =\frac{0.115\text{\hspace{0.17em}cm}}{20.05\text{\hspace{0.17em}cm}\cdot (270-20)°\text{\hspace{0.17em}C}}\\ =23\times {10}^{-6}/°\text{\hspace{0.17em}C}\end{array}$

Hence, the value of the coefficient of linear expansion is$23\times {10}^{-6}/°\text{\hspace{0.17em}C}$