91Ó°ÊÓ

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,$\mathrm{Î\pm }=11×{10}^{−6}/°\text{ C}$
4. Initial length before the expansion,${\mathrm{L}}_1=20.05\text{ c³¾}$
5. Final length after the expansion,${\mathrm{L}}_2=20.11\text{ c³¾}$

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$\mathbf{Δ}\mathbf{\text{T}}$, its length is found to be increased by the amount$\mathit{Δ}\mathbf{\text{L}}\mathbf{=}\mathbf{\text{L}}\mathit{Î\pm }\mathit{Δ}\mathbf{\text{T}}$where,$\mathit{Î\pm }$is the coefficient of linear expansion of the material.

Formula:

The linear expansion of a body, $\mathbf{Δ³¢}\mathbf{=}\mathbf{³¢Î\pm }\mathbf{}\mathbf{Δ°Õ}\mathbf{}$ …(¾±)

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

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:

$\begin{array}{c}\mathrm{Δ³¢}=\left(20.11\text{ c³¾}\right)(11×{10}^{−6}\text{ }/°\text{ C})(270°\text{C}−20°\text{C})\\ =0.0551\text{ c³¾}\end{array}$

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

$\begin{array}{c}\mathrm{Δ}\text{L}=(20.11−20.05)\text{cm}+0.055\text{ c³¾}\\ =0.115\text{ c³¾}\end{array}$

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

$\begin{array}{c}\mathrm{Î\pm }=\frac{\mathrm{Δ³¢}}{\mathrm{L}â‹\dots \mathrm{Δ°Õ}}\\ =\frac{0.115\text{ c³¾}}{20.05\text{ c³¾}â‹\dots (270−20)°\text{ C}}\\ =23×{10}^{−6}/°\text{ C}\end{array}$

Hence, the value of the coefficient of linear expansion is$23×{10}^{−6}/°\text{ C}$