Question

(a) Suppose a meter stick made of steel and one made of invar (an alloy of iron and nickel) are the same length at 0ºC . What is their difference in length at 22.0ºC ? (b) Repeat the calculation for two 30.0-m-long surveyor’s tapes.

Short answer

(a) The difference in length of meter stick made of steel and invar is $2.44\times {10}^{-4}\mathrm{m}$ for 1 metre of length, and

(b) for 30 meters of length.

Step-by-step solution

Introduction

We calculate the change in length of the steel and invar stick by the formula for linear expansion in solids and find the difference in lengths.

Given parameters and formula for linear expansion

Coefficient of linear expansion of steel =$12\times {10}^{-6}/°C$$\mathbf{12}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{/}\mathbf{°}\mathbf{C}$

Formula for length expansion of solids =$∆\mathrm{L}=\mathrm{\alpha L}∆\mathrm{T}$

Here,

$\alpha$- coefficient of linear expansion.

$\mathrm{L}$-original length

$∆\mathrm{T}$- change in temperature.

 Increase in length for steel & invar  of length 1 metre

(a)

For steel

$\begin{array}{rcl}\mathbf{∆}\mathbf{L}& \mathbf{=}& \mathbf{\alpha L}\mathbf{∆}\mathbf{T}\\ & \mathbf{=}& \left(1.2\times {10}^{-5}/°C\right)\mathbf{\times }\mathbf{1}\mathbf{m}\mathbf{\times }\left(22°C\right)\\ & \mathbf{=}& \mathbf{2}\mathbf{.}\mathbf{64}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}\mathbf{m}\end{array}$

For invar

$\begin{array}{rcl}\mathbf{∆}\mathbf{L}& \mathbf{=}& \mathbf{\alpha L}\mathbf{∆}\mathbf{T}\\ & \mathbf{=}& \left(9.0\times {10}^{-6}/°C\right)\mathbf{\times }\mathbf{1}\mathbf{m}\mathbf{\times }\left(22°C\right)\\ & \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{98}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathbf{m}\end{array}$

$\begin{array}{rcl}\mathbf{Difference}\mathbf{}\mathbf{in}\mathbf{}\mathbf{length}\mathbf{}& \mathbf{=}& \left(2.64\times {10}^{-4}m\right)\mathbf{-}\left(1.98\times {10}^{-5}m\right)\\ & \mathbf{=}& \mathbf{2}\mathbf{.}\mathbf{44}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}\mathbf{m}\end{array}$

 Increase in length for steel & invar  of length 30 metres

(b)

For steel

$\begin{array}{rcl}\mathbf{∆}\mathbf{L}& \mathbf{=}& \mathbf{\alpha L}\mathbf{∆}\mathbf{T}\\ & \mathbf{=}& \left(1.2\times {10}^{-5}/°C\right)\mathbf{\times }\mathbf{30}\mathbf{m}\mathbf{\times }\left(22°C\right)\\ & \mathbf{=}& \mathbf{7}\mathbf{.}\mathbf{92}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}\mathbf{}}\mathbf{m}\end{array}$

For invar

$\begin{array}{rcl}\mathbf{∆}\mathbf{L}& \mathbf{=}& \mathbf{\alpha L}\mathbf{∆}\mathbf{T}\\ & \mathbf{=}& \left(9.0\times {10}^{-6}/°C\right)\mathbf{\times }\mathbf{30}\mathbf{m}\mathbf{\times }\left(22°C\right)\\ & \mathbf{=}& \mathbf{5}\mathbf{.}\mathbf{94}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}\mathbf{m}\end{array}$

$\begin{array}{rcl}\mathbf{Difference}\mathbf{}\mathbf{in}\mathbf{}\mathbf{lenght}\mathbf{}& \mathbf{=}& \mathbf{}\left(7.92\times {10}^{-3}m\right)\mathbf{-}\left(5.94\times {10}^{-4}m\right)\\ & \mathbf{=}& \mathbf{7}\mathbf{.}\mathbf{33}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{m}\left(\frac{1\mathrm{mm}}{{10}^{-3}m}\right)\\ & \mathbf{=}& \mathbf{7}\mathbf{.}\mathbf{33}\mathbf{}\mathbf{mm}\end{array}$

The difference in length of meter stick made of steel and invar is 2.44 X 10^-4 m for 1 metre of length and7.33 mmfor 30 meters of length.