Q16PE - College Physics (Urone)

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.

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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

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

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

Step 4: 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 and 7.33 mm for 30 meters of length.