Question

Short answer

The required temperature is$\mathrm{\Delta }T=\pm 0.11{}^{\circ }\mathrm{C}$.

Step-by-step solution

Given data

The standard bar should be within$\mathrm{\Delta }L=\pm 1.0\mu \mathrm{m}$.

The coefficient of linear expansion is$\alpha =9\times {10}^{-6}{/}^{\circ }\mathrm{C}$.

The length is $L=1\mathrm{m}$.

Expression for linear expansion

In this problem, use the expression for the linear expansion of a material to determine the temperature required.

Calculation of the temperature

The relation to find the requiredtemperature can be written as:

$\mathrm{\Delta }L=\alpha L\mathrm{\Delta }T$

Here,$\mathrm{\Delta }T$is the required temperature.

On plugging the values in the above relation, you get:

$\begin{array}{rl}\left(1\mu \mathrm{m}\times \frac{{10}^{-6}\mathrm{m}}{1\mu \mathrm{m}}\right)& =\left(9\times {10}^{-6}{/}^{\circ }\mathrm{C}\right)\left(1\mathrm{m}\right)\mathrm{\Delta }T\\ \mathrm{\Delta }T& =\pm 0.11{}^{\circ }\mathrm{C}\end{array}$

Thus, $\mathrm{\Delta }T=\pm 0.11{}^{\circ }\mathrm{C}$is the required temperature.