# A sidewalk is made in squares of length, 1 meter on each side, with expansion gaps between...

## Question:

A sidewalk is made in squares of length, 1 meter on each side, with expansion gaps between segments to avoid cracking. If sidewalk concrete has a linear expansion coefficient of 15x10{eq}^{-6} {/eq}(1/K), how far apart do the expansion gaps need to be to ensure that the sidewalk doesn't crack? Assume a 50K temperature variation of the sidewalk.

## Thermal Expansion:

When the material is subjected to the change in temperature, then the material undergoes the deformation. Therefore, the rise in the temperature will cause an increase in the dimension of the material, and a fall in the temperature will cause contraction of the material. The thermal expansion or contraction of the material depends upon three factors: one is the original dimension of the material, the second is the linear coefficient of thermal expansion, and the third is the temperature variation.

Given data:

• {eq}L=\rm 1 \ m {/eq} is the length of the concrete sidewalk
• {eq}\alpha=\rm 15 \times 10^{-6} \frac{1}{K} {/eq} is the linear coefficient of expansion of the concrete
• {eq}\Delta T=\rm 50 \ K {/eq} is the change in the temperature

The expansion of the sidewalk is given by the following expression:

{eq}\begin{align} \Delta L&=L \alpha \Delta T \\[0.3cm] \Delta L&=\rm 1 \ m \times 15 \times 10^{-6} \frac{1}{K} \times 50 \ K \\[0.3cm] \Delta L&=\boxed{\rm 7.5 \times 10^{-4} \ m} \end{align} {/eq}

Therefore, the gap between the sidewalk should be {eq}\rm 7.5 \times 10^{-4} \ m {/eq} to avoid cracking.