 # A disk with the hole in the middle is heated. Does the hole get bigger, smaller, or stay the same size?

Can you think of any common applications where this can apply?

Assuming the disc is uniform and isotropic (the same in different directions), the hole will expand in the same ratio as the metal. You can see this because the thermal expansion equation

d𝐿=𝐿𝛼d𝑇dL=LαdT

applies to all lengths associated with the metal, including the circumference of the hole, since the edge of the hole is made out of metal. And if the circumference of the hole expands, so does the diameter.

If you have a disc with different regions that are made of different types of metal, or if the metal that makes up your disc has an anisotropic crystal structure (so that it expands by different factors in different directions), then the analysis is more complicated. But in both cases, I think the hole would still get larger since the overall change in size is still an expansion.

In order to get the hole to shrink, you would need to use a material with a negative thermal expansion coefficient 𝛼<0α<0, which means it gets smaller as the temperature gets higher. In that case the entire disc would shrink as it heats up.

A practical application of this would be with metal shrink fits.

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