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I am asked to find if on heating a metal ring with a small gap, the gap would increase or decrease.

Now, I clearly get that the length of the ring would increase linearly due to coefficient of linear expansion. But shouldn't the gap then decrease in length? Why does my book say that the gap increases?

How is this phenomenon explained? Is there a mathematical relation?

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  • $\begingroup$ Note: Answers below are based on the assumption that the ring is made of some hypothetical, rigid material with no internal stresses, and which will uniformly expand in every direction when heated. An actual, physical ring made of metal may behave differently. Consider, in the extreme case, a ring made of memory wire: en.wikipedia.org/wiki/Shape-memory_alloy $\endgroup$ – Solomon Slow Oct 29 '19 at 12:54
  • $\begingroup$ Related: physics.stackexchange.com/questions/12599/… $\endgroup$ – JMac Oct 29 '19 at 20:02
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The shape grows by the same factor in each direction, i.e., not changing the shape itself.

enter image description here

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  • $\begingroup$ Can you tell me...change in which Direction increases the length of the gap? $\endgroup$ – Techie5879 Oct 29 '19 at 7:46
  • $\begingroup$ as the diameter increases the gap will increase $\endgroup$ – Adrian Howard Oct 29 '19 at 8:24
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A simple way of looking at the problem is to consider the whole ring made up of two parts: the ring with the “gap” and the “gap”.
The whole ring expands and so do two parts which make up the ring and that includes the “gap”.

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The thermal expansion is due to the increased vibration of the particles, thus they are driven further apart. So if you consider the particles on the inner circumference, they too will be driven further apart thus the inner circumference increases thus radius r of the hole increases.

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