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?
The shape grows by the same factor in each direction, i.e., not changing the shape itself.
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”.
Consider a narrow width ring around the gap.Note that,if the hole contracts in size then the length of the ring decreases which is a contradiction to the fact that length of the metal ring increases with temperature.
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.
