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SOme tight coiled springs are made so that they are already under tension - how do they manufacture them?

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I believe you are referring to Extension Springs, which have a certain degree of pre-tension that holds the coils against each other until a certain force has been applied.

As the spring is wound, the wire is under tension. Imagine wrapping a some elastic around your finger a few times while it is stretched, there will be an inwards force, the diameter wants to be smaller, it wants to crush your finger. So too with the spring - the wire wants to relax, and become shorter, but it can't - the other spring coils are in the way.

Take a spring, hold it at opposite ends, and twist - you will see that it naturally wants to get longer and fatter, or shorter and narrower depending on which way you twist. With an extension spring, it can't get any shorter, so the initial tension from winding remains.

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