Radius of a stretched spring

When you stretch a coiled spring, the radius decreases slightly, so slightly that you can’t see the difference unless you stretch the spring so much that you damage it.

The math is essentially the same as in the previous post about wrapping Christmas lights around a tree trunk.

If you have a coiled spring of radius r, the points along the coil can be described by

(r cos t, r sin t, ht/2π)

where h is the spacing between turns. If t runs from 0 to T, the length of the spring is hT/2π and the length of the material in the spring, if it were uncoiled, would be

(r² + h²/4π²)1/2 T.

When we stretch a spring, we increase h. We don’t increase the total amount of material, so the radius must decrease, though not by much.

Suppose the spring initially has radius r1 and coil spacing h1. Then when we stretch it the spring has radius r2 and coil spacing h2. Since we haven’t created new material, we must have

(r1² + h1²/4π²)1/2 T = (r2² + h2²/4π²)1/2 T

and so

r1² + h1²/4π² = r2² + h2²/4π².

A small change in h results in a change in r an order of magnitude smaller, for reasons given in the previous post. Both posts boil down to the observation that for y small relative to x,

(x² + y²)1/2  x  = y² /2x + O(y4).

If we choose our units so that the initial radius is on the order of 1, then a change in length on the order of y results in a change in radius on the order of y².