# The length of an uncoiled spring

by lntz
Tags: length, spring, uncoiled
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,353 A spring is, as zapz suggested, is a helix. It can be modeled as $x= r cos(t)$, $y= r sin(t)$ [/tex]z= (h/2\pi)t[/tex] where "r" is the radius of the helix and "h" is the vertical distance between two consecutive "turns' of the helix. "t" is the parameter giving each point on the heiix as t varies. $dx/dt= -r sin(t)$, $dy/dt= r cos(t)$, and $dz/dt= h/2\pi$. The "differential of arclength is $$\sqrt{(dx/dt)^2+ (dy/dt)^2+ (dz/dt)^2}dt=$$$$\sqrt{r^2sin^2(t)+ r^2cos^2(t)+ r^2sin^2(t)+ h^2/4\pi^2}dt$$$$= \sqrt{r^2+ h^2/4\pi^2}dt$$ The total length is the integral of that from whatever t determines the beginning of the helix to whatever t determines the end of the helix. And, since that differential is a constant, it is just the that constant times the difference in to two "t"s. In particular, if we take z= 0 as the start and z= H as the end, because $z= (h/2\pi)t$ we have $(h/2\pi)t= 0$ at one end and $(h/2\pi)t= H$ at the other so that t= 0 and $t= 2\pi H/h$ at the other. The length of the helix is $(2\pi H/h)\sqrt{r^2+ h^2/4\pi^2}dt$