= Lead
{disambiguate=engineering}
{title2=$L$}
= Screw lead
{synonym}
In <screw thread> geometry, \b[lead] $L$ is the axial advance in one complete relative revolution of the screw and its matching <nut (hardware)>. The term <screw lead> distinguishes this geometric quantity from the chemical element. For $n$ <thread starts> with axial <thread pitch> $p$,
$$
L=np.
$$
A single-start thread has $L=p$, whereas a three-start thread with $p=2\,\mathrm{mm}$ has $L=6\,\mathrm{mm}$.<Selecting the Optimal Screw Technology by Tolomatic>[Screw Terminology, printed pp5–6: lead and the distinction from pitch]{ref}
Assume the screw is restrained axially and the nut is guided against rotation. If the relative <angular displacement> is $\theta$, consistent signs give
$$
x=\frac{L}{2\pi}\theta.
$$
Differentiating gives $v=L\omega/(2\pi)$, where $\omega$ is relative <angular velocity>. Equivalently, rotation at $f$ revolutions per second gives axial speed $v=Lf$. With $L=6\,\mathrm{mm}$ per revolution and $f=5$ revolutions per second, the ideal speed is $30\,\mathrm{mm\,s^{-1}}$; a $90^\circ$ turn advances $1.5\,\mathrm{mm}$.
Unwrapping a cylindrical reference surface of radius $R$ gives a right triangle with circumferential run $2\pi R$ and rise $L$. Its helix inclination $\alpha$, measured relative to a plane perpendicular to the axis, therefore obeys
$$
\tan\alpha=\frac{L}{2\pi R}.
$$
This is a geometric statement about the chosen reference radius. It does not by itself determine the drive <torque>, because contact geometry and <friction> also matter.
A commanded turn is a prediction from geometry, rather than a guarantee of measured travel. <Backlash> can delay reversal motion, and <mechanical compliance> changes displacement under load. These effects remain relevant even when the nominal <screw lead> is known exactly.
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