= Lead screw
A \b[lead screw] converts relative rotation into axial movement through a <screw thread> and matching <nut (hardware)>. In a common arrangement the screw rotates while the nut is restrained from rotating and is guided along the screw axis. The constraint is essential: the useful output is their relative movement, rather than rotation of both parts together.<Selecting the Optimal Screw Technology by Tolomatic>[Introduction, printed p4]{ref}
Let $L$ be the <screw lead>, the axial advance per complete relative turn. With consistent directions and no <backlash>, an <angular displacement> $\theta$ produces
$$
x=\frac{L\theta}{2\pi},\qquad
v=\frac{L\omega}{2\pi},
$$
where $\omega$ is the relative <angular velocity>. A screw with $L=4\,\mathrm{mm}$ per revolution advances $1\,\mathrm{mm}$ in a quarter-turn. This is a geometric prediction; free play, deformation and manufacturing deviations can change the measured motion.
The <thread pitch> $p$ is the axial spacing of adjacent threads. A screw with $n$ thread starts has $L=np$, so a two-start screw with $p=2\,\mathrm{mm}$ has $L=4\,\mathrm{mm}$. Lead and pitch agree for a single-start screw but need not agree otherwise.<Selecting the Optimal Screw Technology by Tolomatic>[Screw Terminology, printed pp5–6: lead, pitch and single/double-start diagrams]{ref}
An ideal slowly moving screw also illustrates the exchange of <torque> for axial <force>. Neglecting <friction>, changes in stored <energy> and other losses, equality of input and output <work> gives
$$
\tau\,d\theta=F\,dx,\qquad
\tau=\frac{FL}{2\pi}.
$$
For $F=500\,\mathrm N$ and $L=0.004\,\mathrm m$, the required ideal torque is about $0.318\,\mathrm{N\,m}$. Real drive torque must also account for <friction> and other loads. Geometry alone does not determine whether an applied axial load can drive the mechanism backwards.
A <lead screw> is therefore both a motion-conversion mechanism and a load path. Its <backlash> affects reversals, while its <mechanical compliance> affects movement under changing <force>; these are distinct contributions to positioning error.
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