= Nut
{disambiguate=hardware}
A \b[nut] is a component with an internal <screw thread> that engages a matching external thread. In a <lead screw> mechanism, it transfers load through thread contact and can provide the translating output. Either the screw or the nut can be the rotating member, provided the other required motions are constrained.<Selecting the Optimal Screw Technology by Tolomatic>[Introduction, printed p4; Acme Screws, printed p7, load transfer through sliding thread surfaces]{ref}
The useful motion depends on relative rotation. Consider an ideal assembly with <screw lead> $L=2\,\mathrm{mm}$ per relative revolution. Restraining the nut against rotation while turning an axially restrained screw through a quarter-turn produces $0.5\,\mathrm{mm}$ of nut travel. If both rotate together through the same angle without relative rotation, that common rotation produces no axial advance through the thread geometry.
An ideal <work> calculation connects the motion conversion to load transfer. For steady slow motion without <friction> or changes in stored <energy>,
$$
\tau\,d\theta=F\,dx,\qquad
dx=\frac{L}{2\pi}\,d\theta,
\qquad
F=\frac{2\pi\tau}{L}.
$$
With $L=0.002\,\mathrm m$ and $\tau=0.10\,\mathrm{N\,m}$, the ideal axial force is approximately $314\,\mathrm N$. Real contact <friction> and other loads enter the drive-torque balance; the ideal calculation is not a torque specification for a manufactured assembly.
A matching thread must establish a suitable contact path, rather than simply fit through the nut's opening. Clearance can permit <backlash> when the loaded contact surfaces change on reversal. After contact is established, further <displacement> under changing <force> includes <mechanical compliance>. Counting rotations alone therefore does not measure the load-dependent position of the nut.
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