= Rack and pinion
A \b[rack and pinion] is a meshing <gear> pair that converts between rotation and straight-line motion. The pinion is the circular gear and the rack has teeth along a straight path. The ideal motion is described using the <gear pitch circle>, rather than the outer tips of the teeth. Manufacturer diagrams compare rack travel over one full pinion revolution and show how tooth spacing affects that travel.<CP Racks and Pinions: KHK Technical Information>[printed pp270–271, Features and Movement of one cycle diagrams]{ref}
For a fixed pinion axis and a guided rack, let $r$ be the pitch-circle radius and choose consistent positive directions. With <angular displacement> $\theta$ in <radians>,
$$
x=r\theta,\qquad v=r\omega,\qquad a=r\alpha,
$$
where $\omega$ and $\alpha$ are <angular velocity> and <angular acceleration>. For an original example, $r=20\,\mathrm{mm}$ and a quarter-turn give $x=10\pi\,\mathrm{mm}\simeq31.4\,\mathrm{mm}$. A rotation rate of2revolutions per second gives $v=80\pi\,\mathrm{mm\,s^{-1}}\simeq251\,\mathrm{mm\,s^{-1}}$.
Ideal lossless <work> transfer also gives
$$
\tau\,d\theta=F\,dx,\qquad F=\frac{\tau}{r}.
$$
Thus a1newton-metre <torque> at the same20millimetre pitch radius corresponds to50newtons of tangential rack force. This is a kinematic and energetic model; it omits tooth deformation and <friction>. <Backlash> makes reversal especially important: a small rotation can take up clearance before the rack moves. The manufacturer's catalogue explicitly describes backlash adjustment through mounting geometry.<CP Racks and Pinions: KHK Technical Information>[printed p271, Features of Tapered Racks and Pinions]{ref}
In a <flywheel>-based <inerter>, a rack and pinion makes relative terminal translation rotate the flywheel. The motion conversion then changes the apparent <mechanical inertance> without requiring the translating assembly to have that same physical mass.
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