Rack and pinion travel and force, explained

A rack and pinion is a gear pair where one gear's radius went to infinity — which collapses all the gearing math into two lines. The pinion's pitch circle rolls along the rack without slipping, so one revolution moves the rack exactly one pitch circumference, and torque becomes force over the pitch radius.

The two relations

travel/rev s = π·d = π·m·z · force F = T / (d/2)

Module m and tooth count z set the pitch diameter d = m·z; everything else follows. Linear speed is travel per rev times RPM; the rack and pinion calculator runs all four outputs from the four inputs. (Inch racks specify diametral pitch — convert with m = 25.4/DP — and the underlying tooth geometry lives in the module vs DP guide.)

Worked example — m2, 20 teeth

A 20-tooth module-2 pinion at 300 RPM with 10 N·m at the shaft:

d = 40 mm · s = π × 40 = 125.7 mm/rev · v = 37.7 m/min · F = 10 / 0.020 = 500 N

Those are ideal numbers — tooth friction, preload and efficiency take their cut in practice. Note what the pinion size did: swap to a 12-tooth pinion and the same torque pushes 833 N, but each revolution only travels 75.4 mm. Pitch diameter is the lever arm.

Where the speed-force tradeoff lives

Because the rack "ratio" is fixed by geometry, the tradeoff happens upstream: put a 2:1 belt or gearbox ahead of the pinion and the example becomes 150 RPM / 20 N·m — 18.8 m/min at 1,000 N. Same motor power, double the push. Sizing that stage is ordinary gear ratio / torque-power-RPM work, and the pinion shaft itself still answers to shaft torsion and keyway checks.

Common mistakes

• Using the pinion OD instead of pitch diameter. The tips stand one addendum (= m) proud per side; travel and force math on OD runs ~10% optimistic on small pinions.
• Forgetting the force is tangential. A 20° pressure angle adds a separating component (F·tan 20° ≈ 0.36·F) pushing pinion and rack apart — the guide rails carry it, and skipping that check is where racks get noisy.
• Chasing force with a tiny pinion. Below ~17 teeth undercut weakens the very teeth carrying the higher load; gear the motor down instead.
• Quoting backlash as a class instead of a number. Rack backlash depends on mounting distance — the pinion-to-rack center distance tolerance matters more than the rack grade in most installs.

Frequently asked questions

How do you calculate rack and pinion travel?

Travel per pinion revolution is the pitch circumference: s = π × module × teeth. An m2, 20-tooth pinion has a 40 mm pitch diameter and moves the rack 125.7 mm per turn — at 300 RPM that is 37.7 m/min of linear speed.

How do you calculate rack and pinion force?

Force = torque ÷ pitch radius. The same m2/20T pinion (20 mm pitch radius) under 10 N·m pushes the rack with 500 N, before friction and tooth losses. Smaller pinion, more force: a 12-tooth version delivers 833 N from the identical torque.

What is the gear ratio of a rack and pinion?

There isn’t one in the usual sense — the rack is a gear of infinite radius, so the "ratio" is a conversion constant: π·m·z millimeters of travel per revolution. Trade speed against force upstream instead, with a gearbox or belt stage ahead of the pinion: gearing 2:1 halves rack speed and doubles rack force.

Why use a small pinion or a large pinion?

Small pinions give more force per N·m and finer positioning per motor step, but fewer teeth in mesh, more sliding, and undercut risk below ~17 teeth. Large pinions run smoother and faster per revolution but demand more torque for the same push. The motor, not the pinion, is usually the right place to change the tradeoff.