How to use this calculator
- Enter the module. Enter the pinion module m (mm). For an imperial gear, m = 25.4 / DP.
- Enter the pinion teeth. Enter the number of teeth z on the pinion. The pitch diameter is d = m · z.
- Enter the pinion speed. Enter the pinion shaft speed in RPM to get the rack linear speed.
- Enter the pinion torque. Enter the torque at the pinion shaft to get the ideal linear force.
- Read the results. Read travel per revolution, linear speed, linear force and pitch diameter.
How it works
A rack and pinion converts rotary motion into linear motion: a round pinion gear
meshes with a straight toothed bar (the rack), so turning the pinion drives the rack
along in a line. The pinion’s pitch diameter follows from its module and tooth count,
d = m · z
and one full revolution rolls the rack along by exactly the pinion’s
pitch circumference:
travel/rev = π · d = π · m · z.
The rack’s linear speed is just that travel times the rotational
speed: v = (π · m · z) · n / 60 (n in RPM → mm/s). The
linear force comes from torque acting at the pitch radius
r = d / 2 = m · z / 2, so F = T / r. A smaller pinion
multiplies a given torque into more force but moves the rack less per turn — the usual
speed-versus-force trade.
If the same axis is being considered as a screw drive instead of a rack drive, compare torque, thrust and critical speed with the lead screw torque calculator.
Worked example
Verified against the live calculator
Take a module-2 pinion with 20 teeth. The pitch diameter is
d = 2 × 20 = 40 mm, so each revolution moves the rack
π × 40 ≈ 125.66 mm. Spinning the pinion at
100 RPM gives a rack speed of
125.66 × 100 / 60 ≈ 209.4 mm/s. With a pinion torque of
10 N·m and a pitch radius of 0.02 m, the ideal linear force is
F = 10 / 0.02 = 500 N. Those are the numbers the calculator shows for
these inputs.
Frequently asked questions
How do I calculate rack-and-pinion travel per revolution?
One pinion revolution moves the rack by the pinion’s pitch circumference: travel/rev = π · d = π · m · z, where m is the module and z the pinion tooth count. For a module-2 pinion with 20 teeth, d = 40 mm and travel/rev = π × 40 ≈ 125.66 mm. The rack advances that distance for every full turn of the pinion.
How fast does the rack move?
The linear speed of the rack is v = travel/rev × n / 60, where n is the pinion speed in RPM. With travel/rev = 125.66 mm at 100 RPM, v = 125.66 × 100 / 60 ≈ 209.4 mm/s. Speed is set entirely by the pinion pitch diameter and its rotational speed.
How much linear force does a rack-and-pinion produce?
The linear (push) force is F = T / r, where T is the pinion torque and r = d/2 = m·z/2 is the pitch radius. A 10 N·m torque on a 40 mm pinion (r = 0.02 m) gives F = 10 / 0.02 = 500 N. A smaller pinion multiplies torque into more force but moves the rack more slowly per turn.
Does a bigger pinion give more force or more speed?
It is a trade-off. A larger pinion (bigger module or more teeth) moves the rack farther per revolution — more speed for a given RPM — but the longer pitch radius reduces the linear force for the same torque. A smaller pinion does the opposite: more force, less travel per turn.
Is the calculated force the real usable force?
No — F = T / r is the ideal tangential force at the pitch line. It ignores tooth-mesh friction, drive-train efficiency and any acceleration load, so the usable push at the rack is somewhat lower. Apply your drive efficiency and a service factor for sizing.
What is the difference between a straight and a helical rack?
A straight (spur) rack has teeth perpendicular to the travel direction; a helical rack has angled teeth. Helical racks run quieter and carry more load because more teeth are in contact, but they generate an axial thrust the bearings must take. Both must share the pinion’s module and pressure angle, and both have a little backlash that limits positioning precision.
Does this work in metric and imperial?
Yes — enter the module in mm (or convert from diametral pitch with m = 25.4/DP) and the torque in N·m or lbf·ft. Travel, speed and force are shown in SI or imperial. Toggle SI/Imperial in the header.
Method & assumptions
- Travel per revolution equals the pinion pitch circumference π·m·z — the rack rolls without slip on the pitch line.
- The linear force F = T / r is the ideal tangential force at the pitch radius; it ignores mesh friction, drive efficiency and acceleration loads, so the usable push is lower.
- Pinion and rack share the same module and pressure angle. Helical racks run quieter and carry more load but add an axial thrust on the bearings.
- Backlash and elastic deflection are not modelled; both slightly limit positioning accuracy in a real drive.