rack and pinion system

There is some web sites which permits you to calculate rack and pinion and other kind of moving part linear systems. HAve a googlew search for rack & pinion calculation and I am pretty sure you will find it. But wht I do not understand is the loads and all other info about the machine weight and reductions.....

You might find something here https://www.sdp-si.com/products/Gears/Index.php

The circumference of the pitch diameter of a gear is how far it will move on a rack.
3.1416 * pitch diameter = linear travel.

I think it would be based on the arc length formulas. i.e, s = rθ

s=arc length
r = radius
θ = is the angle in radians.

example circumference of a 2" circle is PI* d = 3.14 *2 = 6.28
s = 1 * (360 deg) (360 deg = 6.283 radians)
s = 1 * 6.283

You get the same answer calculating the arc length of 360 degrees. radians is a unitless unit. There are 2*PI radians in 360 degrees.