Before the formula can be used to determine torque, there will be certain aspects of the machine that needs to be considered. FIrst, are the lead screw alignments pretty good? Is there anti backlash mechanisms on the machine which will add friction to the linear motion? The next thing to consider is the properties of the machine, because you will need this information in the formula. Ok, so let's give it a try:

To determine torque, lets gather some information. By the way, we are talking about torque during a continual turning motion, not at a holding position. Let's say we are trying to figure out the necessary torque to move the gantry. This motion is on a horizontal plan, so gravity is less important than with a vertical situation, like the z-axis lead screw. We will need to know the number of inches pre revolution. This is pretty easy since it's the screw # of threads per inch. You will also need to know the pi. Pi is half of a circle in radians, or 3.1415927 rounded to the ten millionth. The formula is as follows: Torque = ((weight)(inches/revolution))/2(pi)(efficiency). You are wondering where do I add in things like friction. There is also a formula to determine the weight that considers friction. Should I expose this formula?!? Hmmm... Well, it doesn't hurt... much. Weight = (load being slid)(Coefficient of sliding friction)+force against sliding mechanism. You can use this formula to your hearts content, but I will not use it here for simplicity.

Ok, so, let's say we have a gantry that weighs 50 lbs and the number of inches per revolution is one tenth (1/10 or .1). This is a standard thread pitch for 1/2 ACME screws (sometimes 1/8, depending on your preference). Therefore, the torque (lbs-in or pounds at one inch) requirements will be: torque = (50 (1/10))/2(pi) or 5 / 6.2831854. We will leave out the efficiency of the screw/nut combination for simplicity. We get a value of about .7958 lbs-in. Now your thinking, that number seems WAY low. That's because it's in pound inches and not ounce inches, which is the typical unit in the US for these ratings. There are conversion charts across the internet to convert this number to oz-in, or Nm (Newton Meter). This conversion happens to be 12.7328 oz-in or torque necessary to move the gantry in constant motion. Now remember, this may seem like a low torque requirement, and there is no factor of safety built into this number. There are those characteristics like friction, misalignment, and torque reduction at high speeds, so work in a bit of wiggle room here. The number also shows how powerful a screw mechanism is, and how easy it is to move heavy loads with this mechanism. When looking around for motors, look at their datasheets which should contain a torque curve related to RPM (revolutions per minute). This is where this formula and applying it to the properties of the CNC machine being considered should be used. I would love to receive some feedback on this topic especially if I need correction by the math engineers out there, or if more clarification and simplicity is desired.