Question: Say I have a motor with a certain amount of torque, $T.$ It is turning a ball screw mechanism like [this](http://www.omc-stepperonline.com/threaded-rod-nema-17-external-linear-stepper-motor-34mm-body-100mm-t8-x-4-p-212.html) . 

Say, I mount something on the nut, I want to calculate the mass of an object I can lift upwards, i.e. the force that the actuator can generate.

I think this equation is what I want: $$T=\frac{Fl}{2\pi\nu}$$ [source](https://en.wikipedia.org/wiki/Ball_screw#Equations)

But I don't understand:

$\bullet$ What is the "lead" $l\,?$ Is it the millimeter pitch distance between grooves? 

If so, using that tiny motor above (I have a similar one) it seems that $F = 2\cdot \pi \cdot 0.26~\mathrm{Nm / 8mm}$ which is $204$ Newtons, so equating to $F = mg$ it seems I could lift $204~\mathrm N/9.8~\mathrm{m/s^2} = 20.8~\mathrm{kg}\,?$ This seems really heavy for that small motor, so I figure something is wrong with my interpretations and/or calculations.

$\bullet$ How is the wikipedia formula derived?

Answer: 
I think this is far more a question for EngineeringSE, but just to hopefully clear up two points

[![enter image description here](https://i.stack.imgur.com/ZulZH.png)](https://i.stack.imgur.com/ZulZH.png) 

The lead on a thread.

The derivation follows, but it's not exactly the same as Wikipedia.

[![enter image description here](https://i.stack.imgur.com/5a1iW.jpg)](https://i.stack.imgur.com/5a1iW.jpg)

$\eta\_{thread} \eta\_{thrust}$ might be combined as efficiency in the Wikipedia equation.

> 
> If so, using that tiny motor above (I have a similar one) it seems that F = 2\*pi \* 0.26Nm / 8mm which is 204 Newtons, so equating to F = m\*g it seems I could lift 204N/9.8m/s2 = 20.8 KG?? This seems really heavy for that small motor, so I figure something is wrong with my interpretations and/or calculations.
> 
> 
> 

My apologies, I am an idiot, Martin's answer spells it out.
It is simple:

F = T\*2\*Pi\*eff/l

Where:
F: Force, newtons; 
T: Torque, newton-meters;
l: lead, meters;
eff: ballscrew efficiency

In most cases, efficiency can be set safely to 90% with enough margin.

So, for example, a ball screw with 5mm lead, driven by 2Nm motor will give:

2\*2\*Pi\*0.9/0.005 = 2262N = 231kg