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: 
*lead* is the pitch of the screw,it probably needs to be in meters if the rest of the equation is SI units.

Yes you can lift a very big load with a screw and a small force - that's why your car has a screw thread jack to change wheels.

Really a screw is just a slope (*inclined plane* in physics speak) so the equation should be pretty easy to derive.

edit: An easy way to check is to consider the energy. In fact whenever you aren't sure in a physics calculation ALWAYS consider the energy, it's often the simplest way.

One turn of the thread moves the object 8mm vertically.

If you have a Torque of 0.26Nm then one turn of the motor is like providing a force of 0.26N at a radius of 1m, and energy is force \* distance.

So a force of 0.26N around a circle of 1m radius is 2\*pi\*1.0m\*0.26N = 1.6J

The energy to lift 20kg vertically 8mm = 20kg \* g \* 0.008m = 1.6J
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