I have a question mostly related to mathematics or physics.

I would like to use a stepper motor to move a 2 meter and 10kg aluminum bar.
Like some kind of clock but where the clock hand is 2 meter and 10gkg.
I'm looking for the formula to calculate the torque in order to choose the right motor and gearbox.

That will be quite the load !
Is it going to be on a vertical plane or horizontal ?
Is there going to be any form of counter balance ?
Will there be any runner support at the end ?

Many steppers include the max load in the specifications so using that and adding a safety margin would be the simplest calculation. (+10%)

I suspect a minimum of a NEMA 24 with a suitable rating would be in order.
This may also have a bearing on the driver you need to employ too.

If you have a uniform 2 metre horizontal bar that weighs 10kg and is hinged at one end then you can assume that the weight is concentrated at the mid point and the torque at the hinge would be 10kg m or about 100 Nm

At other angles (assuming the axis is horizontal) the torque will be less.

yes it does indicate the front plate size (mounting) but there are often different body lengths and the longer ones usually indicate that it is capable of more torque.

You can also get them with different shafts and dual shafts which would allow you to add a counter balance on the back side to reduce the torque needed.

I have a question mostly related to mathematics or physics.

I would like to use a stepper motor to move a 2 meter and 10kg aluminum bar.
Like some kind of clock but where the clock hand is 2 meter and 10gkg.
I’m looking for the formula to calculate the torque in order to choose the right motor and gearbox.

Do anyone have an idea on it?

Thank you

Alas steppers are too complex for a simple formula to apply. The best you get are the graphs of
dynamic torque v. speed in the datasheet - there will be several for different supply voltages.

Typically these are only given for the motors nominal max current rating. Reducing current will
reduce torque, but don’t assume its very linear effect.

Resonance effects can strongly affect torque performance in steppers and this is load-dependent too,
so you always have to check a given setup and take the performance graphs with a pinch of salt.

In general microstepping helps by reducing resonance, and torque always drops with speed in
any stepper. High supply voltage improves torque at speed, but has no effect on the hold-in
torque at stationary.

This all assumes the stepper is driven properly with a stepper driver that controls current and
can handle the supply voltage.

In theory yes, in practice you lose power to friction so that, particularly when moving, the output
torque will be somewhat less, but yes gears trade torque for speed. They are just a kind of lever.