What parts would I need?

I want to create a pulley system, this Pulley system will use metal wire. The pulley system needs to pull up and pull to the side and make very small movements as well as being strong. I am not sure of the exact parts.

I believe I need the following?

Two Bipolar Motors with 44 steps (Would Nema 17 Stepper Bipolar Motor do?)
Some kind of wheel to hold and spool the wire
H bridge
Aruino Uno

Wrong type of motor, I think. Stepper motors have low torque and are very inefficient. A geared DC motor might be better, controlled by an H-bridge, with also maybe some rotation feedback from the motor for control.

Look up "Timing belt pulleys". They are commonly used and have a lot of variation regarding size and strength.

The exact parts depend on your project specifications. Nema 17 has 3.2 kg-cm torque, Nema 23 has 19 kg-cm torque, Nema 34 has 42 kg-cm torque and there are probably more that are even stronger. people use these for all kinds of stuff so its pretty easy to find one that meets your specific needs

Nema xx only tells about the mechanical interface, not any electrical data, not torque.
Nema17 uses M3 screws and a certain hole oattern.
Nema23 uses M4 screws and another hole pattern.. .

Well that one can lift 25 pounds, that's enough for what I need. The reason I didn't choose geared DC is because someone told me it must complete a full RPM. I need my pulley to stop at certain points as needed.

That depends on the feedback sensor and where it is attached. Some feedback sensors can provide multiple signals per revolution, e.g. 5 pulses per revolution. Also, if the sensor is attached to the motor rather than the gearbox output, and if the gear ratio is 20x for example, then you could have 5x20 pulses per revolution and be able to register 1% of a revolution.

If this is position control (stopping at particular points) then a stepper motor might be the best approach. However you'll need to calculate the torque required from the geometry of the pulleys and select appropriate gearing.

Not sure when "complete a full RPM" means. Literally you are saying "complete a full revolution per minute" which seems odd to me.

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