I’m working on a prototype and I’m having trouble picking out a motor for that job. I need it to be able to move a rope that is about 2-4 mm thick outward (like a winch in reverse almost) and I also need it to be quick. Quick being about 0.025 mm per second. How would I determine this exactly? Unfortunately, math has never been my strong suit and while I have been looking over datasheets I’m not understanding them too well. What I’m asking is how do you determine if a motor has enough torque AND enough speed for an application.
There are a few things that are important, including the characteristics of the load. Is it a constant load? Does it change slowly? Quickly? Make sure to take this into account.
You can get some mileage out of the standard formula for mechanical power, which is $P=\tau\omega$. In other words, for a constant power motor, you can have high torque & low velocity, or low torque & high velocity. So, you can start by calculating the torque required to drive your load. Then, the angular velocity to move the rope at your desired rate of 250 microns/sec. It's straightforward to compute power from that, but don't forget to add a safety factor.
Once you've got the power requirement, you can consult the datasheets to see which motors can deliver that amount of power. I would also recommend using motor sizing tools provided by vendors. For example, this one from Portescap, or this one from Kollmorgen.