I'm designing a simple robot arm but I don't know how to calculate the torque of the rotating base for the robot arm so I can choose the right gear ratio for the stepper motor that will be in the base, can anyone tell me the easy way to calculate it?
Assuming a horizontal rotating base (you never actually said):
You need enough torque to overcome and cancel out friction so that it behaves like a frictionlesss system for any excess torque beyond that. Measure this with a pull force meter pulling on the rotating base at some radius.
Then for that excess torque beyond that it behaves like there is no friction, so then it's more about how fast you want to accelerate and decelerate the arm when rotating (i.e. how fast is the max rotation you desire and how long are you willing to wait to reach it?)
So for to calculate that excess torque you need moment of inertia (get a crude estimate from tables for geometric shapes + parallel axis thereom) and to decide an angular specification specification.
It's not difficult but you will need to spend an hour reading.
Acceleration Torque = (moment of inertia) x (angular acceleration). Add that to the friction torque. Just like F=ma but for rotation.
Of course your motor needs to spin fast enough to reach your max desired angular velocity. A lot of torque but low max RPM will accelerate quickly but not be able to reach the desired speed.