Max mechanical power of motors, how is it useful?

[Ken Leung]

I understand that the maximum mechanical power output by a DC motor can be obtain from the equation:

Pmax = 1/4 * Torque max. * w max. (Angular velocity)

Now, my question is, what is so important about this equation? Is it only used to calculate available power from motors, or can it be used in some other way?

I remember that while controlling the motors, we should always set the mechanical power output of motor a little less than the Pmax... Well, how exactly do we do that?

Do we use the value of power reflected after setting up the gearing, or in another words, do we figure out the the mechanical power after we attach the gears and arms? If so, do this mean that we should use gear ratio to control the mechanical power and aim to design a ratio to achieve a P close to Pmax?

Should we only include the gears when calculating power output, or should we also include the components we use (such as wheel, or arms)?