When an object is rotating with a constant angular velocity, the relationship between external torque and speed can be described by the following principle: The external torque acting on a rotating object is directly proportional to the angular acceleration produced and, consequently, to the change in angular velocity over time.
Mathematically, this relationship can be expressed as:
τ = Iα
where: τ represents the external torque acting on the object, I denotes the moment of inertia of the object, α signifies the angular acceleration produced by the torque, and the moment of inertia, I, is a measure of an object's resistance to changes in its rotational motion.
When an object is rotating with a constant angular velocity (which means the angular acceleration, α, is zero), it implies that the external torque acting on the object is also zero. In this case, the speed (angular velocity) of the object remains constant, and there is no net external torque causing any change in its rotational motion.