When you push on a rotating object, when are you doing work?
That’s an interesting question and requires two answers. First, if you push on a part of the rotating object and that part moves a distance in the direction of the force you exert, then you do work on it. In principle, it is possible to identify all the work that you do on the rotating object via this approach.
However, it is also possible to determine the work you do entirely in terms of physical quantities of rotation. If you exert a torque on the rotating object and it rotates the an angle in the direction of your torque, you again do work on the object. That’s the rotational version of the work formula: whereas force time distance is the translational work formula, torque times angle is the rotational work formula.
An important complication arises, however, in that you must measure the angle in the appropriate units: radians. The radian is the natural unit of angle and is effectively dimensionless (no units after it). When you multiple the torque times the angle in radians, the resulting units are those of work and energy. If you use a non-natural unit of angle, such as the degree, then you’ll have to deal with presence of the angle unit in your result.