Main Question or Discussion Point
I feel like this is a very simple concept that I seem to confuse more often than I'd like to admit. Namely, if you have a rotating simple pendulum (or really any object), why does it have 0 translational kinetic energy if it is kept rotating around a fixed axis? The centre of mass is constantly changing its position in space, and although this motion is encompassed within the rotational energy term, how is this much different than the case of a rod rolling on a slope? In either case, isn't the centre of mass position changing at each instant while rolling (in some axis)? Why do we neglect translational energy in the case where there is periodic motion even though the centre of mass is constantly changing? Similarly, why do we include a translation energy term for the case of rotating body (e.g. sphere of radius R) on a slope if the rotational energy already encompasses the rolling motion (i.e. ## d = R\theta##)?