Distinction between translational and rotational energy

[TheCanadian]

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. $d = R\theta$)?

 

[andrewkirk]

A pendulum bob is usually assumed to be small enough to be treated as a point mass. Point masses have zero moment of inertia and hence cannot have any rotational energy (relative to their own centre of gravity). You are correct that, if given a problem with a large pendulum bob, it would be necessary to introduce additional terms relating to the rotational energy of the bob around its centre of mass. I think that would complicate things quite a lot.

A cylindrical rod rolling down a slope would normally not be considered to have a zero radius. So its moment of inertia around its axis will be nonzero and rotational energy needs to be taken into account.