First, I thought theorems do not have proofs. Anyway, for parallel axis theorem, I was given that the addition of translational kinetic energy and rotational kinetic energy proves the parallel axis theorem. I am confused how that happens, although I can see it mathematically. But I cannot understand, what does translational kinetic energy has to do with trying to find the equivalent of inertia of an object that rotates in a different that its centre of mass axis. I don't see where the translational kinetic energy occurs in that example! I would expect in that case the object would have a rotational kinetic energy with an Igg different than the regular Igg which is calculated in respect to the centre of mass axis. But how can we add the translational kinetic energy and then assume that the sum of it with the rotational kinetic energy will help us calculate the Izz? Can you clarify this to me please?
[Physics] Parallel axis theorem proof
moment of inertiareference framesrotational-dynamics
Best Answer
I don't know what proof of the PAT you are referring to which involves using translational or kinetic energy.
The PAT can be proved by using the relationship which exists between two different coordinate systems.
This proof is clearly shown here:
https://en.wikipedia.org/wiki/Parallel_axis_theorem
and in many other references on mechancs.