I understand that the Total Kinetic energy can be determined by finding out the rotational kinetic energy about the pivot or by finding the rotational kinetic energy about the centre of mass and then adding the linear kinetic energy about the centre of mass. I have even proven this to myself mathematicaly they are equivalent. I don't understand from a physics point of view why this is the case. Any explanations would be appreciated.
About the pivot, the body can be viewed to be in pure rotational motion since the pivot is the center of rotation of each particle of the rod. Here, the center of mass has no special significance.
On the other hand, if we are not willing to take advantage of the fact that the motion of each particle is perfectly circular, we may view it as combined motion: (about the center of mass) + (of the center of mass).
The second method is particularly useful if it were not possible to classify the motion of the body as circular about some stationary axis. Here the motion of each particle from the ground frame would be really complicated. So it helps to break it down into two parts, which is how we express combined motion. In this case we will have circular motion about an axis through the center of mass(since the body is rigid) and the center of mass could have some arbitrary path.
The two methods are just different ways of describing the motion of the entire body. So one can expect to get the same kinetic energy in both cases.Mathematically, the parallel axis theorem can be used to prove that the two expressions for kinetic energy are indeed equal.