# About which axis the moment of inertia of A body is minimum

I remember my teacher saying it to be the axis along the centre of mass but the centre of mass has all the mass concentrated in it and hence moment of inertia should increase as moment of inertia is proportional to mass?

## Answers and Replies

Ibix
Science Advisor
the centre of mass has all the mass concentrated in it
I'm not sure what this is meant to mean. The mass of a body is the same whatever axis you rotate it about. Why would it change?

Did your teacher mention the parallel axis theorem? https://en.m.wikipedia.org/wiki/Parallel_axis_theorem

I'm not sure what this is meant to mean. The mass of a body is the same whatever axis you rotate it about. Why would it change?

Did your teacher mention the parallel axis theorem? https://en.m.wikipedia.org/wiki/Parallel_axis_theorem
Yeah,he did mention.Centre of mass is where the all the mass is assumed to be concentrated.Just look up for it's definition.

Ibix
Science Advisor
You can treat a body as if it were a point particle of the same mass located at the centre of mass for some purposes. Not all, and rotation is not one if those purposes, since how hard something is to start rotating depends on the mass distribution.

The parallel axis theorem is the rigorous answer to your question - take a look at the derivation in the wiki link. Intuitively, the centre of mass is the place where the average distance to all the little elements of mass making up the body is minimised. If one tries to rotate about an axis that does not pass through the center of mass then the average distance to all the little bits of mass is increased. So the ##mr^2## is increased because the ##r^2## increase, not because the m changes. And the total ##mr^2## is just the moment of inertia.

mfb
Mentor
I remember my teacher saying it to be the axis along the centre of mass but the centre of mass has all the mass concentrated in it and hence moment of inertia should increase as moment of inertia is proportional to mass?
Mass contributes to moment of inertia no matter where your axis is - but mass closer to the axis contributes less (it scales with the squared radius), so a lot of mass close to the axis leads to the smallest moment of inertia. Anyway, you cannot replace an object by a point-mass if you want to calculate its moment of inertia, as discussed in the previous post.

CWatters
One way to clarify the moment of inertia axes ( there are 3 ) is by way if example. Lay a baseball bat on the floor. Now turn the baseball bat around an (vertical) axis with the proviso that the vertical axis is through the center of mass. Next turn the baseball bat around a horizontal axis through the center of mass. The axis you picked will probably be one where the bat rotates in much the same way as the previous case. I takes approximately the same effort to turn the bat in both these cases.

Finally turn the bat through a third axis perpendicular to the first two, but this time turn the bat as though the bat were on a wood lathe. This time much less effort would be required to turn the bat at the same rate. This axis of turning on the wood lathe would be the minimum axis of inertia.