# Why doesn't the moment of inertia of a body equal to the moment of inertia of its centre of mass?

Moment of inertia is the summation (or integration, in case of continous bodies) of the product of the square of perpendicular distance of all individual particles in a system from the axis of rotation and their respective masses. But, for a point sized object, it must be equal to the square of its perpendicular distance from the axis of rotation. So, the moment of inertia of a system should be equal to the the moment of inertia of its centre of mass,since all the mass of the body can be said to be concentrated there,but it doesn't happen. Why is this so ?

• Most times the axea of rotation is through the centre of mass. so you have the moment of all the masses outside the center. Oct 15, 2022 at 14:19

If you place a body with the center of mass on the origin and rotate it, it is still experiencing mass moment of inertia. Why is that?

Because all the individual particles are still moving (having their own momentum) at a distance from the rotation axis.

Mass moment of inertia calculation contains two parts. One is the MMOI due to the offset of the center of mass as you inquired in the question, and the second part is the contribution of the distribution of mass around the center of mass.

The first part is due to the parallel axis theorem, and the second part is from the summation of the perpendicular distances around the center of mass.

$${\bf I}_{\rm origin} =m\, d^2 + {\bf I}_\text{center of mass}$$

I think that your problem is the statement "all the mass of the body can be said to be concentrated there" - in general, this is not always true. When we are dealing with gravitational attractions (assuming that we are using a Newtonian framework), we can say that the bodies behave as if all their mass was concentrated at their center of mass. However, when it comes to rotation, two bodies with the same center of mass but different mass distributions could have very different rotational properties (for instance, a body will have a greater moment of inertia if more of its mass is concentrated farther away from the center). After all, if the body behaved like its center of mass under all circumstances, the moment of inertia would always be $$0$$, which doesn't make much sense at all.

• "When we are dealing with gravitational attractions (assuming that we are using a Newtonian framework), we can say that the bodies behave as if all their mass was concentrated at their center of mass." Not quite; I think you mean if we are dealing with a uniform field. Oct 15, 2022 at 15:28

We are allowed to freely choose the axis of rotation. This axis does not have to go through the center of mass. For such an axis we have to calculate the moment of inertia, and it make perfectly sense that the distance of each infinitesimal mass with respect to this rotation axis is used to calculate the moment of inertia. Don't you agree?