The moment of inertia for a rod that rotates about the axis perpendicular to the rod and passing through one end is: . If the axis of rotation passes through the center of the rod, then the moment of inertia is: . Give a physical explanation for this difference in terms of the way the mass of the rod is distributed with respect to the axis in the two cases.

Explanation:

he moment of inertia for a rod that rotates about the axis perpendicular to the rod and passing through one end is:  m L²/ 3  where m is mass and L is length of rod

If the axis of rotation passes through the center of the rod, then the moment of inertia is:   m L² / 12

So for the former case , moment of inertia is higher that that in the later case .

In the former case , the axis is at one extreme end . Hence range of distance of any point on the rod from axis is from zero to L .

In the second case , as axis passes through middle point , this range of distance of any point on the rod from axis is from zero to L / 2 .

Since range of distance from axis is less , moment of inertia too will be less because

Moment of inertia = Σ m r² where r is distance of mass m from axis .

increase masses, decrease separation