For two identical masses connected by a rod, about which point is the moment of inertia smallest?

The moment of inertia of a system quantifies how difficult it is to change its rotational motion about a given axis. For the scenario with two identical masses connected by a rod, the moment of inertia varies depending on where the rotation axis is located.

When considering the point at one of the masses, this location becomes pivotal. The moment of inertia about this axis accounts for the mass located at that point contributing zero to the moment of inertia since it is at the axis of rotation. In contrast, the other mass, which is a distance equal to the length of the rod away from the axis, contributes to the overall moment of inertia. However, the benefit here is that the contribution of that single mass (which is at one end) is minimal compared to when you account for added distances at other rotational axes.

At other suggested locations, like the midpoint or the center of mass, the mass distribution contributes more significantly, resulting in a higher moment of inertia due to the larger distances from the axis. Thus, when the rotation occurs about the point of one mass, the system exhibits the least resistance to rotational motion, leading to a smaller moment of inertia compared to other configurations.