Final answer:
To maximize the moment of inertia for four identical point masses around an axis, place the masses as far from the axis as possible. Moment of inertia is calculated using the formula I = Σ m*r², and it depends on the distance from the axis. The highest total moment of inertia is achieved when masses are symmetrically positioned at the maximal distance from the axis.
Step-by-step explanation:
To arrange four identical point masses so that the resulting system has the maximum possible moment of inertia around an axis passing through the crossing point of the beams and going into or out of the screen, you would place the masses as far from the axis of rotation as possible. Moment of inertia, given by I = Σ m*r² (where m is the mass and r is the distance from the axis), increases as the distance to the axis of rotation increases.
In the example of two point masses at the ends of a massless rod, the moment of inertia would be I = m(0)² + m(2R)² = 4mR² when rotational axis is through one of the ends. However, this value is different if the axis is through the midpoint of the rod, illustrating how the position of the axis affects the moment of inertia.
If we apply this principle to the crossed beams with point masses, the configuration with the highest total moment of inertia would be the one where the masses are symmetric around the axis and positioned at the maximum distance from it. To rank the configurations from lowest to highest total moment of inertia, simply order them by the increasing distances of masses from the axis, giving more weight (figuratively) to those configurations where the masses are furthest away.