The axis of rotation for the sphere-rod combination to have the greatest moment of inertia should pass through the smaller 5-kilogram sphere, as this will put the larger mass further from the pivot point, increasing the moment of inertia.
Understanding the Moment of Inertia-
To determine through which point the axis should pass for the moment of inertia of the sphere-rod combination to be greatest, we must consider the distribution of mass relative to the axis of rotation. The moment of inertia is a property that measures the tendency of an object to resist angular acceleration, which depends on the distribution of the object's mass around the axis of rotation. For a rod with negligible mass and two spheres with different masses (5 kg and 10 kg in this case), the moment of inertia will be greatest when the axis of rotation is through the point furthest from the center of mass of the system. Given that the larger mass will contribute more to the moment of inertia, the pivot point should be chosen at the end where the smaller mass (5 kg sphere) is located.
The parallel-axis theorem can be applied when shifting the axis of rotation from the center of mass to another point. The formula for the moment of inertia of a rod rotated about an axis through one end is I = ML² /3, which is greater than it would be for a point mass at the center of mass because that formula would be I = ML² /4. This is because the rods mass is distributed over its length, so more mass is farther from the axis than in the point mass scenario. The moment of inertia for a compound object, such as a rod with attached spheres, can be found by summing the moments of inertia of each part about the same axis.
Therefore, to maximize the moment of inertia for the sphere-rod system, the axis of rotation should pass through the point where the 5-kilogram sphere is, as that will place the more significant portion of mass, the 10-kilogram sphere, further away from the rotation axis, and thus, according to the parallel-axis theorem, increase the moment of inertia.