Final answer:
The rotational inertia of a uniform rod rotating about an axis through its center is derived by considering two halved rods, yielding the formula I = (1/12)ML².
Step-by-step explanation:
The question asks about the rotational inertia (also known as moment of inertia) of a uniform rod with a given mass and length, rotating about an axis through its center perpendicular to its length. The moment of inertia is a measure of an object's resistance to changes in its rotational state and depends on the distribution of the object's mass around the axis of rotation. To find the moment of inertia of such a rod, one can start with the known moment of inertia formula for a rod rotated about an axis through one end, which is I = (1/3)ML², where M is the mass of the rod and L is its length.
However, the question pertains to an axis through the center of the rod. To prove that the formula for this is I = (1/12)ML², one can consider the rod as being composed of two rods each of half the length and mass, positioned side by side, so that each contributes (1/3)(M/2)(L/2)² to the total moment of inertia. Summing up the contributions of both halves yields I = 2 * (1/3)(M/2)(L/2)² = (1/12)ML², which is the formula for a rod rotating about an axis through its center.