Final answer:
The moment of inertia for a rod with non-uniform density rotating about one end is most closely approximated by ML^2/3, as this is the moment of inertia for a uniform rod under the same conditions. The correct answer appears to be (a) (13mL^2), but the exact value might vary depending on the specific non-uniform mass distribution.
Step-by-step explanation:
The question relates to the rotational inertia (or moment of inertia) of a rod with a non-uniform mass density about an axis through one end. The moment of inertia is essentially the rotational analogue of mass for linear motion, representing an object's resistance to changes in its rotational motion. According to known physics principles, specifically those pertaining to rotational dynamics, the moment of inertia for a long rod spun around an axis through one end perpendicular to its length is ML2/3.
For a rod with a uniform mass distribution, the moment of inertia about one end is indeed ML2/3, which suggests that the correct answer would be closest to option (a) (13mL2). This is greater than the inertia of a point mass spun at the rod's center of mass because the distribution of mass in a rod means that there is more mass farther away from the axis of rotation, which increases the moment of inertia.
Given that the actual mass distribution along the rod is non-uniform, we would need more information about the density function ρ(x) to determine the exact moment of inertia. However, comparing to the example of a uniformly distributed mass, we can say that the rotational inertia is likely to be ML2/3 for a start, with adjustments depending on the nature of the mass density function ρ(x).