Final answer:
To calculate the moment of inertia of a thin rod about the neutral axis, we start with the existing moment of inertia about an end axis and use the parallel axis theorem. By integrating the mass elements, we can verify the result, which should be consistent with the theorem's prediction.
Step-by-step explanation:
To determine the moment of inertia of the cross section about the neutral axis for a thin rod with a small cross-sectional area, we start with the known moment of inertia for a rod rotated about an end axis (I = ML² / 3). Recognizing that the axis through the center of the rod is at a distance L/2 from the end axis, we apply the parallel axis theorem, which states that Iparallel-axis = Icenter of mass + Md², where d is the distance between axes.
To find the moment of inertia about the center, we take half the mass of the rod and apply it at each end to simulate the mass distribution (since the rod is uniform). The resulting formula becomes I = (1/2)ML² / 3 + (1/2)ML² / 3, which simplifies to I = ML² / 6. Finally, because we have two of these halves, we multiply by 2, giving us the moment of inertia for rotation about the center axis: I = ML² / 3.
Determining the moment of inertia for a rod about a point L/3 from the end is a matter of integrating the mass elements along the length of the rod. Using the parallel axis theorem, we can check the integration's result, which should align with the theorem's prediction.