According to what was given in the lecture, any cross-section of any shape has a set of principal centroidal axes where the product of inertia is zero.

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Consider the concept of planar area inertia. When you compute the product of inertia Ixy in some coordinate system, it may be nonzero for an arbitrary orientation.

Option 1 (True): The inertia tensor for a planar lamina is symmetric, which guarantees the existence of a pair of perpendicular axes—principal centroidal axes—about the centroid in which the product of inertia vanishes (Ixy = 0). By rotating the coordinate axes to align with these principal directions, the inertia matrix becomes diagonal, eliminating the cross-term. This is a standard result in rigid-body and structural analysis: every planar area has principal axes through its centroid where the product of inertia is zero.

Option 2 (False): Saying that a cross-section always has nonzero product of inertia in any orientation would ignore the fundamental fact that a suitable rotation of axes can diagonalize the inertia tensor. While a given, arbitrary coordinate frame may yield Ixy ≠ 0, you can always find the principal centroidal axes by diagonalizing the inertia tensor, at which point Ixy = 0. Therefore, the statement as written is not correct because it ignores the rotational freedom that yields zero product of inertia in principal axes.

Option 3 (False alternative if present): If someone claimed that only certain shapes possess principal centroidal axes with Ixy = 0, that would conflict with the universality of the principal-axis theorem for planar laminae, which holds for any shape with a well-defined centroid and area moment of inertia.

Option 4 (False alternative if present): Claims that the product of inertia cannot be made zero by rotation are incorrect, since diagonalizing the inertia tensor by rotating the axes to principal directions always eliminates the cross-term for planar sections.

BK10A6500 Engineering Mechanics 3 - Blended teaching 6.1.2025-20.4.2025 Quiz 7.1

According to what was given in the lecture, any cross-section of any shape has a set of principal centroidal axes where the product of inertia is zero.

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