For the equations for I given in parts (a) and (b) of Table 9.2 to be valid, must the rod have a circular cross-section? Is there any restriction on the size of the cross-section for these equations to apply? Explain.

The moment of inertia of a rigid body depends on the shape, size, and location of the axis of rotation. It does not depend upon the cross-section area of the body.

The expression for the moment of inertial of the slender rod with the axis passing through the center is given by,

$I=\frac{1}{12}M{L}^2$

Here, M is the mass and L is length.

The expression for moment of inertial of the slender rod with the axis passing through the one end is given by,

$I=\frac{1}{3}M{L}^2$

From the above formulas, it can be observed that inertial depends only on the axis of rotation, length, and mass. It does not depend on the area of cross-section.

Therefore, the inertial will remain the same irrespective of whether the rod has a circular cross-section, and there is no restriction on the size of the cross-section.