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Question

$(a)$ The perpendicular axis theorem $I_z = I_x + I_y$ is only applicable to planar (two-dimensional) objects. A cube is a three-dimensional object, s

Vedclass · Accepted Answer

$b, d$

Vedclass · Answer

$(a)$ The perpendicular axis theorem $I_z = I_x + I_y$ is only applicable to planar (two-dimensional) objects. A cube is a three-dimensional object, so this is incorrect.$(b)$ The $A$-axis is parallel to the $z$-axis passing through the center $O$. The distance between the $z$-axis and the $A$-axis is $d = \sqrt{(\frac{a}{2})^2 + (\frac{a}{2})^2} = \frac{a}{\sqrt{2}}$. By the parallel axis theorem, $I_A = I_{cm} + md^2 = I_z + m(\frac{a}{\sqrt{2}})^2 = I_z + \frac{ma^2}{2}$. This is correct.$(c)$ The $B$-axis is not parallel to the $z$-axis, so the parallel axis theorem cannot be applied in this manner. This is incorrect.$(d)$ Due to the symmetry of the cube about the center $O$, the moment of inertia about the $x, y,$ and $z$ axes passing through the center are equal, i.e., $I_x = I_y = I_z$. Thus, $I_x = I_z$ is correct.

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