State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.

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State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.

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The theorem of parallel axis is applicable to any body of arbitrary shape. The moment of inertia (MI) of the body about an axis through the centre mass should be known, say, I_{CM}. Then, the theorem can be used to find the MI, I, of the body about an axis parallel to the above axis. If the distance between the two axes is h,

**I= I _{CM} + Mh^{2}**

The theorem of perpendicular axes is applicable to a plane lamina only. The moment of inertia of a plane lamina about an axis—the z axis-perpendicular to its plane is equal to the sum of its moments of inertia I_{x} and I_{y} about two mutually perpendicular axes x and y in its plane and through the point of intersection of the perpendicular axis and the lamina.

**I _{z}=I_{x}+I_{y}**

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