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

Parallel axes: any object, two PARALLEL axes, one through the centre of mass -- $I_O=I_C+Mh^2$. Perpendicular axes: a LAMINAR (flat) object only, three mutually perpendicular CONCURRENT axes (two in-plane, one perpendicular) -- $I_z=I_x+I_y$. …

\textbf{Theorem of parallel axes.} Conditions: applies to ANY rigid object, of any shape (laminar or 3-D); needs exactly two axes that are PARALLEL to each other, and (this is essential) one of the two axes must pass through the object's centre of mass. Expression: if $I_C$ is the moment of inertia about the axis through the centre of mass, and $I_O$ is the moment of inertia about a second, parallel axis at perpendicular distance h from the first, then $I_O=I_C+Mh^2$ where M is the total mass of the object. …

State the precise geometric conditions each theorem requires (parallel axes: any object, one axis through the centre of mass; perpendicular axes: laminar object only, three conc …

Applying the perpendicular-axes theorem to a genuinely three-dimensional (non-laminar) solid object, or forgetting that the parallel-axes theorem specifically requires ONE of the two axes to pass through the centre of mass (it c …