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State and prove theorem of perpendicular axes of moment of Inertia.

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Theorem of perpendicular axis: According to this theorem, "the moment of inertia of a planar body (lamina) about an axis OZ Perpendicular to the pane of the lamina (O being a point in this lamina) is the sum of the moments of inertia about any two mutually perpendicular axes OX and OY, both lying in the same plane",

Let Iz = moment of inertia of the lamina about OZ axis.

Ix = moment of inertial about Ox axis.

and ly = moment of inertia about OY axis

Then, Iz = Ix + Iy

Proof : Consider a particle of mass m of the lamina at point p distant r from O. Let (x, y) be point co-ordinates of the point P.(Fig).

The moment of inertia of the particle about z-axis = mr2.

Moment of inertia of the whole lamina about z-axis is,

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