State and prove perpendicular axis theorem.

Explanation

(i) Statement : The moment of inertia of a plane laminar body about an axis perpendicular to its plane is equal to the sum of moments of inertia about two perpendicular axes lying in the plane of the body such that all the three axes are mutually perpendicular and concurrent. To Prove : `I_(z) = I_(x) + I_(y)`
(ii) Consider a plane laminar object of negligible thickness on which the origing O lies. The mutually perpendicular axes X and Y are lying on the plane and Z-axis is perpendicular to plane as shown in figure.

(iii) Let us consider a point mass P of mass m, which as is at a distance r from origin O.
(iv) The moment of inertia of the point mass about the Z-axis is,
`dl_(z) = mr^(2)`
(v) The moment of inertia of the whole body about the Z-axis is,
`I_(z) = sum mr^(2)`
(vi) Here, `{:(r^(2) = x^(2) + y^(2)", So that,"),(I_(z) = sum m (x^(2) + y^(2))),(I_(z) = sum mx^(2) + sum my^(2)):}`
(vii) But `sum mx^(2) = I_(y)`, the moment of inertia of the body about the Y-axis and `sum my^(2) = I_(x)` the moment of inertia about the X-axis. Thus,
Therefore, `I_(z) = I_(x) + I_(y)`
(viii) Hence, the perpendicular axis theorem is proved.