State and Prove Perpendicular axis Theorem

6.A] State and prove perpendicular axes theorem

Answer:-

To prove :- I_{zz} = I_{xx} + I_{yy}

Statement:-

The Moment of inertia of area about an axis perpendicular to its plain at any point ‘o'(Alphabet – o) is equal to the same moment of inertia about any two mutual perpendicular axis through the same point ‘o’ and line in the plane of the area.

From the Statement :-

I_{zz} = I_{xx} + I_{yy}

I_{zz} = r^{2} \sum dA

From Figure:-

r²=x²+y²

I_{zz} = (X^{2}+y^{2}) \sum dA

I_{zz} = x^{2}dA+y^{2} dA

I_{zz} = I_{yy} + I_{xx}

I_{zz} = I_{xx} + I_{yy}

Hence proved..