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Transfer of Axis

AB is the axis passing through any point P. CD is the axis passing through centre

of mass ‘G’ AB is parallel to CD.

We have from trigonometry

Cos (180 –α) = r2G +d2 + r2 / 2rG d

-cos α = r2G +d2 + r2 / 2rG d

2 r2G +d cos α = r2 G / d2 –r2

r2 = r2G + d2 +2 rd cos α

I =ʃ r2 dm

=  ʃ (r2G + d2 +2 rd cos α) dm

= ʃ r2G + dm +d2 ʃ r2G  dm cos α

= IG + d2m +2d  ʃ u dm

= I G +md2 +0

ʃ = u dm = 0

I = IG +md2

(4) is Parallel axis theorem.

Similarly,                    K2 =KG2 +d2

Where ,                       KG = radius  of gyration about G.