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Question Number 26179 by NECx last updated on 21/Dec/17

F o r a n a x i s p a s s i n g t h r o u g h t h e

c e n t r e o f m a s s o f a r e c t a n g u l a r

p l a t e (a l o n g i t s l e n g t h) . S h o w t h a t

i t s m o m e n t o f i n e r t i a i s \frac{{M L}^{2}}{12} a n d

t h e r a d i u s o f g y r a t i o n i s \frac{L}{2 \sqrt{3} .}

Answered by mrW1 last updated on 22/Dec/17

{L e t}^{'} s s a y t h e p l a t e h a s t h e s i z e o f

B \times L, t h e m a s s o f u n i t a r e a i s

ρ = \frac{M}{B \times L}

I = 2 \int_{0}^{\frac{L}{2}} ρ {B x}^{2} d x = 2 ρ B \frac{{(\frac{L}{2})}^{3}}{3} = \frac{ρ B L \times L^{2}}{12} = \frac{{M L}^{2}}{12}

r_{g} = \sqrt{\frac{I}{M}} = \frac{L}{\sqrt{12}} = \frac{L}{2 \sqrt{3}}

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