Consider-a-uniform-square-plate-of-side-a-and-mass-m-The-moment-of-inertia-of-this-plate-about-an-axis-perpendicular-to-its-plane-and-passing-through-one-of-its-corners-is-

Question Number 24730 by Tinkutara last updated on 25/Nov/17

Consider a uniform square plate of side

a and mass m . The moment of inertia

of this plate about an axis perpendicular

to its plane and passing through one of

its corners is

Commented by ajfour last updated on 25/Nov/17

Commented by ajfour last updated on 25/Nov/17

I_{x} = I_{y} = \frac{{m a}^{2}}{12}

s o I_{z} = I_{x} + I_{y} = \frac{{m a}^{2}}{6}

I_{z^{'}} = I_{z} + m {(\frac{a}{\sqrt{2}})}^{2} = \frac{{m a}^{2}}{6} + \frac{{m a}^{2}}{2}

= \frac{2 {m a}^{2}}{3} .

Commented by mrW1 last updated on 25/Nov/17

I_{z^{'}} = \int_{0}^{a} \int_{0}^{a} ρ (x^{2} + y^{2}) d x d y

= ρ \int_{0}^{a} (\frac{a^{3}}{3} + {a y}^{2}) d y

= ρ (\frac{a^{3}}{3} \times a + a \times \frac{a^{3}}{3})

= \frac{2 ρ a^{4}}{3}

= \frac{2 {m a}^{2}}{3}

Commented by Tinkutara last updated on 25/Nov/17

B u t I_{x} = I_{y} = 4 \times \frac{{m a}^{2}}{12} = \frac{{m a}^{2}}{3}

?

Commented by Tinkutara last updated on 25/Nov/17

B u t w h a t s w r o n g w i t h w h i c h I p o s t e d ?

Commented by ajfour last updated on 25/Nov/17

w h y t h e 4 ?

Commented by Tinkutara last updated on 25/Nov/17

B e c a u s e o f s y m m e t r y t h e r e a r e 4

r o d s a n d e a c h h a v e \frac{{m l}^{2}}{12}

Commented by ajfour last updated on 25/Nov/17

i t s a p l a t e o r s l a b, t h e r e a r e n o t

a n y r o d s . f u r t h e r s e e i m a g e b e l o w .

Commented by ajfour last updated on 25/Nov/17

Commented by Tinkutara last updated on 25/Nov/17

Y e s b u t h o w t h i s \frac{{m l}^{2}}{12} ?

Commented by ajfour last updated on 25/Nov/17

Commented by ajfour last updated on 25/Nov/17

d m = \frac{m}{L b t} (b t d r) = \frac{m}{L} d r

I = \int r^{2} d m = \frac{m}{L} \int_{- L / 2}^{L / 2} r^{2} d r

= \frac{m}{L} (\frac{L^{3}}{24} + \frac{L^{3}}{24}) = \frac{{m L}^{2}}{12} .

Commented by Tinkutara last updated on 26/Nov/17

Thank you Sir!