# how is moment of inertia for uniform rectangular lamina i=ml^2/3??

Mass density of Rectangular lamina, $\rho = \frac{M}{2L\times 2B} = \frac{M}{4LB}$

$I_{Y} = \int \mathrm{d}mx^{2}$

Where dm is the mass of the yellow strip with length $\Delta x$

$\mathrm{d}m =\rho 2B\Delta x$

Putting in the above equation,$I_{Y} = \int_{-L}^{+L}\rho 2B\Delta x\times x^{2} = \int_{-L}^{+L}\rho 2B x^{2}\mathrm{d}x$

$I_{Y} = \left [\frac{1}{3}\rho 2B x^{3} \right ]^{+L}_{-L} = \frac{4}{3} \rho BL^{3}$

Putting value of $\rho$$I_{Y} = \frac{4}{3} \frac{M}{4LB} BL^{3} = \frac{1}{3}ML^{2}$

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