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

Answers (1)

 Find MOI about Y axis

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 \rhoI_{Y} = \frac{4}{3} \frac{M}{4LB} BL^{3} = \frac{1}{3}ML^{2}

 

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