Question

Find the moment of inertia of a uniform square plate of mass m and edge a about one of its diagonals.

Answer 1

Let small sectional area is at a distance x from xx` axis therefore mass at that small

section$\frac{m}{a^2}\times a\times dx$

Therefore, moment of inertia about xx` axis

$I_{xx‘}=2{\int }_0^{\frac{a}{2}}\frac{m}{a^2}\times \left(adx\right)\times x^2$

$=2\times \frac{m}{a}{\left[\frac{x^3}{3}\right]}_0^{\frac{a}{2}}$

=$2\times \frac{m}{a}\left[\frac{a^3}{3\times 8}\right]=\frac{m{a}^2}{12}$ Therefore, $I_{zz}=I_{xx}+I_{yy}$

= $2\times \left(\frac{m{a}^2}{12}\right)=\frac{m{a}^2}{6}$

$I_{zz}=I_{xx}+I_{yy}$

$\Rightarrow \frac{m{a}^2}{6}=2\times I_{xx}(I_{xx}=I_{yy})$

$\Rightarrow I_{xx}=\frac{m{a}^2}{12}$