Question

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

Answer 1

Given in the question.
Assume that the distance us x in an small cross sectional area.

Now mass of that cross sectional area would be
$\frac{m}{a^2} * ax dx$
Hence the moment of inertia of that axis will be
$I= \int\limits^ \frac{a}{2} _0 { \frac{m}{a^2} } \, * adx * x^2$
$[ 2 * ( \frac{m}{a}) * (\frac{x^3}{3})]_0 ^{a/2}$
$\boxed{I = ma^2/12}$

Now similarly for the other axis.

I' = 2 × ma²/2
I' = ma²/6
Therefore by perpendicular axis theorem, moment of inertia
I+I =I'
2I = ma²/6
$\boxed{I = ma^2/12}$



Hope it Helps