Question

Find the moment of inertia of system passing about one of its diagonal

Answer 1

Answer:

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