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 there be a small sectional area of width dx at a distance x from the x-axis.

So,
Mass of element \[= \frac{m}{a^2} \times a \times dx\]

Moment of inertia about x-axis,

\[I_{xx}  = 2 \int\limits_0^{a/2} \frac{m}{a^2} \times \left( adx \right) \times  x^2\]

\[\Rightarrow  I_{xx}  = 2 \times \frac{m}{a} \left[ \frac{x^3}{3} \right]_0^{a/2}  = 2\frac{m}{a}\left[ \frac{a^3}{3 \times 8} \right] = \frac{m a^2}{12}\]

\[\text{Similarly, }I_{yy}  = \frac{m a^2}{12}\] 

\[\text{Now, }I_{zz}  =  I_{xx}  +  I_{yy}   ............\left(\text{Pendicular  axis  theorem}\right)\]

\[ \Rightarrow  I_{zz}  = 2 \times \left( \frac{m a^2}{12} \right) = \frac{m a^2}{6}\]

The two diagonals are perpendicular to each other; therefore, we have

\[I_{zz}  =  I_{x'x'}  +  I_{y'y'} \]

Also,

\[ I_{xx}  =  I_{yy} \]

\[ \Rightarrow  I_{zz}  = 2 I_{x'x'}   \]

\[ \Rightarrow  I_{x'x'}  = \frac{m a^2}{12}\]