Question

Four holes of radius $R$ are cut from a thin square plate of side $4R$ and mass $M$. Determine the moment of inertia of the remaining portion about $Z-$axis.

Answer 1

The moment of inertia of four circular plates

$=4\times \frac{1}{2}{M}^{\prime }{R}^2$

Here, $M^{\prime }$ is the mass of each plates

The moment of inertia of square plate about $z-axis$

$\begin{array}{c}\frac{1}{6}M{\left(4R\right)}^2\\ =\frac{8}{3}M{R}^2\end{array}$

Therefore,

$M.I$ of remaining portion

$\begin{array}{c}\frac{8}{3}M{R}^2-2M^{\prime }{R}^2\\ =\left(\frac{8}{3}M-2M^{\prime }\right)R^2\end{array}$