Question

Consider a uniform square plate of side a and mass m. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is:

Answer 1

Step 1: Given information

A uniform square plate,

Side is “a” and mass "m".

According to the question, the diagram will be:

Step 2: Calculation

The moment of inertia of a square plate around an axis perpendicular to the plane going throughout its center of mass is determined by:

$\Rightarrow {\mathrm{I}}_{\mathrm{CM}}=\frac{{\mathrm{ma}}^2}{6}$

The moment of inertia around a parallel axis through one of its corners is determined by:

$\Rightarrow \mathrm{I}={\mathrm{I}}_{\mathrm{CM}}+m{\left(\frac{\mathrm{a}}{\sqrt{2}}\right)}^2$

By substituting the values of ${\mathrm{I}}_{\mathrm{CM}}$,

$$\begin{gathered} =\frac{{\mathrm{ma}}^2}{6}+\frac{{\mathrm{ma}}^2}{2} \\ =\frac{{\mathrm{ma}}^2+3{\mathrm{ma}}^2}{6} \\ =\frac{4}{6}{\mathrm{ma}}^2 \\ =\frac{2}{3}{\mathrm{ma}}^2 \end{gathered}$$

Therefore, the moment of inertia is $\frac{2}{3}{\mathrm{ma}}^2$.