Question

A thin square plate of side $3\text{m}$ has mass $5\text{kg}$. Find the moment of inertia about axis $AB$ as shown in figure.

• A. $15{\text{kg-m}}^2$
• B. $30{\text{kg-m}}^2$
• C. $45{\text{kg-m}}^2$
• D. $7.5{\text{kg-m}}^2$

Answer 1

The correct option is A $15{\text{kg-m}}^2$

Given, mass of square plate $\left(m\right)=5\text{kg}$
Side length of square plate $\left(l\right)=3\text{m}$

To find moment of inertia of square plate about $AB$, let us consider an imaginary square of side $3\text{m}$ adjacent to the given square plate as shown below.


Moment of inertia of complete rectangle about axis $AB=\frac{m{l}^2}{12}$
where $l$ is the length of rectangle $=6\text{m}$
and $m=$ mass of rectangle $=10\text{kg}$

$I_{AB}$ for rectangle $=\frac{10\times (6{)}^2}{12}=30{\text{kg-m}}^2$
Therefore, by symmetry
$I_{AB}$ for square $=30/2=15{\text{kg-m}}^2$

Alternative method:
MoI of square about an axis passing through geometrical centre and parallel to the sides $=\frac{m{l}^2}{12}$
Using parallel axis theorem, MoI about edge $AB$
$I_{AB}=\frac{m{l}^2}{12}+m{\left(\frac{l}{2}\right)}^2$
$=\frac{m{l}^2}{3}$
$=5\times \frac{3^2}{3}=15{\text{kg-m}}^2$