Question

Four equal masses $m$ each are placed at the corners of a square of length $\left(l\right)$ as shown in the figure. The moment of inertia of the system about an axis passing through A and parallel to DB would be:

• A. $m{l}^2$
• B. $3m{l}^2$
• C. $\sqrt{3}m{l}^2$
• D. $2m{l}^2$

Answer 1

The correct option is B $3m{l}^2$


From the figure,
$AC=\sqrt{l^2+l^2}$
or $AC=l\sqrt{2}$
$\Rightarrow d=\frac{l\sqrt{2}}{2}=\frac{l}{\sqrt{2}}$

Moment of inertia about the axis passing through A:
$I=m{\left(0\right)}^2+m(d{)}^2+m(d{)}^2+m(AC{)}^2$
$\Rightarrow I=0+m{\left(\frac{l}{\sqrt{2}}\right)}^2+m{\left(\frac{l}{\sqrt{2}}\right)}^2+m{\left(l\sqrt{2}\right)}^2$
$\Rightarrow I=\frac{m{l}^2}{2}+\frac{m{l}^2}{2}+2m{l}^2$
$\Rightarrow I=3m{l}^2$
Hence, option (b) is correct.