Question

The moment of inertia of a hollow cubical box of mass $M$ and side $a$ about an axis passing through the centres of two opposite faces is equal to

• A. $\frac{5M{a}^2}{3}$
• B. $\frac{5M{a}^2}{6}$
• C. $\frac{5M{a}^2}{12}$
• D. $\frac{5M{a}^2}{18}$

Answer 1

Answer: (D)

Taking mass of plate $m=\frac{m{a}^2}{6}\times 2=\frac{m{a}^2}{3}$
Then MI of two plates through which the axis is passing
M.I of 4 plates having symmetrical position from the axis
$=4\times \left[\frac{m{a}^2}{12}+m{\left(\frac{a}{2}\right)}^2\right]=4\times \left[\frac{m{a}^2}{3}\right]$
Total $MI=\frac{4m{a}^2}{3}+\frac{m{a}^2}{3}=\frac{5m{a}^2}{3}$
using $\frac{M}{6}=m=MI=\frac{5M{a}^2}{18}$