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

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 $M I=\frac{4 m a^{2}}{3}+\frac{m a^{2}}{3}=\frac{5 m a^{2}}{3}$
using $\frac{M}{6}=m=M I=\frac{5 M a^{2}}{18}$