Question

A cube of side length $a$ is shown in the figure. At every corner, particles of mass $m$ are present. Find the MOI of system of particles about $y-$axis.

• A. $6m{a}^2$
• B. $8\sqrt{2}m{a}^2$
• C. $(6+2\sqrt{2})m{a}^2$
• D. $8m{a}^2$

Answer 1

The correct option is D $8m{a}^2$

As we can see from the figure,
Distance of particles 2, 3, 8 and 5 from $y$ - axis is $a$.
and 1 and 4 particles lie on the $y$ axis, hence MOI for 1 and 4 particles is zero.
Distance of particle 6 and 7 from $y$ axis $=\sqrt{a^2+a^2}=\sqrt{2}a$
Hence, moment of inertia of the system is
$I=I_1+I_2+I_3+I_4+I_5+I_6+I_7+I_8$
$I=0+m{a}^2+m{a}^2+0+m{a}^2+m(\sqrt{2}a{)}^2+m(\sqrt{2}a{)}^2+m{a}^2$
$I=8m{a}^2$