Question

Three rings each of mass $m$ kg and radius $a$ are arranged as shown in the figure. The moment of inertia of the arrangement about $x-x^{{}^{\prime }}$ axis will be

• A. $\frac{3}{2}m{a}^2$
• B. $\frac{7}{2}m{a}^2$
• C. $\frac{2}{7}m{a}^2$
• D. $\frac{2}{5}m{a}^2$

Answer 1

The correct option is B $\frac{7}{2}m{a}^2$

The moment of inertia of the ring about is centroidal axis perpendicular to its plane is $m{a}^2$. Now let moment of inertia about any two perpendicular axis in its plane be I. Thus using perpendicular axis theorem we get $2I=m{a}^2$ or $I=\frac{m{a}^2}{2}$.
Now moment of inertia of the ring about a tangential axis in its plane is given using parallel axis theorem as $\frac{1}{2}m{a}^2+m{a}^2=\frac{3}{2}m{a}^2$.
Thus for the system shown we see that moment of inertia is $\frac{1}{2}m{a}^2+\frac{3}{2}m{a}^2+\frac{3}{2}m{a}^2=\frac{7}{2}m{a}^2$