Question

Find the MOI of four identical solid spheres about axis $X{X}^{\prime }$ placed in horizontal plane each of mass $M$ and radius $R$ as shown in figure. Line $X{X}^{\prime }$ touches two spheres (3 and 4) and passes through diameter of other two spheres (1 and 2).

• A. $\frac{9}{10}M{R}^2$
• B. $\frac{5}{18}M{R}^2$
• C. $\frac{18}{5}M{R}^2$
• D. $\frac{9}{5}M{R}^2$

Answer 1

The correct option is C $\frac{18}{5}M{R}^2$



MOI of sphere about its COM is
$[I_{COM}=\frac{2}{5}M{R}^2]$
So, MOI for sphere $1\&2$
$I_1=I_2=\frac{2}{5}M{R}^2$
Now, MOI for sphere $3\&4$ about $X{X}^{\prime }$ by parallel axis theorem,
$I_3=I_{COM}+M{R}^2$
$I_3=\frac{2}{5}M{R}^2+M{R}^2$
$I_3=\frac{7}{5}M{R}^2=I_4$
$\therefore$ MOI of system about $X{X}^{\prime }$ is
$I=I_1+I_2+I_3+I_4$
$I=2(\frac{2}{5}M{R}^2+\frac{7}{5}M{R}^2)$
$I=\frac{18}{5}M{R}^2$