Question

Two spheres each of mass $M$ and radius $R / 2$ are connected with a massless rod of length $2\, R$ as shown in the figure. What will be the moment of inertia of the system about an axis passing through the centre of one of the sphere and perpendicular to the rod?

• A. $\frac{21}{5} M R^{2}$
• B. $\frac{2}{5} M R^{2}$
• C. $\frac{5}{2} M R^{2}$
• D. $\frac{5}{21} M R^{2}$

Answer 1

Answer: (A)

Moment of inertia of the system about $y y'$
$I_{y y'}=$ Moment of inertia of sphere $P$ about $y y'$
+ Moment of inertia of sphere $Q$ about $y y'$
Moment of inertia of sphere $P$ about $y y'$
$=\frac{3}{5} M\left(\frac{R}{2}\right)^{2}+M(x)^{2}$
$=\frac{2}{5} M\left(\frac{R}{2}\right)^{2}+M(2 R)^{2}$
$=\frac{M R^{2}}{10}+4 M R^{2}$
Moment of inertia of sphere $Q$ about $y y'$ is
$\frac{2}{5} M\left(\frac{R}{2}\right)^{2}$
Now, $I_{y y'}=\frac{M R^{2}}{10}+4 M R^{2}+\frac{2}{5} M\left(\frac{R}{2}\right)^{2}=\frac{21}{5} M R^{2}$