Question

Find the moment of inertia of a spherical ball of mass $m$ and radius $r$ attached at the end of a straight rod of mass $M$ and length $l$, if this system is free to rotate about an axis passing through the end of the rod (end of the rod opposite to sphere).

Answer 1

Moment of inertia about $A{A}^{\prime }$ is = Moment of inertia of rod about $A{A}^{\prime }+$ Moment of inertia of spherical ball

Moment of inertia of rod, $I_{rod}=\frac{M{l}^2}{3}$

Moment of inertia of spherical ball, $I_{ball}=\frac{2m{r}^2}{5}$

Moment of inertia about AA' axis $I_{ball}=\frac{2m{r}^2}{5}+m{l}^2$

Total moment of inertia=

$\frac{M{l}^2}{3}+\frac{2m{r}^2}{5}+m{l}^2$

$\therefore$ MOI of the system $=\frac{1}{3}(M+3m)l^2+\frac{2m{r}^2}{5}$