Question

For the thin rod of mass $m$ shown in the figure, choose the correct statement(s) regarding moment of inertia about the axis $1,2,3,4$. Here axis $4$ is passing through the centre of the rod.

• A. $\text{M.O.I}$ about axis $1$ is zero.
• B. $\text{M.O.I}$ about axis $2$ is $\frac{m{L}^2}{3}$
• C. $\text{M.O.I}$ about axis $3$ is $\frac{13m{L}^2}{12}$
• D. $\text{M.O.I}$ axis $4$ is $\frac{m{L}^2}{12}$

Answer 1

Answer: (A), (B), (C), (D)

$\text{M.O.I}$ axis $4$ is $\frac{m{L}^2}{12}$

About axis $1$,
Using $I=m{r}^2$, as $r=0$
$I_1=m{r}^2=0$

About axis $2$,
Using parallel axis theorem
$I=I_0+m{d}^2$
$I=\frac{m{L}^2}{12}+\frac{m{L}^2}{4}$
$\Rightarrow I=\frac{m{L}^2}{3}$
[$\because d=\frac{L}{2}$]



About axis $3$,
Similarly as in case of axis $2$, using the same parallel axis theorem and putting $d=L$
$\Rightarrow I=I_0+m{L}^2=\frac{13m{L}^2}{12}$


About axis $4$,
$\because$ Axis $4$ is through centre of mass of rod.
$I_4=I_0=\frac{m{L}^2}{12}$ [about axis passing through centre of mass]
$\Rightarrow I_4=\frac{m{L}^2}{12}$
$\therefore$ All of the above options are correct.