Question

Find the MOI of the uniform rod about the given axis

• A. $\frac{m{L}^2}{12}$
• B. $\frac{m{L}^2}{12}{\cos }^2\theta$
• C. $\frac{m{L}^2}{12}{\sin }^2\theta$
• D. $\frac{m{L}^2}{12}\sin \theta$

Answer 1

The correct option is C $\frac{m{L}^2}{12}{\sin }^2\theta$


As the rod rotates about the given axis, the element $dm$ goes in a circle of radius $r=x\sin \theta$. Thus,
$dI=dm{r}^2$
$=dm{x}^2{\sin }^2\theta$
$\Rightarrow I={\sin }^2\theta \int dm{x}^2$
Since $dm=\frac{m}{L}dx$
$I=\frac{m}{L}{\sin }^2\theta \underset{-\frac{L}{2}}{\overset{\frac{L}{2}}{\int }}{x}^{2}dx$
$=\frac{m}{L}{\sin }^2\theta \times {\left[\frac{x^3}{3}\right]}_{-\frac{L}{2}}^{\frac{L}{2}}$
$=\frac{m}{3L}{\sin }^2\theta \times [\frac{L^3}{8}-(-\frac{L^3}{8})]$
$I=\frac{m{L}^2}{12}{\sin }^2\theta$