Question

Find the moment of inertia of the rod $AB$ about an axis $yy$ as shown in figure. Mass of the rod is $m$ and length is $l$.

Answer 1

$dm=\frac{m}{l}dr$

$\int dI=\int dm{x}^2$

$x=r\sin \alpha l$

$I_{y{y}^{\prime }}=\frac{m}{l}{\sin }^2\alpha {\int }_0{r}^{2}dr$

$I_{y{y}^{\prime }}=\frac{m}{l}{\sin }^2\alpha \frac{l^3}{3}=\frac{m{l}^2{\sin }^2\alpha }{3}$

Taking an element of the rod, we find its distance from the axis and the use the definition of moment of inertia i.e.$M.I=\int r^{2}dm$we get the moment of inertia of the rod about yy'.