Question

A uniform rod of mass $m$ and length $l_0$ is rotating with a constant angular speed $\omega$ about a vertical axis passing through its point of suspension. Find the moment of inertia of the rod about the axis of rotation if it makes an angle $\theta$ to the vertical (Axis of rotation)

Answer 1

We can observe that each and every element of rod is rotating with different radius about the axis of rotation
Take an elementary mass $dm$ of the rod
$dm=\frac{m}{l_0}dl$
The moment of inertia of the elementary mass is given as $dI=\left(dm\right)r^2$
The moment of inertia of the rod
$=I=\int dI\Rightarrow I=\int r^{2}dm$
Substituting $r=l\sin \theta ;dm=\frac{m}{l_0}$ we obtain
$I=\int \left(l^2{\sin }^2\theta \right)\frac{m}{l_0}dI=\frac{m{\sin }^2\theta }{l_0}{\int }_0^{l_{0}m}{l}^{2}dl=\frac{m{l}_0^3}{3l_0}{\sin }^2\theta$
$\Rightarrow I=\frac{m{l}_0^2{\sin }^2\theta }{3}$