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 make an angle `theta` to the vertical (axis of rotation).

Explanation

Take an elementary mass dm of the rod.
`dm = (m)/(l_(0)) dl`
The moment of inertia of the elementary mass is given as
`dI = (dm)r^(2)`
The moment of inertia of the rod
`I = int dI" "rArr" "I = int r^(2) dm`
Substituting `r = l sin theta & dm = (m)/(l_(0)).dl`, we obtain
`I = int (l^(2) sin^(2) theta) (m)/(l_(0))dl`
`= (m sin^(2)theta)/(l_(0))int_(0)^(2)l^(2) dl = (ml_(0)^(3))/(3l_(0))sin^(2) theta`
`rArr" "I = (ml_(0)^(2) sin^(2)theta)/(3)`