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 make 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=m/(l_(0))dl`
The moment of inertia of the elementary mass is given as `dl=(dm)r^(2)`
The moment of inertia of the rod
`=I=intdIimpliesI=intr^(2)dm`
Substituting `r=lsintheta` and `dm=ml_(0).dl` we obtain
`I=int(l^(2)sin^(2)theta)m/l_(0)dI=(msin^(2)theta)/l_(0)int_0^(l_0)l^(3)dl=(ml_0^(3))/(3l_(0))sin^(2)theta`
`implies l=(ml_(0)^(2)sin^(2)theta)/3`