A uniform rod of mass m and length lo is rotating with a constant angular speed ω about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle θ to the vertical (axis of rotation) is

We can observe each and every element of rod in rotating with different radius about the axis of rotationTake an elementary mass dm of the rod.dm = ml0dlThe moment of inertia of the elementary mass is given as dl = (dm)r2The moment of interita of the rod =  I = ∫dI  ⇒I = ∫r2dmSubstituting r = 1 sin θ and dm = ml0.dl, we obtainI = ∫(l2sin2θ)ml0dI = msin2θl0∫0l0l2dl = ml023l0sin3θ⇒I = ml02sin2θ3