A uniform rod of mass m and lengh I makes a constant angle `theta` with an axis of rotation which passes through one end of therod. Find the moment of inertia about this axis.

Explanation

`theta` - angle that rod makes with the axis (vertical)

Linear mass density `lambda = (m)/(l)`
Consider dx. the part of rod
Mass dx, the part of rod dm `= lambda`.dx
`dm = (m)/(l) dx`. The M.I of the rod about OO'
`I = int r^(2) dm`
Where is the perpendicular distance from the elementary mass dm.
`sin theta = (r)/(x) rArr r = x. sin theta`
`I = underset(-(l)/(2))overset((l)/(2))(int) x^(2) sin^(2) theta.(M)/(l).dx = (M)/(l).sin^(2) theta underset(-(l)/(2))overset((l)/(2))(int) x^(3) dx = (M)/(l).sin^(2)theta [(x^(3))/(3)]_(-(l)/(2))^((l)/(2))`
`= (m)/(l).sin^(2)theta (2l^(3))/(24)=m.sin^(2)theta.(l^(2))/(12)`
M.I about the axis OO' `= (1)/(12) ml^(2).sin^(2) theta`