A conical pendulum, a thin uniform rod of length `l` and mass `m`, rotates uniformly about a vertical axis with angular velocity `omega`(the upper end of the rod is hinged). Find the angle `theta` between the rod and the vertical.

Explanation

Let us consider the system in a frame rotating with the rod. In this frame, the rod is at rest and experiences not only the gravitational force `mvecg` and the reaction force `vecR`, but also the centrifugal force `vecF_(cf)`.
In the considered frame, from the condition of equilibrium i.e. `N_(0z)=0`
or, `N_(cf)=mg1/2sintheta` (1)
where `N_(cf)` is the moment of centrifugal force about O. To calculate `N_(cf)`, let us consider an element of length `dx`, situated at a distance x from the point O. This element is subjected to a horizontal pseudo force `(m/l)dxomega^2xsintheta`.
The moment of this pseudo force about the axis of rotation through the point O is
`dN_(cf)=(m/l)dxomega^2xsinthetaxcostheta`
`=(momega^2)/(l)sin thetacosthetax^2dx`
So `N_(cf)=underset(0)overset(l)int(momega^2)/(l)sinthetacosthetax^2dx=(momega^2l^2)/(3)sinthetacostheta` (2)
It follows from Eqs. (1) and (2) that,
`cos theta=((3g)/(2omega^2l))` or `theta=cos^-1((3g)/(2omega^2l))` (3)