Question

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

Answer 1

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 m vector g and the reaction force vector R, but also the centrifugal force vector F_cf. 

In the considered frame, from the condition of equilibrium i.e. N_oz=0

or, N_cf=mg l/2sinθ   .......(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) dxω²xsinθ.

The moment of this pseudo force about the axis of rotation through the point O is