Applied Maths - Mechanics 61

A light rod OA of length 2a is rigidly fixed at A to a uniform square plate ABCD of mass m and side 2a, so that OAB is a straight line. The end O of the rod is pivoted to a fixed horizontal axis which is perpendicular to the plate of the square and the system can swing freely about this axis. If the system is released from rest when OAB is horizontal and
CD is below Ab, show that when OA has turned through an angle @, the angular verlocity of OA is d@/dt , where

16a(d@/dt)^2 = 3g(3sin@ + cos@ - 1).

Show also that when the centre of gravity of the square plate is vertically below O, the action at O is vertical and of magnitude

mg[ 46 - 3(10)^0.5] / 16 .