(Solved): Figure 2 shows a thin, uniform, circular disc of radius \( R \), centre of mass \( G \) and mass \ ...

Figure 2 shows a thin, uniform, circular disc of radius \( R \), centre of mass \( G \) and mass \( M \). The disc is mounted onto a uniform horizontal slender shaft of negligible mass and negligible diameter and of length \( L \). The disc rotates relative to the horizontal shaft with a constant angular velocity \( \underline{\omega} \), as shown. The shaft is supported by a frictionless pivot at \( O \). Gravity is acting vertically downward and its acceleration is \( g \mathrm{~m} / \mathrm{s}^{2} \). It is observed that when the shaft is released from rest in the horizontal position that the disc precesses about pivot with an angular velocity \( \underline{\Omega} \), as shown. (a) Draw the Free Body Diagrams of the system. [7marks] (b) Write all the equations of motions of the disc. [8marks] (c) Determine the expression for the precessional angular velocity \( \underline{\Omega} \), of the disc. [10marks]