question_answer 40)

A thin circular loop of radius R rotates about its vertical diameter with an angular frequency . Show that a small bead on the wire remains at its lowermost point for . What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for ?  Neglect friction.  

Answer:

Fig. 5.138 shows the free-body diagram of the bead when the radius vector joining the centre to the bead makes an angle  with the vertical downward direction. The normal reaction here is equal to the centrifugal force, i.e.,            Fig. 5.138 Also              or For the bead to remain in the lowermost position,  and . Hence Thus the bead will remain in the lower-most position, it For  we have             .