Final answer:
When a small bead slides without friction on a vertical circular hoop, it undergoes both translational and rotational motion. The system's mechanical energy remains constant throughout the motion.
Step-by-step explanation:
The question asks about a small bead sliding without friction on a circular hoop in a vertical plane. When the bead is released from rest at the top of the hoop, it will undergo both translational and rotational motion. The translational motion is the bead sliding down the hoop due to gravity, while the rotational motion is the hoop rotating about its center.
The bead will continue to slide down the hoop while rotating until it reaches the bottom. At the bottom, the bead will have a maximum velocity due to the combination of gravity and the hoop's rotation. The bead will then start moving up the other side of the hoop, decelerating due to gravity until it comes to rest at the top. At this point, the bead will reverse its direction and the process repeats.
During this motion, the bead's kinetic energy will continually convert between translational kinetic energy and rotational kinetic energy. The total mechanical energy of the system (including both translational and rotational energy) will remain constant, assuming no energy losses due to friction or other factors.