Orbit constraint of a small bead on a rotating large circular hoop in a horizontal plane

Orbit constraint problems can be encountered in mechanical equipment and amusement equipment. Mechanics exercises generally consider the ideal physical model, and the practical problems also consider the influence of friction, which makes the problem more complex and and practical. The problem of the force and oscillation of objects on orbit needs to be deeply discussed. In order to simulate the orbital motion of objects more realistically and help students expand their theoretical mechanics beyond class, we study the orbit constraint of a small bead on a rotating large circular hoop in a horizontal plane about an axis passing through a point on the circumference. The coupling equations followed by the bead on the hoop are derived using Newton's second law in a planar polar coordinate system and solved by numerical methods. We found that under the action of friction, when the initial angular velocity of the bead is greater than the critical angular velocity, the bead will rotate on the ring, and the number of rotations is related to the initial angular velocity and influenced by the friction coefficient. At different initial angular velocities, the number of oscillations of the bead on the hoop is basically the same and ultimately stops near the fixed point.