Circle gymnastics is a form of artistic gymnastics. The equipment used by athletes is a wooden or plastic circle with an inner diameter of 80-90 cm, whose cross-section can be round, oval, etc. The weight of the circle must be at least 300 grams. Circle gymnastics performances consist of rolling, spinning, 8-shaped movements, looping, throwing and catching, spinning, and balancing. In circle gymnastics performances, we often see an interesting phenomenon: when athletes throw the circle in their hands, the circle can automatically return to the athlete's hands after moving forward on the ground for a certain distance. Why is this happening?
When physicists study the motion of objects, they often simplify the actual objects into certain ideal models for discussion. For example, only considering the mass of the object and ignoring its shape and size, they treat it as a "point mass" model; or ignoring the slight changes in the shape of the object during its motion, they treat it as a "rigid body" model whose shape always remains unchanged. The circle in circle gymnastics can be regarded as a rigid body. The motion of the circle after being thrown can be decomposed into the translation of the center of mass (the center of the circle) and the rotation around the center of mass.
At the moment when the circle gymnast athlete throws the circle in her hand (assuming it's to the left), in addition to giving the circle an initial velocity to the left, she also makes the circle rotate clockwise, that is, it has an initial angular velocity. After being thrown, the circle slides to the left on the ground as a whole and has clockwise rolling. At this time, it is subject to the sliding friction force from the ground to the right. Since the direction of the friction force is opposite to the direction of the initial velocity, the center of mass of the circle decelerates to the left, and the rotation around the center of mass also slows down. Whether the circle can successfully return to the athlete's hand, and when and where it starts to return, are closely related to the initial velocity, initial angular velocity, and the two deceleration movements of the circle. There may be three possible situations. The first situation is that when the circle is rolling and sliding to the left, the angular velocity of the circle rotating around the center of mass first decreases to zero, but the velocity of the center of mass moving to the left has not yet decreased to zero. At this time, the circle will continue to move to the left. The second situation is that when the angular velocity of the circle rotating around the center of mass decreases to zero, the velocity of the center of mass moving to the left also happens to decrease to zero. At this time, the circle will stop moving. The third situation is that when the velocity of the center of mass moving to the left first decreases to zero, but the angular velocity of the circle rotating around the center of mass has not yet decreased to zero. At this time, under the action of the sliding friction force, the center of mass of the circle will accelerate in the opposite direction to the right, so that the circle can return to the athlete's hand.