A thin, rigid hoop of negligible thickness is mounted so that it can rotate freely about a fixed vertical axle that passes through its center. The center of mass of the hoop remains at rest throughout the motion.
The hoop has radius R and total mass M. During an experiment the hoop is set spinning about the axle with constant angular speed ω. A small marker is painted on the hoop at a point on its rim.
Answer the following questions about the motion and energy of the system.
(a) Describe, in words, the path followed by the marker and state whether its speed is constant, increasing, or decreasing. Justify your answer.
(b) Explain how the hoop can possess kinetic energy even though its center of mass is stationary. In your explanation, refer to the motion of individual mass elements of the hoop.
(c) Two additional markers are painted on the hoop, one at radius 21R and one at radius 43R. Rank the linear speeds of the three markers (radii 21R,43R,R) from smallest to largest. Provide a clear explanation based on rotational motion.
(d) Suppose the angular speed of the hoop is doubled. Qualitatively describe how its kinetic energy changes. Support your reasoning without using a specific formula.