Final Answer:
The sought equation of motion in a rotating frame corresponds to Equation (7.69). (option b)
Step-by-step explanation:
Equation (7.69) describes the equation of motion for a bead threaded on a circular hoop in a rotating frame. This equation is derived considering the forces acting on the bead in a rotating reference frame and accounts for the centripetal force, gravitational force, and the apparent centrifugal force due to the rotating frame.
In this scenario, the rotating frame introduces a fictitious force, the centrifugal force, which appears to act outward from the center of rotation. The equation of motion (7.69) should agree with the resultant forces experienced by the bead in the rotating frame, considering the bead's radial position, velocity, and the rotational motion of the hoop.
Equation (7.69) should provide a comprehensive representation of the forces and dynamics governing the bead's motion on the circular hoop within the rotating frame, ensuring that the derived equation aligns with the observed behavior of the system.