A carnival ride features a rigid arm that pivots about a fixed vertical axis. A passenger seat is attached to the arm by a telescoping section that can extend or retract along the length of the arm but always stays aligned with it. Starting from rest at point P, the seat undergoes the following sequential motions:
• Segment 1: The arm rotates 60° while the seat is 3.0 m from the axis.
• Segment 2: The seat is instantaneously extended so that it is 4.5 m from the axis. Without pausing, the arm then rotates an additional 100° to reach point Q.
Treat the telescoping change as occurring instantaneously at the junction between segments, and neglect any translation of the seat during that extension.
(a) Calculate the linear distance traveled by the passenger seat during Segment 1.
(b) Calculate the linear distance traveled by the seat during Segment 2.
(c) Determine the total linear distance the passenger covers from point P to point Q.
(d) Suppose the ride is redesigned so that the motion consists of n successive rotational segments, each through an angle θi (in radians) at a corresponding distance ri from the axis. Derive a general algebraic expression for the total linear distance traveled in terms of the sets {ri} and {θi}.