Final answer:
The principle of motion in different frames of reference, including the effect of the Coriolis force in a rotating system like a merry-go-round, explains how an object can follow a spiral path. In a non-inertial frame, the path appears curved, but in an inertial frame, it is straight. This concept is also relevant in astronomy when discussing the enduring spiral structure of galaxies.
Step-by-step explanation:
The question about how motion can occur in a spiral pattern around an origin, such as in the context of a ball moving on a merry-go-round, and conclude in finite time while making infinite turns, relates to the phenomena in physics where an object's movement is perceived differently based on the frame of reference. If you are on the merry-go-round, the Coriolis force—a fictitious force only observed in this non-inertial frame—explains why the ball appears to curve. However, to an outside observer, the ball moves in a straight line while the platform rotates beneath it.
A similar phenomenon can be discussed in astronomy when considering spiral arms of galaxies. These spiral structures, due to differential rotation, should theoretically wind tighter over time, but in reality, the structure remains. This underlies the complexity of astronomical objects and forces at play.