Ring moves on curved path faster

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Discussion Overview

The discussion revolves around the motion of a ring on a curved path compared to a ring moving in a straight line. Participants explore the principles that might explain why the ring on the curved path travels a longer distance in the same time interval as the straight path, raising questions about the underlying physics and mathematical principles involved.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant suggests that the rings are released simultaneously and questions how to explain the difference in distance traveled by the rings on different paths.
  • Another participant asks for clarification and a diagram, questioning the assumption that both rings would travel from point A to B in the same time.
  • A third participant draws a parallel to the Brachistochrone problem, suggesting that the question relates to finding the path that minimizes time under gravity, indicating that the path could be modified to achieve the same travel time.
  • A fourth participant references the Brachistochrone problem and notes that it is typically solved using calculus of variations, emphasizing that the resulting curve is dependent on the specific setup.
  • A fifth participant admits to not having a diagram and expresses uncertainty about the explanation provided, indicating a lack of satisfaction with the response given to the original question.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions regarding the travel times of the rings on different paths. There is no consensus on the explanation or principles that govern the motion of the rings.

Contextual Notes

The discussion highlights the need for a diagram to clarify the setup and assumptions involved in the problem, as well as the complexities of the Brachistochrone problem that may influence the conclusions drawn.

apr
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A ring on curved path (say on bent rod) will move longer distance than the one moves on straight line in same interval of time (the two rings are released simultaneously - one moves on curved path and other one moves horizontally) - How to explain? by which principle one can explain?
 
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apr said:
A ring on curved path (say on bent rod) will move longer distance than the one moves on straight line in same interval of time (the two rings are released simultaneously - one moves on curved path and other one moves horizontally) - How to explain? by which principle one can explain?
Do you have a diagram to illustrate what you mean? Assuming I understand you correctly, why do you think the two rings taking different paths will travel from A to B in the same time?
 
My guess is that the original poster question is similar to which path is best for gravity to pull a ring on a frictionless wire to get from point A to point B in the shortest time. If the goal is to get the same time as a straight line from A to B, then the path of shortest time could be modified in numerous ways to get the time to be the same.
 
This is the so called Brachistochrone problem: http://en.wikipedia.org/wiki/Brachistochrone_curve and is usually solved using the calculus of variations but the resulting curve depends crucially on the setup hence Doc Al's request for a diagram.
 
Sorry I don't have the diagram. The question asked me over the phone by a teenager - I replied as said by rcgldr in previous post here. But not satisfied really.
 

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