Constrained Falling of three balls

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Discussion Overview

The discussion revolves around the mathematical modeling of the constrained falling of three balls, particularly focusing on calculating the time taken for each ball under different paths. The conversation touches on concepts from variational calculus and the Brachistochrone problem, exploring the effects of path length and speed on the motion of the balls.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that the mathematical approach involves variational calculus, requiring knowledge of multivariable calculus and differential equations.
  • There is a discussion about maximizing speed and minimizing path length, with the upper path being shorter but resulting in slower initial acceleration, while the lower path offers a larger mean speed but is longer.
  • One participant references the Brachistochrone curve, noting that it neglects the rotational inertia of the balls.
  • Another participant proposes that if the balls roll without slipping, an "effective inertial mass" can be used to model the motion, which combines the effects of inertial mass and moment of inertia, suggesting that the Brachistochrone remains optimal under this assumption.
  • There is a consideration of the condition that the ball's radius should be small compared to the curvature of the track for the angular velocity to be directly related to the ball's radius and velocity.
  • A participant expresses gratitude for the insights shared and mentions a personal connection to the topic, having previously skipped portions of Classical Mechanics.

Areas of Agreement / Disagreement

Participants express various viewpoints on the mathematical modeling and assumptions involved, indicating that multiple competing views remain without a clear consensus on the optimal approach or solution.

Contextual Notes

Some assumptions regarding the simplifications made in the modeling, such as the neglect of rotational inertia and the conditions under which the Brachistochrone is optimal, remain unresolved.

En Joy
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Can anyone explain this "mathematically"? i.e. how to calculate the time in each cases?
 
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Yes, but not at B level. The math required is called variational calculus. You will need multivariable calculus and differential equations as prerequisites.

At B level, I would say that it is a combination of maximising the speed and minimising the path length. The upper path is shorter, but the ball accelerates slower in the beginning. The lower path gives a larger mean speed, but is longer.
 
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Orodruin said:
Yes, but not at B level. The math required is called variational calculus. You will need multivariable calculus and differential equations as prerequisites.

At B level, I would say that it is a combination of maximising the speed and minimising the path length. The upper path is shorter, but the ball accelerates slower in the beginning. The lower path gives a larger mean speed, but is longer.
Sorry, I did the mistake. It should not be in "B" level. But I've corrected that.
 
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En Joy said:
Can anyone explain this "mathematically"? i.e. how to calculate the time in each cases?
Here is the full clip BTW:

 
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A.T. said:
This is related, but neglects the rotational inertia of the balls:
If one makes the simplifying assumption that the balls roll without slipping then one can model the motion by using an "effective inertial mass" which bundles the effect of the ball's inertial mass together with its moment of inertia. The effect is the same as if the acceleration of gravity was reduced. Unless I am missing something, the Brachistochrone should still be optimal in such a case.
 
jbriggs444 said:
If one makes the simplifying assumption that the balls roll without slipping then one can model the motion by using an "effective inertial mass" which bundles the effect of the ball's inertial mass together with its moment of inertia. The effect is the same as if the acceleration of gravity was reduced. Unless I am missing something, the Brachistochrone should still be optimal in such a case.
I thought about this a bit and I think it is correct as long as the ball radius is small in comparison to the curvature of the track such that the angular velocity of the ball is directly given by the ball's radius and the velocity of the ball.
 
Orodruin said:
I thought about this a bit and I think it is correct as long as the ball radius is small in comparison to the curvature of the track such that the angular velocity of the ball is directly given by the ball's radius and the velocity of the ball.
Good point about the radius. I'd missed that.
 
Thanks to all of you (specially @A.T.). Actually I skipped some portions in Classical Mechanics due to (my personal) exam. strategy in completing my Graduation as I am not a good student. That’s why I could not solve this problem. But now, I have found this (The Brachistochrone) in my textbook! Wow! It’s very interesting topic.
 
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