What Happens to a Ball's Trajectory in a Partial Loop-de-Loop?

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Homework Help Overview

The discussion revolves around the trajectory of a ball in a partial loop-de-loop, specifically focusing on the conditions under which the ball can ascend and descend the loop. Participants explore the dynamics of motion, including velocity, gravitational effects, and the transition from rolling along the track to free fall.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants question the conditions under which the ball maintains contact with the track and the implications of losing contact. There are discussions about the trajectory after the ball falls off the track and the potential discontinuities in its path. Some participants suggest clarifying the problem by using angular measurements to describe the ball's position on the loop.

Discussion Status

The discussion is ongoing, with various interpretations of the problem being explored. Some participants have offered clarifications and suggestions for describing the trajectory, while others express confusion about the initial setup and the implications of the ball's velocity at different points in the loop.

Contextual Notes

There is mention of potential confusion regarding the definitions of the loop's segments and the effects of friction, which may not be part of the original problem context. Participants are encouraged to reconsider their assumptions about the ball's motion and the forces acting on it.

Loren Booda
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How does one describe the trajectory of a ball which has enough velocity to climb the first 1/4 of a circular loop-de-loop, but not enough to smoothly descend the last 1/4 of the loop? E. g., in this situation what discontinuity does the ball path experience in curvature or its derivative?
 
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If the ball has only sufficient speed to climb the first 1/4 of the loop, then obviously it will not make it ot the last 1/4 of the loop.
 
nice name you have for this track, loop de loop. lol
 
Geometrodynamics will throw one for a loop.:bugeye:

e(ho0n3,

Please read the problem again. The ball may have enough velocity to stay on the track approaching the 3/4 mark.
 
Loren Booda said:
Geometrodynamics will throw one for a loop.:bugeye:

e(ho0n3,

Please read the problem again. The ball may have enough velocity to stay on the track approaching the 3/4 mark.

I'm a bit confused as well, if it got to the 3/4 mark (lets call this 3 o clock), the rest is easy as it has both gravity and the support of the track for the rest of the way. Now if it had problems beween say 10:30 and 1:30, I could understand.
 
Loren Booda said:
Please read the problem again. The ball may have enough velocity to stay on the track approaching the 3/4 mark.
Your description of the problem is confusing; why don't you try again. What do you mean by first 1/4 of the loop? Last 1/4? 3/4 mark? Describing the path in terms of angle might be clearer: 0 degrees is at the bottom; 180 at the top; etc.

In any case, unless the ball has sufficient speed to maintain contact throughout the loop, it will at some point lose contact and become a projectile in free fall. If it maintains contact at the top part of the loop then it will maintain contact to the bottom again, ignoring losses.
 
OK. The ball has at least enough velocity to stay on track past 90o from the bottom, but at most not enough to stay on until 270o. How can one generally describe its trajectory after falling off the track under such conditions?

I guess there are two components to this "free fall" trajectory: the momentum due to its angular velocity when leaving the track, and that due to gravitational potential from the point of departure. I thought there might be some discontinuity in the path of the ball at this point.
 
Loren Booda said:
OK. The ball has at least enough velocity to stay on track past 90o from the bottom, but at most not enough to stay on until 270o.
Again, I do not understand. 0 degrees is at the bottom; 180 degrees is at the top. I can understand having enough speed to make it past 90 degrees, but not enough to make it to the top (180 degrees); but if it does make it to 180, it will surely make it past 270 degrees as well (ignoring losses).
 
Therein may lie the fault to my initial reasoning, Doc Al (I mentally added friction to the situation). What are the governing equations to describe the frictionless ball falling off between 90o and 180o?
 

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