How is this length R(Theta)? I don't see it....

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Discussion Overview

The discussion revolves around the geometric relationship involving the length labeled as Rθ in a triangle related to a block rocking on a cylinder. Participants explore the validity of this labeling and the implications of angles and lengths in the context of the problem.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how the edge of the triangle can be Rθ, suggesting confusion over the definition of arc length and its application in this context.
  • Another participant expresses skepticism about the assumption that Rθ is a given, indicating a potential misinterpretation of angles in the diagram.
  • Some participants propose that the block's initial horizontal position and its rocking motion without slipping on the cylinder leads to a relationship between certain lengths in the diagram.
  • There is a challenge regarding whether the green line can be equated to Rθ, with one participant noting that the extension of the radius R does not align with the line from the top of the cylinder to a specific point.
  • A later reply acknowledges a potential error in the diagram, suggesting that a specific segment should be omitted to clarify the relationship when theta is 90 degrees.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the Rθ labeling and the geometric relationships involved. There is no consensus on the correctness of the assumptions made regarding the triangle and the lengths involved.

Contextual Notes

Participants highlight potential issues with the assumptions made about angles and lengths, particularly when theta approaches 90 degrees, indicating that the situation may not be as straightforward as initially presented.

MienTommy
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?temp_hash=66cabd9882823a10f05c3cfc502cfb5c.png


That's the image. I can't see how the edge of that middle triangle's length is Rθ at all. I've tried using similar triangles and law of cosines to figure that out but I'm missing lengths. I thought Rθ was the arc length of a piece of a circle? So if Rθ is a length of a piece of a circle, how is Rθ equal to that edge of the triangle?
 

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MienTommy said:
I can't see how the edge of that middle triangle's length is Rθ at all.
It looks to me like this is a "given." You might be seeing right angles where none exist.
 
Took me a few moments to figure it out. The answer is that the block started off horizontal then rocks to the left without slipping on the cylinder ... so the red and green lines in this version must be the same length...

Figure.png
 
CWatters said:
Took me a few moments to figure it out. The answer is that the block started off horizontal then rocks to the left without slipping on the cylinder ... so the red and green lines in this version must be the same length...

View attachment 93508
But your green line doesn't equal R*theta as labeled, right?
 
insightful said:
But your green line doesn't equal R*theta as labeled, right?
I.e., the extension of the radius R to the mid-point line of the bar is not parallel to the line from the top of the cylinder to the big black dot.
 
Why not?
 
I had another think about this. I think you are right. I think the bit in yellow should have been omitted.

Figure.png


It becomes obvious something is wrong when theta is 90 degrees.
 
CWatters said:
It becomes obvious something is wrong when theta is 90 degrees.
Bingo; that was my "aha" moment also.
 

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