Radius required when only Chord Length is Known

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OB1M

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Think of a rocking chair and a container whose contents I don't want to settle. Easy, just place it on my lap and rock backwards and forwards. It would get a bit tedious after a while so I need to dream up some sort of mechanism to do this for me. Thought of using a cam but the trig feels like a simultaneous equation might be involved.

The shape of the cam is a circle segment that when moved along the horizontal will tilt the angle of the chord to the horizontal. Only two things are known: the length of the chord some 400mm and the maximum tilt angle of 7 degrees. In order to construct the cam I need the radius associated with the arc. Would very much welcome some guidance on how the radius might be obtained. Thx.
 

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If the maximum tilt angle is equal to the angle between the chord and the tangent line at A then 1/2 arc lengths (C1,C2) correspond to that angle, i.e. [imath]\phi = C_1 / R[/imath].
 
blamocur said:
If the maximum tilt angle is equal to the angle between the chord and the tangent line at A then 1/2 arc lengths (C1,C2) correspond to that angle, i.e. [imath]\phi = C_1 / R[/imath].
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I know from my working on other forums that it takes both time and commitment and readers can rely on what is said which is scary so thank you for taking the time to reply. As an angle is involved I expected to see transcendental or trigonometric use as part of your reply. I hope you don't mind but I put the following to the AI ChatGPT. The answer that came back for the radius was: r = 200 * tan(7).

The following conversation took place:

Question: The length of the chord to a circle segment is 400mm. It starts at A and ends at B. The chord is initially positioned parallel to a tangent to the circle, its baseline. The segment is rotated along the baseline until chord A touches the baseline at an angle of 7 degrees. What is the formula that will give the radius associated with the segment's circumference when A touches the baseline?

Answer: Let's call the center of the circle O, the point where the chord intersects the circle C, and the midpoint of the chord M. Since the chord is initially parallel to the tangent to the circle, we know that angle OMC is a right angle.

When the segment is rotated along the baseline until chord A touches the baseline at an angle of 7 degrees, the angle OCA is also 7 degrees. Let's call the length of the radius OC r.

Since OM is the perpendicular bisector of the chord AB, we know that OM is half the length of AB, which is 200 mm. Therefore, we have:

OM = 200 mm

We can also use trigonometry to find the length of OC in terms of r and the angle OCA:

sin(OCA) = OC / AC

sin(7) = r / (r + 200)

r = 200 * tan(7)

r ≈ 25.55 mm

Therefore, the formula for the radius associated with the segment's circumference when A touches the baseline is:

r = 200 * tan(7)

I hope that helps! Let me know if you have any further questions.
 
I believe it should be [imath]r=\frac{200}{\sin (7)} \approx 1641[/imath]. Indeed, I would expect smaller angles have larger radii for the same chord length.
 
OB1M said:
I hope you don't mind but I put the following to the AI ChatGPT. The answer that came back for the radius was: r = 200 * tan(7).
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Hi OB1M. Personally, I don't mind references to ChatGPT (or its upgrades) in your own threads, as long as you're not posting links or promoting its use beyond your own.

I would not like to see suggestions or solicitations for AI in student threads. Most members here who need help ought to be using their course materials and doing the majority of their own thinking. Cheers :)
[imath]\;[/imath]
 
blamocur said:
I believe it should be [imath]r=\frac{200}{\sin (7)} \approx 1641[/imath]. Indeed, I would expect smaller angles have larger radii for the same chord length.
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Yes I agree. As the math relating to segments is well established I will try and tease out the height from mid chord to the circumference this I feel would give emphasis to raddii that are too small.
 
Otis said:
Hi OB1M. Personally, I don't mind references to ChatGPT (or its upgrades) in your own threads, as long as you're not posting links or promoting its use beyond your own.

I would not like to see suggestions or solicitations for AI in student threads. Most members here who need help ought to be using their course materials and doing the majority of their own thinking. Cheers :)
[imath]\;[/imath]
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Thank you for raising these issues. The AI is writing some of its own code and at least one developer has publicly said that he checks over the generated code before allowing it to become part of the library. The message for students is that ChatGBT output can appear very convincing but don't rely on it. By posting its output here it is subjected to peer review. Because developers very carefully check code most are not experienced mathematicians so comments on this forum good or bad are helpful. I am not in any way connected with OPenAI.

Back around 1992 I found myself in a job where I had to do a lot of technical writing. I was very inexperienced and using Microsoft Word v2.0, it had a spelling and grammar checker that I relied upon. The outcome of all of this was the quality of my written work improved enormously, an early AI had taught me how to write properly. This is role of ChatGPT it teaches how to do math by improving awareness however mistakes will happen. I remember adding semicolons to my prose on one occasion to improve its readability score and getting an almighty ticking off from the consultant in charge. As you can see that embarrassment still haunts me today.
 
Otis said:
Hi. Are you thinking that OpenAI developers will be reviewing threads at this forum?


I don't think academia is anywhere near that point in time, yet! :)
[imath]\;[/imath]
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The OpenAI GPT project is open source that publishes on GitHub so its difficult to say. I came here because my trigonometry leaves a lot to be desired and I cannot get my head around this segment problem. GPT opened it up at least here by describing at length the process that led to its conclusion even though that looks wrong. This style of of play is what the best of academia does when teaching subjects to students. Its more likely that GPT will visit threads on this site prompted by students who use the site however it cannot access pictures as a login is required.

The current online version of GPT is 3.5 and can be accessed freely online. I'm very sensitive about this because I don't want academics or you for that matter to be uncomfortable. Everybody appears to have accepted the now ubiquitous desktop calculator perhaps it time to accept that neural nets are here to stay. What I am equally uncomfortable about is using GPT in an open loop fashion. It needs a closed loop relationship that feeds back to improve its open source neural net.

GPT 4.0 is coming with its ability to understand pictures including diagrams and very shaky hand writing. Developers are having a tough time getting the neural net to correctly interpret what it sees so the release is delayed. Remember, developers are not blindly moving forward they need to understand whats going on and some of this is mind blowing, something that slows development down.

Forum owners can add the subscription version to their websites and GPT has to be a useful tool to moderators. When I get my answers I will feed back here and to GPT. As for diagrams I will provide GPT with a link to any forum picture via an imgbb.com link.
 
Otis said:
If ChatGPT could inform moderators how to get in touch with the forum's owner, then that would be awesome! ;)
[imath]\;[/imath]
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GPT said:

"Freemathhelp.com is a website that provides free online math resources and tutoring services. It was founded in 2008 by John Zimmerman, who is also the current owner. The website offers various resources such as lessons, calculators, worksheets, and forums where students can get help from volunteer tutors. The website also has a premium service called "Mathway" which provides step-by-step solutions to math problems for a fee. Overall, Freemathhelp.com has been recognized as a useful and accessible resource for students of all ages and levels who are seeking help with math."

DomainTools suggests that technical support is based in Iceland.
 
blamocur said:
I believe it should be [imath]r=\frac{200}{\sin (7)} \approx 1641[/imath]. Indeed, I would expect smaller angles have larger radii for the same chord length.
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GPT was made aware of your expectation and said:

"I apologize for any confusion caused by my previous response. You are correct that the radius should be larger when the angle is smaller. The formula I provided for the radius was incorrect and I apologize for the mistake.

The correct formula for the radius associated with the segment's circumference when A touches the baseline at an angle of 7 degrees is:

r = 200 / sin(7)

This formula can be derived using the law of sines. Let's call the length of AC x. Then we have:

sin(7) = r / (r + 200)

sin(7) = x / 2r (since AM is the perpendicular bisector of AB and hence x = AM = BM = AB/2 = 200/2 = 100)

Using the fact that x = r * sin(7), we can substitute into the second equation to get:

sin(7) = r * sin(7) / 2r

Simplifying this equation yields:

1 = 2 * cos(7)

Using the identity sin^2(7) + cos^2(7) = 1, we can solve for sin(7) and get:

sin(7) = 1 / sqrt(1 + cot^2(7))

sin(7) = 1 / sqrt(1 + (1/tan^2(7)))

sin(7) = 1 / sqrt(1 + (1/(tan^2(7))))

sin(7) = 1 / sqrt(1 + (1/(200^2)))

sin(7) ≈ 0.1228

Substituting this value back into the equation sin(7) = r / (r + 200), we get:

r = 200 / sin(7)

r ≈ 1625.7 mm

Therefore, the correct formula for the radius associated with the segment's circumference when A touches the baseline at an angle of 7 degrees is:

r = 200 / sin(7)

I hope this clears up any confusion and please let me know if you have any further questions or concerns."
 
OB1M said:
GPT was made aware of your expectation and said:

"I apologize for any confusion caused by my previous response. You are correct that the radius should be larger when the angle is smaller. The formula I provided for the radius was incorrect and I apologize for the mistake.

The correct formula for the radius associated with the segment's circumference when A touches the baseline at an angle of 7 degrees is:

r = 200 / sin(7)

This formula can be derived using the law of sines. Let's call the length of AC x. Then we have:

sin(7) = r / (r + 200)

sin(7) = x / 2r (since AM is the perpendicular bisector of AB and hence x = AM = BM = AB/2 = 200/2 = 100)

Using the fact that x = r * sin(7), we can substitute into the second equation to get:

sin(7) = r * sin(7) / 2r

Simplifying this equation yields:

1 = 2 * cos(7)

Using the identity sin^2(7) + cos^2(7) = 1, we can solve for sin(7) and get:

sin(7) = 1 / sqrt(1 + cot^2(7))

sin(7) = 1 / sqrt(1 + (1/tan^2(7)))

sin(7) = 1 / sqrt(1 + (1/(tan^2(7))))

sin(7) = 1 / sqrt(1 + (1/(200^2)))

sin(7) ≈ 0.1228

Substituting this value back into the equation sin(7) = r / (r + 200), we get:

r = 200 / sin(7)

r ≈ 1625.7 mm

Therefore, the correct formula for the radius associated with the segment's circumference when A touches the baseline at an angle of 7 degrees is:

r = 200 / sin(7)

I hope this clears up any confusion and please let me know if you have any further questions or concerns."
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I know I can't beat AI in chess or Go, but I am happy to beat it in math :)
 
blamocur said:
I know I can't beat AI in chess or Go, but I am happy to beat it in math :)
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Amazing work on your part, thank you. As part of the journey I turned to website omnicalculator. One of their lead mathematicians gave me some help with the calculator and drew a sketch describing the problem. In their answer they gave the solution as 400/(2 Sin(7)) which strictly speaking was more relevant as the chord length was originally given as 400. Both you and ChatGPT had factored by 2 to give your respective answers. However, if a more general formula was to be adopted would it be L/(2 Sin(angle))? Sorry about the lack imath HTML I simply don't know how to use it.
 
OB1M said:
GPT said:

[freemathhelp] was founded in 2008 …
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Incorrect. This site was founded in 2002. (I'd registered my first username in late 2003 or early 2004.)

OB1M said:
… by John Zimmerman
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Incorrect. This site was founded by Ted Wilcox.

OB1M said:
who is also the current owner.
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The "current" owner is debatable; could be the most-recent hacker(s).

OB1M said:
provides step-by-step solutions … for a fee.
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:LOL:

OB1M said:
technical support is based in Iceland
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I find that peculiar, but I'd asked about contact info. ;)


I confirmed accurate information with a simple Internet search (albeit no contact info). As a fancy "search engine", ChatGPT ought to start by finding a better drawing board.
[imath]\;[/imath]
 
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