O-31 half curve same as O-34?

[Bob.M]

 After successfully completing the job of replacing some O-31 curve track with O-42 on an incline which previously would derail some cars, I noticed that I had some half curve track left over from a previous project. Just out of curiosity, I measured a couple of them to see what the equivalent diameter would be. I could see that 2 half curves were longer than one O-31 full curve. The chord length of the two is 12". Using the tried and true method detailed by Mr. Lionelsoni, I divide this by 2, getting 6" then divide that by the Sine of half the included angle of 2 of these half tracks, or 45 degrees/2 = 22.5 degrees. The sine is 0.382683. The radius is 6/0.382683 or 15.6788". Diameter is twice this or 31.3575. The conventional diameter includes the ties, so add 2.25: to this. The result is 33.61". You could round this off to O-34.
I probably could have used these half tracks instead of buying the O-42. Hope this info helps someone with similar derailment problems.

[lionelsoni]

Sorry to burst the bubble, Bob; but the radius of the O31 "half" sections is the same as the full section.  Here's the problem:  You did everything right, except for using the sine of the angle of 45 degrees.  Because the "half" sections are slightly longer than half of a full section, they together amount to more than 45 degrees.  So you should have used half of that (unknown) angle, not 22.5 degrees.

So at least you know that your purchase of the O42 was necessary, if that's any consolation. 

Since changing the length of the section can't alter the radius, we can take advantage of your measurements and work backwards to find out what the angle of one of those "half" sections really is.  The radius multiplied by the sine of that unknown angle equals the 12-inch chord that you measured.  So the unknown angle is the arc sine of the quotient of 6 divided by the radius, which is 14.142.  The angle is 25 degrees.

[Bob.M]

lionelsoni

Sorry to burst the bubble, Bob; but the radius of the O31 "half" sections is the same as the full section. ...

 

As usual, you are right. Thanks for 'splaining it. I went back and made the Subject a question, not a statement of "fact". On closer examination, it does appear the half curves are the same radius. Here is how I was misled: basically it is an example of garbage in, garbage out.

1.The use of the word "half" is incorrect, but I knew that.

2. The number of tracks to make a circle was incorrectly stated at a supplier's website as 16.  That information was from LegacyKLine.com. The Lionel/K-Line catalog makes no statement as to the number for a circle. (Another supplier actually says it takes 8 !!)

3. Using real tracks on my bench, comparing 4 half tracks with 2 full length looks like the halves are about an inch or so wider than the full. I need to apply considerable bending force to get them to conform.

IF I used 16 tracks to make a circle, my guess it would have an overall diameter as I calculated, but it would be composed of a series of O-31 arcs, not a smooth circle.  Net result would be the same sharp turn transitioning from straight to curve that was derailing the cars. It would not have fixed the problem. According to your 25 degree calculation, it would take 360/25 = 14.4 "half" curves to make an O-31 circle. I don't think I am going to try it, but it is most likely true.