Re: Help on Radius of Arc

[Leigh Palmer <palmer@sfu.ca>]

> I am having my students design model roller coasters.  Is there a way we
> can determine the radius of an arc if we know the length of the arc and
> the length of the cord of that arc?

Yes. Let S = length of the circular arc and L = the length of its chord.
Let r = the radius of the arc and a = the angle subtended by the arc
expressed in radians. Inspection of your drawing will show that

         L          a                            S
       ----- = sin ---          and         a = ---
        2 r         2                            r

We have two equations in two unknowns. Combining these to eliminate a:

         L           S
       ----- = sin -----
        2 r         2 r

Now simply solve for r. This is a transcendental equation which has no
solution for r in closed form, but a calculator may be used to solve it
iteratively, something that comes as a revelation to the students and
they usually think it's pretty neat (I do, too). A specific example will
be necessary to demonstrate this method.
                                                      1
First let's simplify this equation. If I define p = ----- and rewrite:
                                                     2 r
        L p = sin (S p)

I can now solve this simpler equation for p (which is upside down for
d = 2 r = diameter*) by iteration. This converges very slowly, so I
advise the use of a programable calculator. The expresion to iterate is

             sin (S p)
        p = -----------
                 L

Make a guess at the value of p and plug it into the right side of the
equation. This will give you a refined value, nearer the value sought.
Repeating this process will converge on the desired answer.

I know that Bob has already posted suggesting the use of Mathcad. I
only add my version because it can be done with a hand calculator. I
did try it out using a non-programmable, but I soon switched!

Leigh

* You probably thought I just wanted to get my initials in there!