Okay, here's a spreadsheet solution using Excel. You'll have to be a
little patient to follow along on this procedure.
The idea here is that the answer approaches the "exact" value, without
actually getting there. But for your purposes it is more than close
enough.
In cell A1 of the spreadsheet, put 600 (or whatever mast length you want)
In cell A2, put 2 (or whatever displacement you want)
Next come the formulas
In cell A3, put a starting guess for the radius. I used:
=10*A1
(That's 10 times whatever the mast length is)
In cell A4, we use this guess to get closer to the correct value.
Here is the formula to put in A4:
=$A$1/ACOS(1-A$2/A3)
Now COPY the formula from A4 down to the cells in the next 20 or so
rows, i.e. cells A5 through A25. You'll see the number approach a
nearly constant value.
To see how close the answer is, we can use a formula to see the actual
displacement for the different values of radius. Starting in cell B3,
put this formula:
=A3*(1-COS($A$1/A3))
Then copy the formula in B3 down to cells B4 through B25. At some
point the number will become VERY CLOSE to the desired displacement
entered in cell A2.
For example, here are is the result for a 600 mm mast length and 2 mm displacement:
600
2
6000 29.97500833
23237.25455 7.745751517
45730.98413 3.936005782
64154.17813 2.805720426
75985.82783 2.36885048
82696.39959 2.17662693
86270.7712 2.08644564
88115.48903 2.042765767
89052.59128 2.021269864
89524.87484 2.010606846
89761.95595 2.00529643
89880.73228 2.002646474
89940.17959 2.001322804
89969.9181 2.000661295
89984.7911 2.000330622
89992.22855 2.000165305
89995.94752 2.000082651
89997.80706 2.000041325
89998.73686 2.000020663
89999.20176 2.000010331
89999.43421 2.000005166
89999.55044 2.000002583
89999.60855 2.000001291
The starting guess of 6000 mm radius would give an actual displacement
of 29.97...mm, which is not at all close to the 2mm we want.
BUT: once the radius gets to be 89940.1..., the actual displacement is
already 2.0013..., more than close enough to the required 2 mm.
I hope this is clear enough to follow. FYI, for your other numbers I get
600 mm mast length:
90,000 mm (or 90 m) radius for 2 mm displacement
36 m radius for 5 mm displacement
22.5 m radius for 8 mm displacement
768 mm mast length:
59 m radius for 5 mm displacement
36.9 m radius for 8 mm displacement
29.5 m radius for 10 mm displacement
Regards,
Mark |