Let's say you have two full bobbins, and two empty bobbins (because, like me, you have plenty of bobbins...). So, you want to make a nice two-ply, and you figure you can do that onto the two empty bobbins.
How can you get the same plied yardage (or, close to the same) on the bobbins? By stopping when your bobbins of singles are half-full, of course. Okay, I'll give you time ... go and google "when is a cylinder half full".
Lots of tanks of water draining out, aren't there? Sigh.
So I pulled out the math for the volume of a cylinder ... volume = PI (3.14159...) * height * (radius-squared). Eek. Height, we have. Radius, is half the diameter.
Now, let's call volume V, height H, PI 3.14, and radius R ... so that's:
V = 3.14 * H * R^2
When the volume's at half, and height stays the same, what's R at?
V/2 = 3.14 * H * R^2/2
and R^2/2 = (R/SQRT(2))^2
See that? we pushed the 2 into the squaring by taking its squareroot -- SQRT(2) squared is, you got it, 2. This math stuff is so easy, right? (I'm helping DS with 7th grade pre-algebra this year -- boy am I glad I took Calculus in college!)
So, the new R would be 1/SQRT(2) the size of the original R, which works out to 70.71% (and a bit that's mostly inconsequential).
But, as vampy and jammam on Ravelry pointed out, we also need to worry about the center pole of the bobbin. So, 70% isn't quite accurate ... and to make matters even more complex, every brand of bobbin can have a different center pole size (sigh).
So, the math gets even more complex (sorry) ...
We have the volume of the center pole (V1) based on its radius (height is the same):
V1 = 3.14 * H * (R1)^2
And we want to know the radius when the total remaining volume, V - V1, is at half:
(V - V1)/2 = ( (3.14 * H * R^2 ) - (3.14 * H * R1^2) ) /2
Looking to the radii, we can get the magic half-way point with this fun formula:
R-half = SQRT( ((R^2) - (R1)^2) / 2)
Oh joy. No wonder there is no magic number for all bobbins.
Here are the numbers for bobbins I can pass along (page down ... blogger still likes biiig gaps before tables...):
|outer radius (in.)||inner radius (in.)||half radius (in.)||in. from outer edge||% whole||bobbin|
|1.78125||0.353045||1.234547||0.55||69.3%||majacraft plastic (does not account for thicker center at ends)|
|1.78125||0.353045||1.234547||0.55||69.3%||Majacraft WooLee Winder|
|2.204724||0.40625||1.532281||0.67||69.5%||Majacraft new plying|
|1.625||0.875||0.968246||0.66||59.6%||Pocket bobbin - large center core, relatively|
|1.8125||0.3125||1.262438||0.55||69.7%||Van Eaton (larger end outer rad)|
|1.4375||0.375||0.98127||0.46||68.3%||Journey wheel (one whorl end-piece slopes, so half is slightly smaller than it ought to be)|
|1.496063||0.3125||1.034541||0.46||69.2%||Butterfly/Lendrum WooLee Winder|
|1.830709||0.364173||1.268635||0.56||69.3%||Jensen Tina II (larger end outer radius)|
|1.496||0.3937||1.020543||0.48||68.2%||Ashford regular bobbin (larger end outer radius)|
The first two columns are measured on the bobbins, the third is plugged into our formula (in excel), then I thought, maybe some other numbers may be helpful, so I played around. The percentage is the total percent full a 1/2 full bobbin is, including the core. Then I thought, hmmm, how useful is that? What can I see when the bobbin is emptying? The empty part! So, a quick addition of a column to excel, and wa-la, how much empty space is showing when the bobbin is half full, in inches ("in. from outer edge").
As you can see, for almost all the bobbins, there's over 1/2 inch of empty space when it's half full. Only Journey Wheel and Butterfly/Lendrum WooLee Winder have less than 1/2 inch -- that's due to their larger center core relative to the overall bobbin radius.
I sure hope I've done the algebra correctly, the numbers all seem to be in the right ball-park.
This is why we all buy plying or jumbo bobbins rather than calculators :-) so we can ply two full normal sized bobbins into one lovely skein on a single bobbin!