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When is my bobbin half full?

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) 2 0.275591 1.400723 0.60 70.0% Mach 1 2.165354 0.393701 1.505616 0.66 69.5% Ashford/Butterfly Jumbo 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!

10 comments:

Rebekkah said...

Hah!

I thought of trying to solve this problem the other day, and realized that it wasn't worth the headache. I'd rather deal with unequal sized skeins. Though the geek in me wants to do the math for my Matchless, just 'cause.

My "solution" to the full bobbin problem for my upcoming sweater project is that I will spin 9 full bobbins of singles, and ply the 3 ply on one of the leftover bobbins, taking a break to wind off each time it fills up. I'll keep on going, replacing bobbins of singles as old ones empty (I'm sure they won't be even), hopefully ending up with 9 very similar skeins, plus or minus one different skein.

Of course, I can't wait until Schacht comes out with that jumbo flyer/bobbin...

Amelia of Ask The Bellwether said...

That's terrific! And you've answered my earlier post too -- you must have at least 10 bobbins for your Schacht. :-)

Ann said...

You have waaaaay too much time on your hands. I agree with Rebekkah. I just don't worry about it. But a simpler method would be:
1. weigh the 2 empty bobbins and divide by 2 (average empty weight).
2. weigh the 2 full bobbins, subtract the weight of the 2 empty bobbins, divide by 2 (average weight of fiber)
3. periodically take one of the bobbins of singles off the kate and weigh it. subtract the average empty bobbin weight and you'll get an approximate idea of whether you are half way through or not.

Leigh said...

Wow, what a post! I think you've gone where no spinner has ever gone before! Inspirational for us techies. :)

Amelia of Ask The Bellwether said...

@Ann, I like the weight approach :-) cool way to do it!

@Leigh, the scariest thing was that I had the calipers in my pencil pot by my computer (grin)!

Dave said...

Hmm, I think I need a drink. :-)

Ok, so half-ers. How do you divide a roving/top in half if you don't have a scale?

Dave said...

Oh, and thirds, too, please. :-) LOL!

I've tried just folding them in half or thirds and ripping it apart, and I've been way off both times. Any advice would be gratefully appreciated.

Anonymous said...

Boy, that makes my head hurt!

Molly said...

That is extremely impressive math work. However, I think I'm going to stick with my usual method: fill one bobbin, finish on another, splice together, divide in equal halves with the help of a ballwinder and a kitchen scale.

I'm willing to do a fair bit of math in my spinning (and a lot more in my knitting), but I'm not sure 3D geometry is going to become one of those things. :)

Marcy said...

Ha! Amelia, you have a kindred soul in Etherknitter. She wanted to best utilize her full bobbins, so she and her computer geek husband invented "Etherply": http://etherply.sourceforge.net/etherply.html