Reel Fill Calculation - Calculating line counter reading

[Carpedium]

Hello,

I have a real world application that I need some help with. I am sensitive to the fact that this is not the space for homework questions, or homework style questions. I feel that this is neither. If you disagree, please let me know and I will post it elsewhere.

It has been a while since college, any help would be appreciated. I believe that this is an integral calculus problem. The reel spool diameter changes as line is wound on the reel. I have posted this on other math forums and have not received input.

I have a fishing reel that has a line counter on it. As you reel in or let line out, it counts the number of revolutions of the spool and mechanically translates it into feet. The line counter is designed such that it is only accurate when the spool is full of line.

When the reel is empty, one foot on the line counter equals four inches of line on the spool. The spool is 0.5 inches in diameter with no line.

The reel is capable of holding 310 yards of line that is 0.019" in diameter, or 3.164 cubic inches of line.

The line that I intend to use is 0.0157 inches in diameter.

I have three reels and I want to fill each with 1000 feet line from a 1000 yard spool.

The problem:

Need to calculate the ending line counter reading when 1000 feet of line has been spooled up on the reel from empty.


Thank you in advance for your help.

 

[berkeman]

Hello,

I have a real world application that I need some help with. I am sensitive to the fact that this is not the space for homework questions, or homework style questions. I feel that this is neither. If you disagree, please let me know and I will post it elsewhere.

It has been a while since college, any help would be appreciated. I believe that this is an integral calculus problem. The reel spool diameter changes as line is wound on the reel. I have posted this on other math forums and have not received input.

I have a fishing reel that has a line counter on it. As you reel in or let line out, it counts the number of revolutions of the spool and mechanically translates it into feet. The line counter is designed such that it is only accurate when the spool is full of line.

When the reel is empty, one foot on the line counter equals four inches of line on the spool. The spool is 0.5 inches in diameter with no line.

The reel is capable of holding 310 yards of line that is 0.019" in diameter, or 3.164 cubic inches of line.

The line that I intend to use is 0.0157 inches in diameter.

I have three reels and I want to fill each with 1000 feet line from a 1000 yard spool.

The problem:

Need to calculate the ending line counter reading when 1000 feet of line has been spooled up on the reel from empty.


Thank you in advance for your help.

Welcome to the PF.

I don't think you need Calculus for this problem. You can just use the old trick for summing up all the numbers from 1 to 100...

1   +   2 +  3 + ... +  98  + 99 + 100
100 + 99 + 98 + ... +    3 +    2 +    1
---------------------------------------------
101 + 101 + 101 + ...+ 101 + 101 + 101 = 100*101 = 10100

Which is twice the single sum, so the sum is 10100/2 = 5050

So you have an inner and outer circumference to your reel of line. Call the inner winding circumference C1 for the inner winding, and Cn for the outermost winding. If you add those two circumferences, you get the same sum as you do when you add the circumference of the 2nd layer and the n-1st layer. And so on.

You should be able to write an equation for the total sum of the circumferences, based on the inner radius and your total 1000 feet of line. You will need to figure out how many turns of line you get per layer, and also write an equation that relates the total line volume to the outermost radius. The number of layers and the number of turns per layer gives you the total number of rotations of the reel.

I think that should get you started. Have a go at it, and post your work so we can see how it is going...

 

[berkeman]

BTW, there will be some packing efficiency, obviously, of winding the real line onto the reel. I'd guess it's in the 75% range if the fishing reel is of good quality, but that's just a guess.

So it would be good to have a way to validate your calculations in the end. One way would be to have an independent way of measuring the line as it is being taken up onto the fishing reel. One way to do this would be if you have a 2nd fishing reel with a turns counter on it. Make a pinch roller that holds the line against the other reel (at the constant inner diameter), so that as you wind the line up on your main reel, the line is pulled past the first reel which counts the number of turns (so distance = # turns * inner reel diameter).

By getting that independent measurement of the line length, you will be able to quantify the packing factor.