Jim...
Good analysis. I had only a few comments in your text in my usual, BOLD CAPS.
Gordy
From: "JFV" <jfv@xxxxxxxxxxxx>
To: "'Gordon Hill'" <hillshead@xxxxxxx>
Subject: RE: Casting math & brain teasing
Date: Tue, 21 Feb 2006 12:56:14 -0500
Hi Gordy,
Did some work on this over the weekend in the interest of my understanding, let me know what you think about the following:
First of all a few definitions and assumptions;
Kinetic Energy = 1/2mv2 and describes the total potential energy in a system.
Fly Leg the leg attached to the fly.
Rod Leg the leg of the fly line closest to the rod tip.
Shooting line is not considered ONE WAY TO CONSIDER THE WHOLE THING WHEN SHOOTING LINE IS TO IMAGINE THE WHOLE COMPLEX MOVING IN THE DIRECTION OF THE CAST. THE MECHANICS CHANGE ADDITIONALLY BY THE FACT THAT THE RESISTANCE TO THAT FORWARD MOTION OF THE LOOP IS DRAMATICALLY LESSENED, SINCE THE ROD LEG IS NO LONGER "FIXED" AT THE ROD TIP. SINCE THAT RESISTANCE IS CONSTANTLY CHANGING AS THE LOOP IS SHOT FORWARD, THE CALCULATIONS BECOME THOSE OF A VIRTUAL, "MOON SHOT". DEFINATELY SOMETHING THE PHYSICISTS COULD DO......BUT, AS YET, HAVEN'T.
In the past we have described the rolling loop as a type of pulley. This example may yield a wrong impression.
A line around a pulley is a 1::1 relationship. That is, move the bottom line (the rod leg for our purposes) 1? and the top or Fly leg will move 1?.
A more appropriate explanation is a ?Moving Pulley? where the ratio is in fact 1::2. ANALOGY OK.....BUT MORE OF A METAPHOR, SINCE THERE IS NO MECHANICAL ADVANTAGE DUE TO NO ROTATIONAL FULCRUM.
When we cast a line the ?Loop? is moving forward? the pulley is moving forward and this is exactly what Mac Brown in Casting Angles refers to as a ?Vector Pull?.
For every 1? the pulley moves forward it lays down 1? of rod leg AND 1? of fly leg. 2? of fly line has been used. The rod leg is stationary (held by the rod tip).
Prove it to yourself.
Lay a loop on a flat surface with the fly end at a marked position.
Now while holding the rod leg stationary move the loop 1? forward.
Measure the distance the fly has moved.
The fly has moved 2?.
The fly leg has covered Twice (2x) the Distance in the same amount of Time? it must be traveling at twice the speed of the loop!
Thus the fly travels at 2x the speed of the loop. A loop speed of 80?/sec = fly speed of 160?/sec
Now for the issue of Mass and its relationship to the velocity.
When we make a forward cast, (no wind, air resistance, friction etc. considered here) we are applying force, transferring energy through our muscles, body, arm, hand, to a flexible lever and to the mass of the line.
The total energy we impart to the system is equal to the total kinetic energy in the casting system. Once a loop is formed whatever its components of mass and velocity, the total Kinetic energy is fixed. Written in the history books. The loop is launched with the formula (KE=1/2mv2 ) so to speak, attached.
Assume 60? of aerialized Level Line, 7.5? of leader and fly (weightless)
At Loop formation and Energy Transfer the total mass (67.5?) of the line is launched with a loop velocity (whatever the caster/ rod applied) and let?s say the total KE = Z.
At mid point of the roll out the KE of the system remains the Same! However the ?active mass? is ½ of what it was. The KE of the system still =Z so the velocity of the fly is forced to increase. THE, "ACTIVE MASS" IS 1/2 OF WHAT IT WAS ONLY WITH THE LEVEL LINE. WITH A TAPERED LINE IT'S A VARIABLE, SINCE THE FORWARD HALF OF THAT LINE WILL WEIGH LESS THAN THE REAR HALF ON A SLIDING SCALE DEPENDENT UPON THE MASS PROFILE OF THE TAPER ITSELF.
Just before turnover the total KE of the system still remains unchanged =Z (again, friction, air resistance, wind etc not considered here), the ?active mass? is now almost 0 and therefore the speed must be higher.
So just as a taper can reduce the mass of the line causing an acceleration and heat loss so does the reduction of mass to what I am calling the ?active mass? increase the velocity. YES.....THAT VELOCITY INCREASES EXPONENTIALLY........(SQUARED).
In Summary it would appear that there are two types of physics working on the unrolling line velocity. One the mechanical advantage of a moving, unrolling loop and Second the reduction in mass of the unrolling line yielding an additional positive change in fly leg velocity. As a result of this I may not be able to quantify the initial velocity, the mid cast velocity or the terminal velocity however I now understand more of what is happening to the flight of the fly.
Therefore in answer to the questions posed earlier I would propose the answers are more correctly stated as:
2x the loop speed Plus an acceleration from the continuing mass reduction of the unrolling line and Less all the real world factors. (wind, air resistance, friction, heat, density, etc.)
Jim GOOD BRAIN TEASER !
Gordy
From: Gordon Hill [mailto:hillshead@xxxxxxx]
Sent: Thursday, February 16, 2006 8:02 PM
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Subject: RE: Casting math & brain teasing
Jim....
My comments in BOLD CAPS.
This was a tough quiz !!!! You did best of all answers I've gotten so far.
Gordy
From: "JFV" <jfv@xxxxxxxxxxxx>
To: "'Gordon Hill'" <hillshead@xxxxxxx>
Subject: RE: Casting math & brain teasing
Date: Wed, 15 Feb 2006 12:55:45 -0500Hi Gordy,
My answers in blue. Think I am going to learn something here.
Jim
From: Gordon Hill [mailto:hillshead@xxxxxxx]
Sent: Monday, February 13, 2006 11:40 AM
To: flysoup@xxxxxxxxxx; sobbobfish@xxxxxxx; rtab@xxxxxxx; CAPTPERMIT@xxxxxxx; creangler@xxxxxxxxxxxxxx; dwright@xxxxxxxxxxxx; daver@xxxxxxxxxx; dennisg@xxxxxxxxxxxx; captdoug@xxxxxxxxxxxxxxxxxxxxxx; dsprague01@xxxxxxxxxxx; ephemera@xxxxxxx; brushycreekfc@xxxxxxxxx; barefootj@xxxxxxx; bradyir@xxxxxxxxxxxxx; flyfishar@xxxxxxxxxxx; ken.cole@xxxxxxxxxxxxx; captkirk@xxxxxxxxxxxxxxxxxx; glbaggett@xxxxxxxxx; mkreider1@xxxxxxx; martyt@xxxxxxxxxx; niallogan@xxxxxxxxxx; pminnick@xxxxxxxxxxxxx; bigfly@xxxxxxxxx; whorwood@xxxxxxxxx; flycasts@xxxxxxxxxxxxx; sheila@xxxxxxxxxx; scjacobs@xxxxxxxxxxxx; cooper@xxxxxxxxxxxxxxxxxx; tharper@xxxxxxxxxxx
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Subject: Casting math & brain teasing
Hi, Group.
I've just finished re-reading Greg Spolek's article, THE MECHANICS OF FLYCASTING:THE FLYLINE (Am. J. Physics. 54(9) 9/86) pp 832-836.)
Some questions come to mind: ( If only the answers would come to mind!) ( Feels Very lonely ?out here in space!)
1.) A standard straight line overhead cast is made without shooting line. There is no wind. The rod is a 7wt fast action 9'. The line is WF7F, long belly. The diameter of the loop is 3'.
Just prior to loop turnover, the speed of the forward travel of that loop was calculated by time/frame video to be 80'/sec. The total line carried out of the rod tip is 60'. For practical purposes, the fly is a nearly weightless 1" yarn, "fly". An 8' tapered leader is used.
Question: How fast is the fly traveling ?
The fly is traveling at 80 ft/sec just prior to the turnover. However a tapered fly line and tapered leader will increase line/ leader/ fly speed as the mass is reduced.
NO. THE FLY IS TRAVELING AT 160'/SEC. (SEE COMMENTS, BELOW.)
2.) The same cast is made with a loop diameter of 1 1/2'. (18") The loop speeed is now 90'/sec.
How fast is the fly traveling, now ?
90 ft/sec just prior to the turnover . The smaller loop will increase the velocity of the fly (the tangential velocity) through turnover since the overall force applied remains unchanged and only the radius of the circle has been decreased.
F=ma=mv2/r Circular Motion Formula (see http:/tutor4physics.com/motioncircular.htm) r = radius, F= Centripital Force, a =centripetal acceleration, v = tangential velocity THE CIRCULAR MOTION FORMULA APPLIES TO TANGENTIAL VELOCITY WHICH IS REALLY THE SAME AS THE SPEED OF ROTATION AT THE EXTREMITY OF A THEORETICAL CYLINDER. (SLOWER AT THE CENTER OF THAT CIRCLE, MAX. AT IT'S OUTER EDGE.)
HERE, WE ARE LOOKING FOR THE FORWARD MOTION OF THE FLY AS RELATED TO THAT OF THE LOOP. THIS IS INDEPENDENT OF LOOP DIAMETER, AND IS TWICE THE VELOCITY OF THE LOOP, SO THE ANSWER IS, 2x 90 =180'/SEC.
3.) The loop diameter remains at 1 1/2 ' ( 18"). The loop speed has now decreased to 80'/sec.
How fast is the fly traveling ? 80 ft/sec NO. THE FLY IS TRAVELING AT TWICE THE FORWARD SPEED OF THE LOOP : 160'/SEC.
4.) Can you come up with a formula which mathematically relates the forward velocity of the loop with that of it's fly leg for varying loop diameters ?
F=ma=mv2/r Circular Motion Formula Bottom line summary velocity is inversely proportional to the radius of the loop. v2 = 1/r THIS ONLY APPLIES TO THE ROTATIONAL VELOCITY OF A THEORETICAL CYLINDER. HERE WE ARE LOOKING FOR THE RELATIONSHIP OF THE FORWARD VELOCITY OF THE LOOP WITH THAT OF THE FLY LEG FOR VARIOUS LOOP DIAMETERS. THE ACTUAL FORMULA IS VERY SIMPLE :-
THE VELOCITY OF THE FLY LEG IS TWICE THAT OF THE FORWARD VELOCITY OF THE LOOP IRRESPECTIVE OF LOOP DIAMETER.
(THIS FORMULA IS LESS TRUE AT THE EXTREMES OF LOOP DIAMETER....SAY WITH A LOOP DIAMETER WHICH IS NEGLIGIBLE, OR ONE WHICH IS SO LARGE AS TO FALL ALMOST IN THE, "NO LOOP" OR, "OPEN LOOP" CATAGORY.)
5.) We now switch to a LEVEL L7F line. Does the loop decelerate and then pick up speed near the end of loop turnover ? or does it accelerate until near the turnover point and then decelerate ?
Once formed the loop does not accelerate, it is traveling at a velocity. It is subject to forces of drag, friction, gravity which will slow it down. A level line will simply release its? energy all at once. Kick! YES.
6.) Now let's go back to making the same cast with a WF7F longbelly XXD line. Does the loop continue to accelerate throughout the most of its travel and then rapidly decelerate as it reaches the point of loop turnover ? Or does it maintain or increase its velocity at that point ?
Loop will maintain it?s velocity except for the forces of drag, friction, gravity acting against it. The tapered portion will accelerate as it trades decreased mass for increased velocity and heat. THE SIMPLE ANSWER IS, YES.
7.) Does air resistance to the forward motion of the loop have a greater effect just after loop formation, (RSP+) or near the end of loop turnover ?
Air resistance is directly proportional to speed YES....PLUS THE DIAMETER OF THE LOOP....WHICH IS WHY AIR RESISTANCE HAS A GREATER EFFECT NEAR THE END OF LOOP TURNOVER.
8.) Air density has now increased due to decrease in air temperature and high atmospheric pressure. Does this situation offer more resistance to the forward travel of the loop ? (Assume no wind).
Yes more resistance. Density the number of air molecules in a given volume will affect all movement through that medium. TRUE.
9.) Let's accept as fact that: a.) Air/Oncoming wind can provide more resistance to a wide loop than to a narrow one, and that: b.) A narrow loop provides better concentration of energy in the direction of the cast than a wide one.
Which of the two has the greatest effect on loop speed in the direction of the cast ?
Narrow loop has more effect as it decreases by the square as radius of the loop is increased. YES. THE SIZE OF THE LOOP RESULTING IN GREATER CONCENTRATION OF ENERGY IN THE DIRECTION OF THE CAST IS MORE IMPORTANT THAN THE AMOUNT OF VISCOUS RESISTANCE (AIR / WIND RESISTANCE)
FOR MOST CASTS :-
WHEN CASTING INTO A HEAD WIND THE ABOVE IS TRUE UNTIL YOU REACH A POINT WHERE THE VELOCITY OF THE WIND MAKES A FORWARD CAST IMPOSSIBLE. AT THAT POINT, THE WIND IS OF OVERRIDING IMPORTANCE. PRIOR TO THAT, AS THE WIND INCREASES, THE SIZE OF THE LOOP'S EFFECT OF CONCENTRATION OF ENERGY IN THE DIRECTION OF THE CAST, WHILE ALWAYS IMPORTANT, BECOMES SUBORDINATED TO THE EFFECT OF THE MOUNTING WIND VELOCITY YIELDING EVER INCREASING RESISTANCE TO THIS FORWARD MOTION.
10.) Does loop speed increase at a greater rate near the end of the cast when a tapered line is used than with a level line ?
Loop speed increases with a tapered line not so with a level line. THEORETICALLY, YES. SPOLEK'S STUDIES SHOWED, HOWEVER, THAT LOOP SPEED AND FLY VELOCITY DECREASED AS THE LOOP PROGRESSED.....AND THEN RAPIDLY INCREASED AT THE END OF THE CAST AS THE LOOP UNROLLED WHEN A LEVEL LINE WAS USED. THE OPPOSITE WAS TRUE OF THE LONG TAPER LINES WHICH INCREASED IN LOOP VELOCITY UNTIL JUST PRIOR TO THE END OF THE CAST WHENCE THIS VELOCITY DECREASED RAPIDLY. (SEE HIS FIG. 5, P.834 OF HIS ARTICLE.)
11.) Why ?
Taper is a decrease in mass and since Kinetic energy remains the same something has to give, therefore as the line mass decreases the speed and heat increase. YES.
12.) Let's say our goal is to have a final loop velocity at or greater than 30m/sec.
Which line requires the GREATEST initial velocity ot the newly formed loop (right after RSP) ?...
a.) A WF7F XXD b.) DT7F c.) L7F
The WF long belly and the DT are virtually the same on the front portion the only difference is the diameter of the running line on the WF will be smaller and thus require less energy on a distance cast. TRUE.
On shorter casts the energy requirement will be the same, on longer casts the DT will require more energy to accelerate the larger mass. YES !
13.) Which of the above lines requires the LEAST initial loop velocity ? YOUR ANSWER, BELOW, IS GOOD....BUT IS AN ANSWER TO A DIFFERENT QUESTION. GREG SPOLEK'S FINDINGS WERE AS FOLLOWS: THE, "TRICK" IN DESIGNING A PERFECT FLY LINE FOR MAKING THE DESIRED DISTANCE AND AT THE SAME TIME HAVING A CONTROLLED PRESENTATION LIES IN THE MASS PROFILE OF THE TAPER. AN, "IDEAL" SCENARIO WOULD BE TO HAVE ABOUT 6M/SEC INITIAL VELOCITY TO END UP WITH 30M/SEC NEAR LOOP TURNOVER, ENDING UP WITH 0 m/SEC. WITH FULL DISSIPATION OF ENERGY AS THE LEADER COMES STRAIGHT FOR A SOFT FLY LANDING.
HIS EXPERIMENTS SHOWED THAT THE DOUBLE TAPER REQUIRED THE GREATEST INITIAL VELOCITY (HE USED ONE WITH A SHORT FORWARD TAPER), AND THE LONG TAPERED LINE REQUIRED THE LEAST.
Level line will carry the energy further due to less dissipation of energy. TRUE....(GLAD YOU USED THE WORD, "FURTHER"....NOT, "FARTHER"). THE CAST DOESN'T GO FARTHER. WHY ?.....BECAUSE OF SUDDEN VIOLENT DISSIPATION OF ENERGY RESULTING IN WHAT FLY CASTERS CALL A "KICK" (SEE BRUCE RICHARDS', "MODERN FLY LINES", P 72.) THIS MAKES FOR A POOR DELIVERY....UNLESS YOU ARE PURPOSELY CRASHING A BASS BUG DOWN ON THE WATER.
14.) With the cast described in 1). at what point does gravity begin to make the fly line fall ?
Gravity is constant throughout the cast. TRUE...FOR THIS REASON, THE FLY LINE ACTUALLY BEGINS TO FALL AS SOON AS THE LOOP IS FORMED, RIGHT AFTER RSP. (IT USED TO BE TAUGHT THAT THE LINE DIDN'T START TO FALL UNTIL THE LOOP HAD UNROLLED.) PHYSICIST NOEL PERKINS DID A VERY DETAILED PAPER ON THAT SUBJECT, AND QUANTIFIED THE RATE OF DESCENT WITH MORE CALCULUS THAN I CAN FOLLOW.
15.) At what point does gravity have maximal effect ?
Gravity is constant throughout the cast YES !
16.) Does the formula: E= 1/2 mV-squared apply in any way to the dynamics of a fly line during the cast ?
Formula is for Kinetic Energy and represents the sum total of energy that is transferred from the body/arm / rod / casting action to the mass of the fly line giving it the ability to perform Work. Usually also called Fun as in ?casting a fly? except when all bound up in physics. (just a bit of humor!) TRUE. THIS FORMULA ALSO EXPLAINS THE MUCH GREATER ACCELERATION AND ULTIMATE SPEED OF THE FLY LEG OF THE LOOP, SINCE, AS IT UNROLLS, THE MASS DECREASES AND THE VELOCITY INCREASES AS ITS SQUARE, THE TOTAL ENERGY REMAINING A CONSTANT UNTIL DISSIPATION. THIS IS DUE TO THE TAPER. MUCH LESS OF AN EFFECT WITH A LEVEL LINE.
I realize full well that this gets away from the, "poetry" and even further from the, "zen" of fly casting.....and it won't result in one more fish caught. For those who really wish to get into the real beef of how things work, however, it can be fun.
Gordy