Spool and Distance of casting
First approximation

There are reasonings that the reels with a nondisengaging levelwind "throw not worse" and even that the heavy spools "helps casting, as submits a line better"...
Let's try, even and as a first approximation, but from strict physical and mathematical items, to evaluate influencing mass and construction of a spool on range of casting.


We throw two blobs
We have two blobs of mass  m1  and  m2 ,  bound by springing hairline. One in a hand, other on table. The hairline with looseness, it is stretched only after departure of a blob from a hand.
By dush of the first blob we add him speed  v1  and, thereby, kinetic energy:
E1 = m1 v12 / 2
When the hairline is stretched and the transients on equalizing speeds of blobs are ended, for the first blob is taken the part of energy on acceleration of second blob,
DE1 = E1 m2 / ( m1 + m2 )
Accordingly, remains of energy of the first blob:
E1' = E1 - E1 m2 / ( m1 + m2 )
On the other hand, now blob represents freely flying body with some speed  v2  (apparently, that   v2  less than  v1 ). Its kinetic energy:
E1' = m1 v22 / 2
From the last two equations and first in section is received:
v22 = v12 m1 / ( m1 + m2 ) (1)  
First "blob" - understandably - is a bait. And second blob for us no, there is a spool. And now we shall attend to transformation it in the "second blob".
We make a blob from a spool
Key item is the moment of inertia of a spool - measure of its inertness "in general". We quote the tutorials:
Moment of inertia - value describing a mass distributions in a body and being alongside with mass a measure of inertness of a body at a nonprogressive headway. The moment of inertia of a body concerning a spin axis depends on mass of a body and on distribution of this mass. The more mass of a body and the further it lies from an envisioned axis, the major moment of inertia has the body. The moment of inertia of elementary (dot) mass  m,  remote from an axis on distance  r,  is so:
I = mi ri2 (2)  
Moment of inertia of all body concerning an axis is [the sum by  i ]:
I = å mi ri2
The last formula only looks simply, actually in practice does not attempted at all to calculate moments of inertia of composite bodies by it, does used miscellaneous experimental techniques and appropriate equipment. For example, so-called "Atwood machine".
Nevertheless, for "simple" bodies of rotation the formulas are too simple. For example, for a thin-wall tube, which is spinned about the axis of a symmetry and coherent with it by "unweighable spokes", formula such:
I = m R2
For the continuous homogeneous cylinder - such:
I = m R2 / 2 (3)  
- where  m  - mass of all body,  R  - radius of an external surface.
As it is visible from the basic formula, the essence in distribution of "slices" of total mass on radiuses of their gyration... Let's see at a representative spool.
On the radiuses, essential for the contribution to a moment of inertia, it is a thin-wall cup of 2-3cm width, filled by line. It is possible to consider this part of a spool as the continuous homogeneous cylinder with average density about 1g/cm.cub. (the line is spooled by cross turns, with splits, but they are filled in with water, plus beads of spool is a little, but magnify average density).
Under a cup (sometimes) there is a "leg" and, for some spools, side makeweight by the way of rotor-inductor or/and centrifugal system. On the average on width some millimeters of metal are received. In view of bottom of a cup this part can be equated to "prolongation" of the cylinder with average density 1g/cm.cub. (or a bit less).
Shaft and "about it" the weather do not make - radiuses are small, elementary mass of density 7g/cm.cub. at the radius 2mm influences a moment of inertia in 8 times more feeble, than elementary mass of density 1g/cm.cub. at the radius 15mm. Let's "spread" heightened density of a shaft on a "bit less than 1" density on a part of a leg and - as a first approximation we can consider a spool as the continuous homogeneous "exact" cylinder of average density about 1g/cm.cub. (concrete digit here does not play role any more). Accordingly moment of inertia under the formula 3.
Further - how to use knowledge of a moment of inertia of a spool? - Let's apply an impulse of force there, where it and is applied at casting - to a upper turn of line.
Our cylinder and certain "dot" blob (i.e. of necessary density in a point and on a unweighable rod pairing it to a spin axis) can have an identical moment of inertia. That is (see formula 2):
ISpool = mBlob RBlob2
Let's neglect a difference of radius of a upper turn of a line and radius of a spool and equate radius of a spool to a radius of gyration of a blob. Knowing  ISpool  (formula 3) and radius of gyration of "equivalent" blob, we receive its mass - so-called "reduced mass of a spool" (RMS),  mR :
mR = mS / 2 (4)  
It just is mass of ours "second blob" (hereinafter  mS  - mass of a spool  with a line ).
That the spool physically does not depart anywhere, it is unimportant. Here problem in introducing it from a spacehold and to cause to spin with peripheral speed  v2  on a upper turn. It is the same as to cause a blob with mass  mR  to fly with such speed.
At last, we throw a bait
So, "the system of two blobs" have thrown. How far it will fly?
As the baits very much differ by lift-to-drag ratios, it is expedient "as a first approximation" to consider range of flight in ideal conditions: an environmental resistance we leave out in general, we throw under an optimum angle to horizon. Then
s = v22 / g (5)  
- where  g  - free fall acceleration (about 9.8 m/s2).
In practice the environmental resistance effect on range highly, and very much variously for different baits. We leave out here and that even at close to an ideal set-up of brakes, the spool even a little and sometimes, but "jerks" a bait back; on-minimum, but owes to be braked in leading manner comparatively to a bait speed drop. We leave out friction of a line in turns on a spool and about a ring... Though just these losses are insignificant - in matching with "inertialess" tackle (see  Appendix ).

All this requires "the second approximation", but our problem now to evaluate not exactly a range of casting, but how it depends on properties of a spool. These proportions we quite can receive and at the made assumptions.
Nevertheless, let's list these assumptions and estimate their influencing on practical results.
  •  We used as mathematical model of casting the model without any losing of kinetic energy of a bait except for on acceleration of a spool up to equal peripheral speed. In practice it is accessible not absolutely, but is close. That is  v2  (and, accordingly, the range) always will be a little less, than calculated by the formula 1.
  •  For calculation of RMS we have accepted a spool as the exact homogeneous cylinder. For calculation of theoretical ranges it doesn't matter. In practice, because of a different construction, the spools of identical mass can have essentially different RMSs, essentially distinct from our "average" one.
  •  We leave outed an air resistance. It is rather essential, first of all because of it the digits of range cannot be perceived as accessible. Only for matching influencing of masses of spools.
  •  We leave outed an other frictions and action of the brake. At "exact" tackle and competent application the influencing of it can be rather insignificant.

    First approximation in the diagrams
    From the formulas 5, 4 and 1 is received:
    s = v12 m1 / g ( m1 + mS / 2 ) (6)  
    Let's suspect, that  before release  of a spool we accelerate baits up to speed of 60m/s or 40m/s. The dependences of range of casting on mass of a spool  with a line  and mass of a bait are  as a first approximation  rotined on the following diagrams.
    Faintly red and blue lines - theoretical limit (spool and line are "unweighable") for concrete values of bait's speed  v1 , 60 and 40m/s.
    The upper curve corresponds to some spools from Avail and sporting spools. Second - approximately to Presso spool; within the limits of third and fourth "are arranged" Alphas, TD-Z and Fuego.
    Apparently, the light spools always give a scoring in range - though it is most significant on smaller masses of baits, and on more weights in percentage terms it descends. Why to not use them always and everywhere, most light on any weights?
    Strength of spool, line capacity. Overage here, as is visible, only in a harm (plus to inert mass), but that is necessary for fishing in pleasure - it is necessary so much to spool, such spool on capacity and safety factor and should be selected.
  •  All same: there is no universal tackle.
  •  As though opposite: it is possible to use a little more "heavy" tackle as more-less universal, - if there is an overage of own forces and it is not a pity to spend them vainly. That is, putting superfluous energy in a cast, forgoing comfort also sensitivity.
    As far as more energy? - From the diagrams it is visible that, for example, the 10-gramme spool overthrows the 30-gramme spool even at acceleration by 2.5-gramme bait up to 40m/s against 60m/s, that is at our power inputs in 2.25 times less. That most at attempt to throw a 10-gramme bait with spools of mass 10 and 50 grammes.
    But it too only in the "first approximation", disregarding air resistance - in this case to movement by a rod. Actually difference in power inputs it is even more.
    Appendix.  Whether the "inertialess" reel is inertialess? (*)
    The records in casting distance are cracked with inertial reels. Why? You see at identical acceleration of a bait before release, after release of a line for an inertialess reel the initial speed of bait's "free fly" remains same, and for inertial reel at once impinges...
    Let's consider an example.
    We use an inertialess reel and inertial reel with a spool of 14 grammes mass (with a line). On both 150m of 0.25mm line is spooled, the bait of 14 grammes mass is accelerated up to 60m/s. Without the consideration of an air resistance and losses on friction, with an inertialess reel the bait  seemed should  (**) fly on 367m. The speed of the bait with an inertial reel will fall up to 49m/s and it should fly on 245m. Actually with an inertial reel the bait can fly on about 150 meters, and with an inertialess reel it is hardly. A reason?..
    At first, in frictions. Dragging of the next turn through turns of winding and bead by "sideways", beating of a spiral of a line about a rings plus a heightened air resistance to this spiral - for an inertialess reel.
    Photo: on a baitcasting and spinning rods the light emitting tubes are installed, on reels the fluorescent lines are spooled. It is well visible, that even by transversing through a lengthy tube, the line from inertialess reel in any way can not be quietened.
    Plus one more factor.  150m of 0.25mm line - it is 7 grammes. On an inertial reel these grammes are already accelerated up to the necessary speed, about  half  of losses of initial speed (from 60m/s to 49m/s) just on it and has left. And on an inertialess reel the line "stands", mass of its each meter the taking place in flight should accelerate from zero, losing on it the own speed... So not such "inertialess" it appears.  Half-inertialess.  With the postponed inertance.

    (*)  In Russia usually the baitcasting (and centerpin) reels are named "inertial reels", and the reels with fixed spool (spinning reels) - "inertialess reels". In-particular, in this Appendix it is just underlined the incorrectness of such terminology.
    (**)  That is the bait in our example that with one, that with other reel can fly on as though 367m only in one case: if at once will be torn off. Differently this theoretical maxima and with an "inertialess" reel theoretically is inaccessible: the inertia of a line should be taken into account in mathematical model of casting by spinning tackle, even in the "first approximation".


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    Posted 11.03.07
    Translated to English 12.07.07

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