Welcome, Guest. Please Login
SLINGING.ORG
 
Home Help Search Login


Pages: 1 2 3 4 5
Send Topic Print
The physics of the sling (Read 34852 times)
Hondero
Senior Member
****
Offline



Posts: 463
Madrid-SPAIN
Gender: male
Re: The physics of sling
Reply #15 - Aug 10th, 2004 at 5:47pm
 
Don´t worry, Yurek, about your lazy holiday, I´m sure I can´t finish the task alone  Grin. Autumn will be a good time to work on physics.  Thinking of different ways to measure the initial speed of the projectile, as you and Matthias are talking, I see that another way, less accurate than the ballistic pendulum (wich is exact if well made), could be one working on the penetration of the projectile into a soft target. It coul be a first aproach, simple to make for  anybody. The kynetic energy of the projectile will be fully spent making the work of penetration. This work is the product of the penetration length and the resistance force of the material. This force would be calculated for a projectile from the resistance pressure (Kg/cm^2, etc) of the material, wich is a data of the manufacturer. It could be also easily calculated loading weights on the material until overcoming its resistance. From the equivalence "penetration work = kynetic energy" we extrac the speed.

A cheap an easy to get material would be the expanded polyethylene, that is used for modern targets in archery, though a material of very fine grain would be better and would provide more accurate messurements. If the plasticine were enough cheap as to make a big target...  Grin
Back to top
 

He brought a conquering sword..., a shield..., a spear... , a sling from which no erring shot was discharged.&&
WWW  
IP Logged
 
Yurek
Funditor
Past Moderator
****
Offline


The best thing, is a good
sling.

Posts: 942
Poland
Gender: male
Re: The physics of sling
Reply #16 - Aug 10th, 2004 at 7:28pm
 
Mechanical adventage of a sling.

We often can meet the opinion, that mechanical adventage of a sling relays on its ability to progresive accumulating the kinetic energy of the projectile in the rotary motion. In other words, one is able to get some relatively rapid turns of the sling in some unlimited time. Next, the sped up stone can be released in a suitable moment. Of course, that reasoning makes some sense, but doesn't explain the reason of the real efficiency of a sling.

Let's look on it closer. During a throw of a stone with the naked hand, we have the limited range of the hand motion. Assume, that the palm of our imagined thrower, during the throw, is able to travel the 1.8 m (S) along some curvelinear trajectory. He is very ambitious and want to beat the famous pitcher Nolan Ryan and accelerate the stone at the 55 m/s velocity (ca. 200 km/h)!.

Let's play a bit with a bit of physics.

For simplicity, let's assume that his hand is uniform accelerated (a = constans), from immobility at the 55 m/s. Since the increase of the velocity is lineal, then mean velocity (Vm) is equal to the arithmetical mean of the initial and final velocity:

Vm = (Vi + Vf)/2 = (0 + 55)/2 = 27.5 m/s

So we can calculate, what time of that throw (t) should be:

t = S/Vm = 1.8/ 27.5 = 0.065 s  !!!

As we can see, our thrower has very limited time for that acceleration.That time is terribly short! I'm not sure, if it is possible for human body, but maybe (we must verify it somewhere).

So, what can he do for geting the assumed final velocity (or even a bigger one)?


S = Vi*t + (a*t^2)/2        this is the well known eqution for the uniformly accelerated motion

Where:

t^2 = t*t
Vi - initial velocity
a - acceleration      

From this, we get:


a = 2 * (S - Vi*t)/t^2

For Vi = 0:

a = 2*S/t^2

So, since, he isn't able to shorten the time (it's pity, bacause it is a very strong factor, as we can see above), due to the limited mechanical power of his body, he may only lenghten the path of the acceleration. However, his hands are inextensible, so he will have to use just the sling.

Ineffective way of a sling using.

Our imagined thrower is an inexperienced slinger, so he will try to reach the assumed linear veloicty by progressive rotational speeding up of the sling. Saying simply, he will try to make the sling very fast whirled.

Let's check, how fast he should rotate his sling, in order to get the velocty of stone at the 55 m/s.

He has unlimited time for doing it (but only theoretically, he may die a natural death in the mean time). Starts slowly by a spacious arched movement of his whole arm. The faster rotations, the smaller circles are traced by the palm. When the sling stops to accelerate, palm still must rotate around the center of the stone (pouch) trajectory, because of, it must pull the cords a litlle agley to radius of sling rotation, in order to overcoming of the air drag of the sling. The picture below shows it.

...

Let's forsake a more penetraive analisis of that, but say only, that the smaller Driving/Centripetal force ratio, the smaller circle of the palm must be, for keeping the contant rotations up. The heavier stone, shorter and thinner is a sling, the more slight the palm movement can be, and vice versa. If our green slinger wants to keep really fast rotations, he shouldn't make spacious movement of his arm, due to its big mass (inertia).

Assumptions:

R = 0.06 m     - radius of the palm rotation
L = 1 m        - lenght of the sling
m = 0.1 kg     - mass of the stone
Vf = 55 m/s    
T - time of the single sling turn


T = 2*pi*(R + L)/Vf = 2*3.14*(1 + 0.06) = 0.12 s


So, for keeping the stone velocity at the 55 m/s he should make the one turn during the 0.12 s.

This gives the speed of rotation:


1/0.12 = 8.3 turns/s    !!!


The centripetal force (Fc) for keeping up that orbit of the stome, amounts:


Fc = m*Vr^2/((R + L) = 0.1*55^2/(0.06 + 1) = 302.5 N = 30.8 KG   !!!


The 8.3 turns/s of the 30.8 KG force whirling around the palm, for sure, this is what, no normal man can do it! Unless, he is the Gigant, who has just got convulsions Grin I think, that average man can make mentioned sling rotated at the 4 turns/s, but it will be a big effort for his arm, and let him reach only the 26 m/s, what is very poor result. If he even use a double lenght (2 m) sling, for sure, want't be able to whirl it so rapidly.

We must also say a few words about accuracy of that method of slinging. Assume, that our slinger wants to hit the 1m x 1m target from the 20 m distance. His sling is rotating at 4 turns/s in the horizontal plane. In this case, the released projectile must be contained in the 2.9 deg angle. The sling runs that angle during 0.002 s (2.9/(360*4))! We can imagine, how it is difficult to feel so short moment during furious brandising with the tense and tired hand, practicaly it is beyond of human perception.

As we see, that way of slinging is full of faults. Although, let us to accelerate projectiles on the very long track (multi-rotations), but our hands are not able to manage so very quick and circular (recurent) motion, which causes so big stress.

We know, that some man (Larry Bray, David Engvall, for example) could throw projectiles with sling at quite a lot above 400 m, what require velocities at least 100 m/s, saying safely. They used sling not longer than 1.3 m and are not the Gigants Wink So we should think about some diferent, much more efficient way of using sling...

================================================================================
================================

Uf! Enough for today. This is the first, short part of my virtual draft. It is very raw and probably contain quite a lot spelling and grammtical mistakes, if so, excuse me it. I hope there aren't content-related mistakes. This part rather treats about disadventages of a sling... or rather that method of slingin. Coments welcomed.

Jurek
Back to top
« Last Edit: May 7th, 2005 at 6:04pm by Yurek »  

In the shape, structure and position of each stone, there is recorded a small piece of history. So, slinging them, we add a bit of our history to them.
 
IP Logged
 
Matthias
Past Moderator
*
Offline



Posts: 1418
Gatineau/Ottawa QC, Canada
Gender: male
Re: The physics of sling
Reply #17 - Aug 11th, 2004 at 1:20am
 
Looks like you are on the right track Yurek. I certainly have the feeling while slinging that it is strength rather than speed that is limiting my range. Given a sling with little or no drag (there's that vacuum again) The amount of force you can put into the snap should be the limiting factor. I'd love to see a very high speed fram by frame analysis of a well executed throw - I think the forces/tensions achieved would surprise many of us...

I like the idea of projectile penetration too, but am hesitant based on too many years setting up experimental apparatus.

I think I "might" be able to swing a high speed strobe without spending any money. I'll build an LED enabled glans first. The unfortunate part is that with me as the only model, we won't be capturing the pinnacle of modern slinging by any means!

Matthias


Back to top
 
 
IP Logged
 
Hondero
Senior Member
****
Offline



Posts: 463
Madrid-SPAIN
Gender: male
Re: The physics of sling
Reply #18 - Aug 11th, 2004 at 5:43pm
 
Yurek, your first work is very clear and didactic, an example to follow for all us. I´m sure that beginners fond of that way of spinning will forget it quickly, or will change to other sport like artistic skating  Grin.
I´m looking forward to read the next part about the dynamics of the efficient way of using the sling.

I´m preparing something about "optimal projectil weight". Maybe tomorrow I´ll post part of it.
Back to top
 

He brought a conquering sword..., a shield..., a spear... , a sling from which no erring shot was discharged.&&
WWW  
IP Logged
 
Chris
Slinging.org Administrator
*****
Offline


Slinging Rocks!

Posts: 1284
USA
Gender: male
Re: The physics of sling
Reply #19 - Aug 11th, 2004 at 11:42pm
 
Can you guys write up a little outline for what you will cover in the Physics section.  i.e.:

PHYSICS SECTION
- Mechanical Advantage (intro, why a sling works).
- Orbital/Rotational Physics (tangential releases, etc.)
- Drag
- Trajectories
- ....
- ....
etc etc.

I'm not a physics buff, so my outline isn't meant to be serious, but perhaps something like it.  I think will help you focus the section and help you divide up the topics among your selves (and have clean divisions).  I can also update the master outline with all the subtopics that will be included.

Chris
Back to top
 
WWW  
IP Logged
 
Hondero
Senior Member
****
Offline



Posts: 463
Madrid-SPAIN
Gender: male
Re: The physics of sling
Reply #20 - Aug 12th, 2004 at 12:39am
 
Chris, I think that all this will come out a little anarchic at first, following the  interest each one has at each moment, like always happens in a forum debate. I think the debate is important even in this task, as a means to contrast ideas and theories with others. This way is also more fun and productive than to set up an index and "work" systematically on it Undecided.  At the end all will be said and we can reorganize the subjet in chapters and to make a definitive structure and index.
In any case the points to work on, as we´ve said before, could include two main parts: Physics of the throw and physics of the proyectile fly. The first one, will consider the biomechanical aspects, the mechanic of the throw, the styles of throwing, the air drag on the sling, etc.
The second part will include the estudy of trajectories, the air drag on the projectile, aerodynamics, ballistic coefficients, maximun ranges, etc.
As a combination of the two parts, it can be considered the optimal weight and shape of the projectil, the optimal designe of the sling depending of the objet of its use, the tecnics of throwing to get range, penetration or impact, and so many subjets that are very interrelated and difficult to place in a precise place.
The organization work at the end will be interesting and valuable.
Well, this is only my opinion, and my way of work (only in these situations)  Wink
Back to top
 

He brought a conquering sword..., a shield..., a spear... , a sling from which no erring shot was discharged.&&
WWW  
IP Logged
 
Matthias
Past Moderator
*
Offline



Posts: 1418
Gatineau/Ottawa QC, Canada
Gender: male
Re: The physics of sling
Reply #21 - Aug 12th, 2004 at 2:03am
 
Agreed. I think we are still in the stages of figuring out what we know (we know, what we don't know we know, known unknowns etc... Tongue) This one is going to be a fair deal tougher than just going out and collecting published work, citing some references and writing it up. We're in a position where we need to do some basic science before we can choose our targets (let alone hit them).

Chris' outline points are good ones, as are Hondero's up-thread. I think we'll get there, but it is going to take some time. If all the pieces that we're talking about come together this is going to be a heck of a seminal reference.

Matthias

Back to top
 
 
IP Logged
 
Hobb
Senior Member
****
Offline



Posts: 329
Denver, CO
Gender: male
Re: The physics of sling
Reply #22 - Aug 12th, 2004 at 12:02pm
 
Quote:
This one is going to be a fair deal tougher than just going out and collecting published work, citing some references and writing it up. We're in a position where we need to do some basic science


I'm lousy at physics, but I'm a statistician by trade.  If anybody ends up with some raw data in need of statistical analysis, or just needs some advice about experimental design, give me a buzz.
Back to top
 
 
IP Logged
 
Hondero
Senior Member
****
Offline



Posts: 463
Madrid-SPAIN
Gender: male
Re: The physics of sling
Reply #23 - Aug 12th, 2004 at 1:03pm
 
OPTIMAL WEIGHT OF THE PROJECTILE

The common sense says to us that the heavier is a sling projectile at smaller speed we will be able to throw it and smaller range it will get, reason why we will use light projectiles to increase the range. It is the effect of "greater reach by speed".
But on the other hand, if we throw several projectiles of different weight at the same speed, the heaviest will reach greater distance since its stored energy will be greater, and the air drag will take more time in restraining it. It is the effect of "greater range by weight". As we see there are two opposed effects dealing with the weight, being more important one or another according to the weight that we are considering. If we suppose that we are using all our power of launching, an optimal weight will exist over wich the range will be smaller due to the superiority of the factor "greater range by speed", and also if the weight is below it, the range wiil be smaller too due to the factor "greater range by weight".
Evidently this optimal weight will be different for each slinger, since his muscular power will be different in general, and different therefore the relation weight-speed that he can obtain. This relation weight-speed of the projectile is related in the first place to the muscular behavior, it is to say to the own relation force-speed of contraction that the muscle is able to develop. This relation is expressed by the Hill´s formula, whose graphical representation is this:

...


As we see, as greater is the contraction speed of the muscle, less  is the force that it can develop. Evidently, both the speed and the force of contraction  will be transmited in certain proportion through the osseous system of the arm to the sling. One could think that the muscle is able to always develop the same maximum power, it is to say W=F*V = constant. Thus a diminution in the speed would make a proportional increase in the force. But the reality is more complex, like it can be observed drawing up the curve of power from the curve force-speed, that would be this:

...

We see that the maximum power is obtained for contraction speeds very reduced, of the order of 30% of the maximum speed. Nevertheless this fact does not worry us too much since in the sport we are considering we are located in the last third of the curve, that can be considered straight in first approach.

We have landed in the biomechanics. The arm, and in general the bones that take part in the launching are driven by muscles. The basic mechanical scheme is that of a lever leant on a joint and driven by a force applied near the joint that makes move the distant end and take to it a demultiplied force. It´s somehow like a little staff-sling. But in addition, the slinger model of propulsion is a set of conected levers that act in sequence, well coordinated, adding speed over speed to the hand. The hand, in turn, communicates this speed to the sling. And the question would be if we can substitute this compound lever for a single one at the time of making calculations to simplify them.

Well, this is the general exposition of the optimal weight problem. And for today it is enough  ???. It will continue.
Back to top
 

He brought a conquering sword..., a shield..., a spear... , a sling from which no erring shot was discharged.&&
WWW  
IP Logged
 
Hondero
Senior Member
****
Offline



Posts: 463
Madrid-SPAIN
Gender: male
Re: The physics of sling
Reply #24 - Aug 19th, 2004 at 4:53pm
 
The graph of normalized Muscle Force vs Velocity of contraction we have seen before shows the values of force and speed relative to the maximum values of these variables. We consider only the part of the curve between 0 and 1. The maximum force is the static weight that can support a muscle, and the maximum speed is the one at wich the muscle would contract without supporting any weight. Naturally, these maximums will be different for each slinger depending on their muscular structure. When we are considering not a single muscle but all the muscular structure implied in the launching with a sling, there are a serie of muscles and lever bones acting, being complex to establish an equivalent and simplified mechanics model, but in any case, given the muscular nature driving these levers, it is possible to accept as a first approach that the hand, driven by the shoulder and arm, follows a similar curve to the one considered for a single muscle,  been now the maximum values:

1- Maximum speed at which the hand can move without any weight;
2- Greatest weight supported at null speed (static weight in throwing position).

These maximum values can be messured or approached, like for example the launching speed of best pitchers (160 Km/h). For the calculations that follows I ´ve considered an average slinger, with a hand velocity of 100 Km/h and a static force in the hand of 5 kg (throwing position).
The useful range of the graph for our purpose will be the corresponding to projectile weights up to 250 gr., that is to say, for relative values of the force between 0,05 and 0. We are placed at the right end of the curve. If we expand this part of the graph we see that it can be considered a straight line, and its equation would be:

X = 130 - 3,25 * Y

And in addition:

Y (mm) = Y (gr)/6,25
X (Km/h) = 80 + 2/13 * X(mm)

With these simple formulas and the Simulator of Trajectories we can make several interesting simulations:


1- RANGE DEPENDING ON THE SLING LENGTH
Let us take a 80 gr standard stone projectile (spherical shape for simplicity) and three slings:  long, medium and short, of lengths 105, 75 and 45 cm. respectively. Also let us consider the torque on the shoulder, being the shoulder the fulcrum, and the equivalent weights on the hand according to the length of each sling. In fact, in the launching, and depending on the style, the arm is partly folded at the beginning of snap, and soon it is stretched completely, that is to say, the length shoulder-hand is variable. In order to simplify the calculation we considered the lenght constant, about 70 cm, which corresponds with the arm length at the final powerful part of snap
Applying the previous equations and the Simulator we obtain the following values.

Sling-Length (cm).... Speed (Km/h)..... Range (m)
 105..............................210 ......................175,5
   75............................ 174,4................ ....140,5
   45 ............................146,7................ ....112.1

As we knew, to greater sling length greater range, but the increase in range is not as great as it could be thought. In addition, the air drag is different on the three slings and  difference in the ranges would be even smaller.
The long sling with respect to the short one has an increase in length of 133 %, but the increase in range is only of 56 %, that is to say, less than the half.




2-OPTIMAL PROJECTILE WEIGHT
This optimal weight, as it´s said, could vary for each slinger. The slinger considered is the average one we have previously considered. We will choose the long sling, of greater reach. We´ll try with projectiles of weight between 50 and 100 gr.. since we suppose that among them it will be the optimal one. We´ll apply the previous equations and the Simulator:

Weight (gr).......... Speed (Km/h)................Range (m)
 45................................. 227.5........................ 178.2
  50................................. 225........................... 179,3  
 55................................. 222,5........................ 184,3
 60..................................220........................... 182.3
 75..................................212,5.........................179.6
  80..................................210............................179.1
  90..................................205............................177


We see that for this slinger the optimal weight would be around 55 gr.
Curiously this weight corresponds enough with the projectiles of the last Guiness records, in spite of being the holders of greater power and range that the average slinger considered.


P.D. All this doesn´t  atempt to be exact calculations, but a general approach to the facts.
Back to top
« Last Edit: Aug 23rd, 2004 at 5:58pm by Hondero »  

He brought a conquering sword..., a shield..., a spear... , a sling from which no erring shot was discharged.&&
WWW  
IP Logged
 
Yurek
Funditor
Past Moderator
****
Offline


The best thing, is a good
sling.

Posts: 942
Poland
Gender: male
Re: The physics of the sling
Reply #25 - Aug 22nd, 2004 at 6:19pm
 
PART 2

EFFICIENT WAY OF SLINGING


As Hondero mentined, Mother Nature has bestowed us with long bones of hand and tendons attached close joints. It has let us to make the rapid movements of hand and made us a quite good throwers, but, of not too heavy stuff. Fortunately, we have got also strong muscles of legs and trunk, of which we have learned use for more efficient throwing, over centuries of huntings, wars, plays and sports. So, there are the tested and efficient ways of throwing different things.

Therefore, it will be quite resonable, if our slinger would use one of these ways of throwing just for slingnig. For instance, the basebal pitcher's way (maybe, a bit modified, adapted to a sling motion).

Now, I try to explain how it working, using the simplified model.

Usually, the pitcher's palm is accelerated along some curved trajectory, because it mostly is a result of interfering rotatory motions of the hips, shoulders, arm, forearm and wrist. The acceleration surely changes along that composite curve.

For our purpose, let's assume, that that trajectory is circular and is a half of the full circle (180 degs).

Before we start our investigation of that throw, we should understand, how the cords behave during either a uniform rotation or eccelerated one, what forces act on a sling. That lets us to imagine how the trajectory of the stone looks.

Forces during a uniform rotation of the sling.

Here is the picture showing the forces during uniform rotations of the sling in the vacum (the air drag and gravity omitted):

...

Pic. 2


Practically, there is only a one external force, just the Centripetal Force, needed to keeping the circular trajectory of the stone (the Centrifugal Force isn't an external force, is only an inertial reaction to the curving the stone trajectory by the Centripetal Force).  

So, only job the palm must do, is pulling the cords with the Centripetal Force during the uniform whirling. No perpendicular to the radius, a driving force (which gives a torque) is needed, because there isn't an acceleration and air drag). After the release, the stone will start its stright travel along a tanget direction, with the same linear velocity, that had during the rotations (we know it from the Newton's First Law).

Angular acceleration = moment of interia * torque = 0  ---> torque = 0

In this case

radius of the stone orbit = radius of the palm orbit + lenght of the sling



Forces during an accelerated rotation of the sling

Now the case is a bit more complcated. When the palm accelerates, the cords become deviated to backward. The most short way of explanation, how it comes up to that deviation is, that a inertia of the stone (its mass) restrains its acceleration (First Newton's Law), saing simply, the stone "tries to delay itself". A deeper analysis of it would be very interesting, however for our purpose we skip it, as it is a bit complicated. Let's say only, the angle of the cords deviation results from a balance of forces. The picture below shows just these forces.

...

Pic. 3

Despite of common sense, tense sling works very similar to a stiff and light stick with a stone on the tip. But, there is one diference. We are able to provide the additional significant torque to a stick, by a suitable action of the fingers and wrist. In case of sling, it is almost impossible, bacause cords can't transfer any torque, due to its flexibility. In the other words, we are able to make the single stroke with stick using only our fingers and wrist, but try to do that with a relative long and sagging statically sling!
       
A stick is able to transfer either the torque or pulling or even pushing force, but cords only the pulling force! However, if one attaches a stick to fingers by a soft and short loop (or simply keeps the stick only beetwen the thumb and index finger), a stick would work almost exactly like a sling. In both cases, his palm only can pull these ones!

Consider, that just the pulling force is the only possible kind of interaction between slinger's palm and a stone! For lashing with a sitck, it is the strongest driving factor too, the figers and wrist action adds only some edge, in comparison to a sling.

So, there is the questions, how it is possible, that we are able to make so dynamic and spacious lash of a sling, only by pulling of the cords?

Look at the above picture (Pic. 2). During the acceleraton, the Pulling Force (Fp) transfered by the cords to the stone,  has the two component forces: the tangential one (Fa) and the radial one (Fcp). The first one (Fa) just transfers the torque to the stone and make it accelerated. The second one is the Centripetal Force, which keeps the stone on the curved trajectory. The forces Fai and Fcf are the interial reactions, the Fa "tries to restrain" the speeding up the stone, Fcf is the Centrifugal Force, which "tries" to make the stone trajectory just stright. They give the resultant force, the Fpi. Just this inertial resultant force (Fpi) determines the angle of the cords deviation from the radial direction, and a value and direction of the Pulling Force of the palm (Fp).

The mentioned forces are in a dynamic balance. The Fdi "tries" to deviate the cords to backward, but the Fcf "tries" to push them outside (erect them). They are changing their direction and value during the stone acceleration. It coerces a continous change of the cord deviations (the angle FI) and value of the Pull (Fp), acting on our fingers.

From the below formula (for the centrifugal force):

Fcf = [mass * (tangent velocity)^2]/radius

We see, that a value of the Fcf force is strongly dependend on the tangent velocity (^2). Therefore, that one must become higher and higher, since the stone is accelerated by the (Fa) force all the time. It causes, that the stone tends to go outside, what results in the additional rotation of the sling around the palm. As a result, the motion of the stone is compounded of the accelerated stone rotation around the centre of the circular orbit of the palm (our earlier assumptions), and of the accelerated rotation around the the palm (grip point).

The next picture shows what's going on.

...

Pic. 4

This is a quite likely sample, how the sling can run during a efficient stroke in a real conditions. W can see, that the stone accelerates more and more just in the last phase of the stroke, due to its high velocities (Fcf grows like tangent velocity^2) and due to decreasing the tangent stone acceleration (Fa decreases, bacause the arm and wrist become more and more erected, and additionaly more laden by the big Centrifugal Force). We can compare the Fcf to a (more and more stiff) spring, which is indirectly diffracted by the Fa, and which gives the kinetic energy back just before the release. That gives an additional spur for the stone, and makes a shape and lenght of the stone trajectory more beneficial, bacause lets us to release the stone in a much more profitable moment (I mean, more to forward, when the slinger's arm is alredy streched). You can see it on the above picture, that trajectory is a kid of a spiral. Of course, it isn't only possible variant in practice. It depends on a slinger's motion, his quickness, a leght of a sling, mass of a stone, and a few more factors. But, let's skip discussing it for the present, bacause, now we should start explore the adventage of our sling.


The adventage of the mentioned slinging way

Now let's play a bit with numbers again, and check what that adventage could be. The folowing analysis arises out of the assumption, that proceeding of the sling on the picture (Pic. 4) is similar to a real one during an efficient stroke.

Our assumptions:

R = 0.6 m              - radius of the palm circular trajectory;
                               (an result of the entire body motion);
L = 1 m                  - lenght of the sling;
THETA = 180 deg   - angle traced by the palm along the trajectory;
Vpi = 0                   - initial linear velocity of the palm;
Vsi = 0                   - initial linear velocity of the stone (assumption,
                               that the stone starts from the immobility is unrealistic,
                               especially for longer slings, because for doing an efficient stroke
                               we need some initial velocity for a proper tension of the cords);
t = 0.15 s               - time of the stroke (acceleration);

At my guess, that time practically could be even almost 50% shorter. This is the next parameter, we need to measure in a practice.

Vpf                 - final linear (tangent) velocity of the palm;
Vsf                 - final linear (tangent) velocity of the stone;
ap                  - mean tangent (along the trajectory) acceleration of the palm;
                      (constant equivalent of a variable acceleration);
as                  - mean tangent acceleration of the stone.

               
Let's calculate the mean tangent acceleration of the slinger's palm:

ap = 2*(S - Vpi*t)/t^2        

where Vpi = 0

and

Sp = 2*pi*R*(THETA/360)    - lenght of the palm arc

Sp = 2*3.14*0.6*(180/360) = 1.88 m

we get

ap = 2*1.88/0.15^2 = 167.1 m/s^2

Now we are able to calculate the final velocity of the palm:

From

ap = (Vpf - Vpi)/t         where Vpi = 0

we get

Vpf = ap * t = 167.1 * 0.15 = 25 m/s

Well, that velocity doesn't seem to be exaggerated, even for an average thrower.
Back to top
« Last Edit: May 7th, 2005 at 6:01pm by Yurek »  

In the shape, structure and position of each stone, there is recorded a small piece of history. So, slinging them, we add a bit of our history to them.
 
IP Logged
 
Yurek
Funditor
Past Moderator
****
Offline


The best thing, is a good
sling.

Posts: 942
Poland
Gender: male
Re: The physics of the sling
Reply #26 - Aug 22nd, 2004 at 6:37pm
 
Consider, that the stone traces a longer trajectory than the palm in the same time. On the above picture, we see, that the angle traced by the stone amounts 223 degs. As we know, the stone trajectory is composed of the circular motion around the slinger's pivot, and the circular motion around the grip of the cords.

Let's calculate a lenght of the first one.

If the sling doesn't rotate aroud the grip, the deviaton angle of the cords would be constant all the time. So, the stone would trace the half-circle. The radius of that half-circle we can calculate as one of the sides of the triangle designed of the arms R and L and the angle 180-120 deg (look at the start position of the sling on the above picture).

From the Law of Cosines (a bit trigonometry) we have:

radius^2 = R^2 + L^2 - 2*L*R*cos(180-120) = 0.6^2 + 1^2 - 2*0.6*1*cos(60) = 0.76 m

radius = 0.91 m

Then we have:

S1 = (180/360)*2*pi*radius = 3.14 * 0.91 = 2.8 m

Now, the lenght of the second "component" of the trajectory.

The angle of the cords rotation around the grip (palm) amounts (accordingly to the picture):

120 - 60 = 60 deg  

S2 = (60/360)*2*pi*L = 3.14*1/3 = 1.1 m

Then we have the entire lengh of the stone path:

Ss = S1 + S2 = 2.8 + 1.1 = 3.9 m

The acceleration of the stone we calculate in the same way like for the palm:

as = 2*(Ss - Vsi*t)/t^2          where Vsi = 0

as = 2*Ss/t^2 = 2*3.9/.015^2 = 346.7 m/s^2

So, the final tangent (muzzle) velocity is:

Vsf = as*t = 346.7 * 0.15 = 52 m/s = 187.2 km/h

So, now our slinger is able surpass pitcher Nolan Ryan's 45 m/s quite easy! All what he needs, is a 1 m long sling, nice stone and quite easy smooth stroke.

The adventage of the sling in this case amounts:

Adventage = as/ap = (2*Ss/t^2)/(2*Sp/t^2) = Ss/Sp

Adventage = 3.9/1.88 = 2.07

From the above contemplations and calculations we can conlude, that during the stroke a sling doesn't work exactly in the same way like a simple stright lever. The trajectory of the stone has the variable radius, so isn't circular (even if the palm traces a circle path). Despite the mean radius of the stone trajectory is smaller than the sum the L and R, the lenght of the path of the acceleration is significantly extended due to the additional rotation of the sling around the grip point. It is very efficient, when she sling is markedly deviated to backward in the first stage of the stroke and strighten itself just before the release. It lets to release a bit later, and better utilize the full motion of the forearm and wrist for the stone acceleration. Therefore, it is very important to drive the palm smoothly (it doesn't mean slowly) along a properly curved trajectory and keep a suitable ratio of Fa/Fcp. In the other words, The bigger a stone velocity, the bigger should be the palm acceleraton, and/or the more extended motion of the palm, the sling shouldn't "escape" permaturely. Fortunately, it naturally results from our's body mechanics. Describing of that questions by analytic equations would be complicated, as we should take into consideration a lot factors. The best way to find our OPTIMUM is understending how it works, and of course, a lot of practice, that gives just the feelling. Proper shots are felt as springy (just result of proper using of the "centrifugal spring") and pleasant, don't hurt our body (of course, a bit trained body). We get impression, that our forearm is just shot out to forward.

Now we try to check, what our slinger should do for... surpassing the curent record holders in slinging for  range...

++++++++++++++++++++++++++++++++++++++++++++++++++

Well, I'm not sure if I should reveal that secret here. So maybe it will be continued Grin Now I'm tired. Hondero and Matthias, thanks for the encouraging words. I'm happy, that there are as much as three person interested in that topic at least. So it is worth of our effort Cheesy I was a bit off lately, so I have some arrears on the forum. I must yet read your Hondero last posts in this topic. I have find out some new interesting things in your post, especially the Hill's fromula charts. You both make a good job trying to measure slinging parameters. I would love, we get to know the exact trajectory of the palm and pouch, and the velocities and acceleratons in diferent points of these, during a efficient slinging.


Jurek
Back to top
« Last Edit: Sep 10th, 2004 at 2:29pm by Yurek »  

In the shape, structure and position of each stone, there is recorded a small piece of history. So, slinging them, we add a bit of our history to them.
 
IP Logged
 
Hondero
Senior Member
****
Offline



Posts: 463
Madrid-SPAIN
Gender: male
Re: The physics of the sling
Reply #27 - Aug 23rd, 2004 at 1:26pm
 
Very interesting your analysis, Yurek!!. I think the position of the sling at the final stroke is quite accurate, though maybe it depends a little on the style you use. I´m addicted to accuracy and so my final stroke is very similar to a particular baseball pitch, describing the hand an almost straight trajectory from behind and ending with the arm streched towards the target. Thus the sling is more retarded at first and at the end it describe this aditional turn around the hand you say in a marked way. Other styles like Newmanitou or other guy whose name I don´t remember and that posted a good video, semms to have a snap with the sling almost like a continuation of the arm, in the upwards position.

The subjet is very interesting though it isn´t of general audience, but I  have to confess that I´m hooked.
I´have to modify a little my previous calculations considering this angle of the sling and the arm at the release in the general throwing style  Wink.
Back to top
 

He brought a conquering sword..., a shield..., a spear... , a sling from which no erring shot was discharged.&&
WWW  
IP Logged
 
Yurek
Funditor
Past Moderator
****
Offline


The best thing, is a good
sling.

Posts: 942
Poland
Gender: male
Re: The physics of the sling
Reply #28 - Aug 23rd, 2004 at 6:13pm
 
Thanks Hondero,

Your posts I read with a great interest too.

Quote:
though maybe it depends a little on the style you use


For sure, you are right. I think it also depends on the lenght of a sling, curvate of the palm trajectory, initial velocity of the pouch, distribution of the acceleration on the trajectory, air drag and indireclty on the mass (that slows down a slinger). So, there are possible different behaviours of a sling during the stroke. I believe there is the optimal one for a given L/R ratio, cosidering the velocity adventage.

Quote:
I´m addicted to accuracy and so my final stroke is very similar to a particular baseball pitch, describing the hand an almost straight trajectory from behind and ending with the arm streched towards the target.


Yeah, it seems to match to the theory. I know you like rather shorter slings, as more accurate. Shorter ones more tend to escape ouside, because the cetrifugal force is inversely proportional to the radius of the trajectory (for a given tangent velocity, the smaller radius, the bigger Fcf). So, if your palm runs along rather flat curve, then it's radius is a bigger and the Fcf is smaller. Therefore the sling doesn't erect itself too early, what would extort a permature release, or stone would go more to the left (down) from target.

Quote:
Other styles like Newmanitou or other guy whose name I don´t remember and that posted a good video, semms to have a snap with the sling almost like a continuation of the arm, in the upwards position.


If so, they would have to release too early (nearly the 3 o'clock or under the shoulder) or to send the stone sideward or into the ground Smiley

Quote:
I  have to confess that I´m hooked


I'm afraid I'm too Cheesy Does anybody know where I could find an Anonymous Slingsters Grin

Jurek
Back to top
 

In the shape, structure and position of each stone, there is recorded a small piece of history. So, slinging them, we add a bit of our history to them.
 
IP Logged
 
Hondero
Senior Member
****
Offline



Posts: 463
Madrid-SPAIN
Gender: male
Re: The physics of the sling
Reply #29 - Aug 24th, 2004 at 11:13am
 
OPTIMAL WEIGHT OF THE PROJECTILE (Final part)

We have seen that for the average slinger, throwing with spherical stone projectile, the optimal weight was of 55 gr. Nevertheless there are not much difference using others near weights. In fact, between 50 and 70 gr. the difference in range is smaller of 2%, barelly about four meters, which almost makes indifferent the use of anyone of these projectiles since other factors as small differences in the shape, rugosity of their surface or position of the projectile in the flight can compensate this difference.
I´m going to define a parameter Grin that I´ll call 2% RWI (Interval of Weights for Range variations of 2 %). For the average slinger this parameter would be then 50-70.

Now let us see the behavior of the lead projectiles, also of spherical shape at first,  in the hands of the average slinger:

weight (gr).....velocity (m/s)......range (m)
   20...................... 66.66................... 255
   30...................... 65.27....................262.5
   40...................... 63.88....................264
   50...................... 62.5..................... 262
   60...................... 61.11................... 258



We see that now the optimal weight is 40 gr. and the 2 % RWI is 30-50. The light lead projectiles are then more advantageous, and in addition we even see that the famous 20 grams small glandes reach a distance smaller only a 3,5 % than the optimal weight, which explains its frequent presence among archaeological glandes.

We have spoken until now of the optimal weight for the average slinger... but will be it the same one for a more powerful slinger? The common sense says to us that existing enough throwing power, the heavy projectiles are more advantageous due to their greater dynamic inertia and greater difficulty to be restrained by the air. The results of the simulation for a slinger with speed in the empty hand of 130 Km/h and a static force in throwing position of 10 kg are:


weight (gr).......velocity (m/s)........range (m)
    50....................... 85.76....................391
    60....................... 84.86....................395.8
    70....................... 83.95................... 399.2
    90........................82.15................... 401
  100....................... 81.25................... 401.4
  120....................... 79.44................... 398.2
  130....................... 78.54................... 396
  150....................... 76.73....................389


We see that the optimal weight is now of 100 gr. and the 2% RWI are 60-130. The evident conclusion is that the more powerful is the slinger more advantages will have using heavy projectiles, although we see that the margin of weights is rather wide and that it will be more important than the optimal weight the other mentioned factors that take part in the range, as the shape, the rugosity of the surface and the position of the projectile in the flight. These points will be the following ones to analyze (HELP, Matthias...  Grin)
Back to top
« Last Edit: Sep 3rd, 2004 at 4:50am by Hondero »  

He brought a conquering sword..., a shield..., a spear... , a sling from which no erring shot was discharged.&&
WWW  
IP Logged
 
Pages: 1 2 3 4 5
Send Topic Print
(Moderators: Bill Skinner, Mauro Fiorentini, Rat Man, Curious Aardvark, Chris, Masiakasaurus, David Morningstar) - (Moderator Group: Forum Moderator)