**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.