Using Moment of inertia in sports performance

We have seen that a body’s inertia is its resistance to change in motion, and with linear movement mass is the sole measure of that inertia; the greater mass sets up the greater resistance and vice versa.

However, with angular motion the resistance offered to acceleration depends not only upon mass but also upon its distribution about the axis, i.e. the moment of inertia; the closer the mass to the axis, the easier it is to turn. This principle, of great practical application in many problems of human locomotion, explains why one can rotate a limb much more easily about its longitudinal axis than a transverse axis of the same articulation.

In athletics, the distribution of mass can obviously be varied by changing position, by changing shape, about the different axes. For example, flexed or straight, the mass of the arm is the same in both positions and yet it is easier to move in the first position (a) for there its mass is closer to the shoulder-axis. Its moment of inertia is reduced. Likewise, the flexor muscles which pull the recovery leg forward and upward in running have an easier task with the leg bent than with a greater angle at the knee.

For the same reason the total body mass can be turned more easily about / Axis f Axis its longitudinal axis than a transverse one, and with most difficulty about a medial axis; all these axes pass through the Centre of Gravity. Again, an athlete’s resistance to rotation (i.e. his moment of inertia) in pole vaulting is greater with the body extended p. immediately after take off b than in the swing-up, where the legs are well flexed. To find, accurately, the moment of inertia of the body in any position about a given axis, it would be necessary to take each particle separately, multiplying its mass by the square of the distance (measured perpendicularly) to the axis, i.e. its radius; finally, all the separate results should be added together. But since this is obviously not possible, useful if not wholly accurate estimates can be made by using the figures already given for the relative average masses of the human body; in estimating the moment of inertia for each of the body units it must be remembered that each separate part has its own Centre of Gravity, lying fairly exactly along a longitudinal line but always slightly nearer the proximal joint (i.e. the joint nearer the trunk).

A person standing rigidly upright on a frictionless turntable with arms extended horizontally has, about a vertical axis, approximately three times the moment of inertia than in a position where the arms are held to the sides. In the first position he is three times more difficult to turn; i.e. to produce an equal angular acceleration, and with the turning force applied at the same point in both positions, the impulse (force X time) must be three times greater; or an equal impulse must be applied at three times the distance to the axis.

In the arabesque allonge skating and dancing position resistance to turning will be about six times greater and when the body lies on the table horizontally , with the vertical axis passing through its Centre of Gravity, it will be about fourteen times greater. In fact, relative to an axis passing through its Centre of Gravity, the moment of inertia of the body is least about a longitudinal axis, with the arms extended and close together above the head and greatest when turning in a similar position about a medial axis but with arms parallel. But it is greatest of all in a fully extended position with the hands as axis , as in handspring take-off (when, usually, the moment of inertia of the whole body is found by adding (1) the moment of inertia of its mass, supposedly concentrated at its Centre of Gravity, about the main axis and (2) its moment of inertia about a parallel axis through its Centre of Gravity). In fact, the body’s moment of inertia is always greater about an axis not passing through its Centre of Gravity than that about a parallel axis that does.

Angular velocity

The angular velocity of a body moving uniformly is the angle through which it turns in a second. If, for example, an arm moves through a right angle in one second, its angular velocity about the shoulder axis is 90 deg. per second. Likewise, if the somersaulting diver turns completely (i.e. through 360 deg.) in two seconds, his angular velocity will be 180 deg. per second. Angular velocity may also be thought of in terms of revolutions per second, and there are other units in which it can be measured.

Angular velocities, like velocities, are rarely uniform in athletic movement, but if we wish to determine the angular velocity of a body at a given instant of acceleration, we must assume it to be moving uniformly for a short time; the shorter thejperiod, the more accurate the calculation will be.

When a turning force ceases to act then, from Newton’s First Law, the body to which it has given velocity will continue to revolve at a uniform rate, or, if brought to rest by force, the body will remain at rest. For example, the diver turns in the air as a result of forces applied on the springboard.

It follows, therefore, that an arm or leg can be moving relative to other parts of an athlete’s body without force acting simultaneously. However, with the human body, because of internal resistances due to opposing muscle forces and the elastic tensions of fascia, ligaments and tendons, where there is motion of such a kind there is usually force acting also. These internal resistances must always be overcome before force can set the various body levers in motion.

As previously mentioned in connection with levers , the linear velocity of a point on a turning body is directly proportional to its distance from the axis. Hence, when the hammer-thrower’s hands are two feet from the axis passing through his base and Centre of Gravity, and the hammer head is six feet away, the hands have only one-third of the linear velocity of the hammer. Again, if two discus throwers turn with equal angular velocities, the athlete with the greater radius of discus movement gives greater speed to the missile.

When the body has angular velocity, its motion may be considered as the linear movement of any point, plus a turning about that point with the same angular velocity. For example: the angular velocity with which a long-jumper rotates about his jumping foot as it rests, momentarily, on the board is equal to that with which he can be considered to turn during the take-off about his moving Centre of Gravity; if he does nothing to counteract it before leaving the ground (and maintains this position in the air), it will be the angular velocity with which he will continue to turn about his Centre of Gravity in flight.

The conservation of angular momentum

The product of a turning body’s moment of inertia and its angular velocity is called its momentum, which, like momentum, is a vector quantity possessing both magnitude and direction and bears the same relationship to impulse as does momentum to impulse.. It is a concept of the utmost importance in analysing turning movement in sport.

According to the law of the conservation of angular momentum, a turning body isolated from external forces, i.e. left completely by itself, will have a constant angular momentum; that is to say, the product of moment of inertia and angular velocity about the axis of momentum is constant.

If it were possible to make a turntable with frictionless ball-bearings and a man standing on it were set in motion by a push, he would continue for ever to revolve with angular momentum, if we ignore air resistance. By the same token, the angular momenta of divers, high-and long-jumpers, etc., whose body masses turn in the air, ‘free’ in space, may, for all practical purposes, be considered constant in magnitude and direction (for the force of gravity, which acts equally on all parts of a revolving body, can be ignored).

Under these conditions the total angular momentum is entirely unaffected by any rotational movements made with the legs, arms or some other part of an athlete. This will be discussed in more detail later.

However, by changing the moment of inertia of body position during spins and rotations it is possible, correspondingly, to speed up or slow down the turning rate. If, for example, the man standing on the revolving turntable increases his resistance to turning three-fold by stretching his arms sideways , then his angular velocity will be three times slower, and if he resumes his first position he will spin at the original rate; but total angular momentum throughout will be unchanged.

The principle is well exemplified in diving. Accepting that, in re lation to a horizontal axis passing through the Centre of Gravity, a ‘straight’ position has approximately three and a half times the moment of inertia of a ‘tuck’ position , if the diver leaves the springboard with just sufficient angular momentum for a complete somersault in the ‘straight’ position it will enable him, should he so choose, to spin from two to two and a half times ‘tucked’. Here allowance is made for the time it takes to adopt the more compact position after leaving the board and a final straightening out prior to entering the water.

This example from diving points to one way in which turning during flight can be controlled in long-jumping. Should the jumper leave the board rotating forward and then ‘jack’ or ‘tuck’ prematurely , his angular velocity is markedly increased thereby and his feet are driven down and back in relation to his Centre of Gravity, for a poor landing. On the other hand, by keeping the body extended in the air, ‘jacking’ at the last moment, this can be averted. Again, a high-jumper twists quickly in the air when his body’s longitudinal axis corresponds closely to his axis of momentum.

Taking another event, a pole vaulter swings slower with his body stretched out just after take-off than when tucked up at the end of the swing, in readiness for his pull-push action.

Ballet dancers and skaters frequently spin at very high speeds. While they are building up angular momentum their arms are stretched out and the free (non-supporting) leg is permitted to swing wide of the body’s vertical axis then, suddenly, the arms and leg are brought in with tremendous effect. A graceful finish and slowing down or stopping are then cunningly combined by again extending the arm and free leg masses.

Good hammer and discus throwers, like the pirouetting dancers and skaters, keep the free leg close to the supporting leg during their turns, thus getting the hips and feet in position quickly, ahead of the rest of the body and the missile.

Again, maintaining balance on ice is the more difficult because of a tendency to rotate (i.e., in a vertical plane), about the body’s Centre of Gravity, as opposed to the feet—where, in the latter case, the moment of inertia is greater.

Determining the axis of momentum

Reference has been made already to an axis of momentum, fixed in direction and about which jumpers, divers, etc., will possess unchanging angular momentum in the air. The position of this axis can be calculated provided the body’s separate momenta about a vertical, transverse-horizontal and medial-horizontal axis at the instant of take-offarc known. Here the principle is applied to a high-jump take-off.

As angular momentum is a vector quantity, each of these three separate turns is represented by a straight line which, in length, is equal to the magnitude of the corresponding angular momentum; all three meet at the athlete’s Centre of Gravity. It has been assumed, for this particular jump, that there is least angular momentum about the vertical axis and most about a medial-horizontal axis; with ackward rotation, also, about a transverse-horizontal axis. However, the turning combinations will vary from style to style, and even from one jump to another by the same athlete.

For a resultant to be found, the direction of each separate turn must also be known. Conventionally, positive direction is that which makes the turning look clockwise and, therefore, each axis is looked along so that this is so , and arrow-heads are then added to point accordingly.

By using the parallelogram method in three dimensions the magnitude and direction of the total angular momentum and the position of the axis of momentum can then be established. The latter, in the case of our high-jumper, will be slightly at an angle to the horizontal and approximately 45 deg. to the crossbar, fixed in direction throughout the jump.

In a somersault dive or long-jump where take-offs are properly balanced, it is not difficult to estimate, from the movements in the air, that the axis of momentum is horizontal. But in many other activities, e.g. high-jumping, it is frequently not perceptible in any but a purely theoretical sense and, in the air, the athlete appears to turn separately, but simultaneously, about his longest and shortest axes; i.e. his axes of minimum and maximum moment of inertia, respectively. In theory it is possible to resolve an athlete’s total angular momentum at any time when he is in the air in space—about either the vertical or horizontal axes previously referred to which remain fixed in direction, or his body’s longitudinal, medial and transverse axes, which change their position relative to the ground as the athlete moves in the air.

As a matter of interest (for it does not arise in track or field events), given sufficient time the human body moving about a prescribed fixed axis will finally settle down to spin with stable equilibrium, selecting its axis of greatest moment of inertia. This, apart perhaps from a small wobbling called nutation (Lat. nutare, to nod), then coincides with the axis of momentum.

An illustration of this principle can sometimes be seen in the circus or on the variety stage where one performer, suspended from a cord gripped by the teeth, is turned by a partner hanging upside down from a trapeze. At first the suspended person revolves rapidly about the body’s long axis but quickly (and automatically) assumes a horizontal position, turning about the body’s ‘preferred’ axis of greatest moment of inertia.

Turns originating on the ground

A careful analysis of track and field techniques shows that, to greater or lesser degree, almost all athletes break contact with the ground during performance, turning momentarily about an axis of momentum. However, it by no means follows that they are always conscious of it, or should be so.

Imperceptible in good running, it nevertheless occurs on each stride and also during well-executed high-hurdle clearances. More obviously it occurs in the jumping events and in the pole vault, as the athlete rotates about the crossbar on releasing the pole.

There are few, if any, good hammer throwers whose two feet do not leave the ground, simultaneously, during the last turn, though doubtless this is something not attempted. In all three of the other throwing events, when well performed, the breaking of contact, with a turn in the air, is important at least to the regaining of balance after the throw if not to the throw itself.

Although, as we shall see later, some turns in athletic movement can originate in the air, nevertheless in track and field athletics most are built up while the runner, jumper or thrower is in contact with the ground, when angular momentum can be acquired in the following three ways: checking linear movement, transference and eccentric thrust. Usually, at least two of these sources are combined.

Checking linear movement. When a body, moving in a straight path, is suddenly checked at an extremity, a hinged moment results and angular momentum is developed. For example: 1. In vaulting over a box the gymnast, after a preliminary run-up, fixes both feet momentarily at take-off, while the rest of his body rotates over and beyond. This turning continues in the air, bringing the head and shoulders down and feet up in relation to his Centre of Gravity. The clockwise body turn illustrated is then reversed when the hands strike the far end. 2. As a result of planting the pole at the end of his approach, the pole vaulter develops angular momentum. He rotates simultaneously about his hands and the end of the pole in the box. 3. The diver’s somersaults can in part be due to checking the feet at take-off on the springboard, as already described. 4. The three previous examples are concerned with angular motion about horizontal axes but, in point of fact, the principle holds for turning about all axes, and sometimes about more than one at a time.

In the javelin event, for example, the thrower checks with his front foot and turns, simultaneously, about a horizontal axis (at the point where this foot meets the ground) and a near-vertical axis (passing through the throwing base and Centre of Gravity). This imparts considerable linear velocity to the throwing shoulder and, subsequently, to the javelin.

As the linear movement of a point on a turning body is directly proportional to its distance from the axis, height can be of particular advantage to a thrower. Assuming, for the moment, that after checking with the front foot the thrower’s Centre of Gravity continues at its previous linear velocity, then the shoulder above it must move considerably faster.

However, checking must cause some loss in forward speed and the more acute the angle between ground and body at the first instant of contact (or, in the pole vault, the smaller the angle between pole and ground) the greater the loss will be. The advantage must be weighed against the disadvantage, as so often in track and field techniques.

Under hinged moment conditions, an athlete’s angular velocity at take-off is identical about both base and Centre of Gravity axes. However, this must not be taken to mean that the angular momenta about these axes are also the same; they are not, for the moments of inertia differ.

In analysing athletic techniques it is usually more convenient to consider take-off angular momentum about the Centre of Gravity as, afterwards, its flight path is more regular than that of the feet and is unaffected by movement of other parts of the body.

Transference. Momentum, linear or angular, can be transferred from one object to another and from a part of an athlete to his whole body.

An example of the transference of linear momentum is if a snooker ball, rolling without ‘side’, strikes one end of a row of balls, its momentum is transferred to the ball at the far end; if two balls are used then, on impact, two balls roll at the other end.

Transference of angular momentum from a part to the whole of the human body can be simply demonstrated. The legs are swung from the hips, unjacking the body and when they are checked in their movement angular momentum transfers to the whole body, which then turns into a sitting position on the table.

Transference can be sudden, as happens when a rotating body part is checked abruptly, or it can be gradual, as the arm, leg, etc., only gradually loses its angular speed. The phenomenon is common to dancing and athletic movement where, often, angular momentum is first developed and stored ‘locally’ while in contact with the ground and used, later, to turn the whole body in the air. The following are examples: (i). In a tour en Pair, the ballet dancer first rotates the arms, shoulders and head in a horizontal plane, with the feet firmly fixed on the floor. Then, as the legs drive him into the air, the effect transfers and his whole body twists. (ii) A good high-jumper’s pronounced free leg swing at take-off builds up much of the angular momentum he will need subsequently for the lay-out over the crossbar. (iii) On the springboard the diver develops angular momentum by swinging his arms. This helps to turn his body in the air. ‘Local’ angular momentum is best built up with the body part extended and accelerating through a considerable arc; maximum angular velocity should be developed before contact with the ground is broken. On transference, the effect on the whole body depends on the distance of the axis of ‘localised’ movement from the body’s main axis, as well as the magnitude of the ‘local’ angular momentum. The turning effect of the arms on the body is all the more powerful because of the distance between the shoulder line and the body’s main axis passing through the diver’s Centre of Gravity. For, in effect, it is the arms’ angular momentum about the main axis which is important.

Eccentric thrust. As, on successive take-offs, an athlete’s line of thrust moves progressively farther from his Centre of Gravity, so will body rotation be easier and the projection of his Centre of Gravity be more difficult. Thus, angular momentum can also be acquired by driving ‘off centre’, i.e. eccentrically.

In track and field athletics this type of thrust is usually applied to an athlete at a point where one foot is in contact with the ground— when its line of action is often at a considerable perpendicular distance to his Centre of Gravity. However, the transmitting of a targe force from the ground to the athlete can take place only when the thrust passes close to his hip, knee and ankle joint and to his Centre of Gravity. Otherwise there is a tendency for these joints to collapse and the thrust develops angular momentum beyond the athlete’s ability to control it. The following are examples of eccentric thrust: (i) By bringing his head and arms forward, as a preliminary to his downward thrust against the springboard, the diver moves his Centre of Gravity slightly forward of his feet. In a good dive the required turning effect is combined with sufficient height off the board. (ii) Although not perceptible in good jumping, part of a high-jumper’s lay-out over the crossbar is due to an eccentric leg thrust at take-off. But the best jumpers sacrifice only a fraction of their upward spring to obtain sufficient rotation. They gain more from their position at the high point than they lose in take-off velocity. (iii) In a gainer (i.e. reverse) dive angular momentum is imparted to the body by means of a final foot thrust, often made with the toes gripping the end of the springboard. This is all the more effective because of its distance from the diver’s Centre of Gravity. (iv) In initiating a front somersault, with no travel, on a trampoline the performer ‘breaks’ at the hips. The reaction of the trampoline thus projects him vertically, with a forward somersault. (Here, the total horizontal momentum generated while the feet are in contact with the trampoline must be zero. This means that if, at this time, he imparts momentum backward to some part of his body, e.g., in rotating the arms clockwise, he must give the rest of his body equal momentum forward, e.g., by shifting his hips horizontally in an opposite direction. Thus the trampoline will have exerted no residual force horizontally.) (The checking of linear momentum is, in fact, caused by an eccentric thrust, whereas a transference of momentum produces such a thrust.)

Turns originating in the air

Action and reaction. While Newton’s Third Law applies to all motion, it is particularly important to an understanding of human and animal movement which begins in the air, ‘free’ in space. For whereas take-off surfaces can ‘absorb’ the reactions to turns originated on the ground, the body alone can do this when initiating movement in the air.

By way of comparison and illustration, we will consider two phases of a pike dive. When the diver drives down on the springboard it reacts with equal force by thrusting upward against his feet; when, as a result, his body is thrown into the air, its linear and angular momenta are equal but opposite to that given to the earth; for impulse and angular impulse are common to both.

But later, when the diver ‘pikes’ in the air , the muscles pulling the trunk down and forward, clockwise, act simultaneously on the thighs, pulling the legs upward in a counter-clockwise direction. Expressed in terms of the contraction of a single muscle, the fibres exert their force equally on origin and insertion, producing rotations on either side of the hip joint that are equal but opposite in their angular momenta. The muscle forces required to stop these rotations are also equal and opposite.

Here are some other examples of this phenomenon. If, the outstretched arm is swung horizontally across the body , the reaction moves the turntable and the standing athlete towards the arm; when the same action is executed after an upward jump the body’s reaction is even more pronounced. (This technique—with the arm lowered in front of the body and raised sideways again to repeat, if necessary— is sometimes used to create twist in dives, so ‘squaring’ the body for entry.)

As the long-jumper brings his legs forward and upward for the landing the hip flexor and abdominal muscles pull his upper body forward and downward. When the hurdler pulls the rear leg across the hurdle rail the contracting muscles simultaneously twist the trunk towards the trailing leg. And when the discus thrower, with both feet momentarily off the ground, twists his hips and legs in advance of his shoulders and the throwing arm , the acting muscles pull the upper parts of his body in an opposite direction.

With such movement, action and reaction occur in the same plane or in planes that are parallel. They will be about an axis, an axis of displacement always at right angles and passing through the athlete’s Centre of Gravity (but not necessarily coinciding with his axis of momentum). Clockwise action produces counter-clockwise reaction, and vice versa, though it is possible that certain very minor reactions to movement can be taken up internally and, therefore, invisibly.

Relative moments of inertia. The angular velocity of the two moving body parts is inversely proportional to their moments of inertia about their common axis. Referring again to the pike dive ; assuming the moment of inertia of arms, head and trunk about the hips to be three times greater than that of the legs, the contracting muscles will impart angular velocities in the ratio of 1:3, i.e., while the trunk rotates through 30 deg. the legs describe a right angle.

If the man on the turntable adopts a sitting position and then repeats the arm experiment, his rotation towards the extended arm is reduced in comparison with the previous example because of his body’s greater moment of inertia about a vertical axis. By the same token, a high-hurdler’s body is less affected by the movements of his trailing leg when he leans forward. On the other hand, a long-jumper preparing for landing will get his heels higher, through quicker and with the minimum of trunk reaction by first bringing the legs forward flexed before extending them. It is therefore possible to control the angular range and speed of reaction to some extent.

We have seen that movement originated in the air cannot disturb an athlete’s Centre of Gravity; thus, in piking , when the upper body and arms and lower legs are brought forward in relation to the Centre of Gravity, the hips, lower trunk and upper parts of the thighs move back, and the product of the masses and their distances about the Centre of Gravity is unchanged.

Control of reaction-location. Within certain limits an athlete can control the location of body reaction. We have seen how the turntable reacts to an arm movement in a horizontal plane; but when both arms are swung in opposite directions simultaneously , the reaction to the movement of one arm is ‘absorbed’ by the other and the turntable does not move. A similar action of the arms after an upward jump will not affect the body. Again, a high-jumper’s clockwise arm sweep could produce a counter-clockwise reaction of the body, or it could be ‘absorbed’ by the other arm or some other part. When a hurdling rear leg movement is performed on a turntable, the reaction swings the table in an opposite direction but if the arms are used correctly, they can absorb this reaction and the table remains still (c). In hurdling, this arm action keeps the trunk facing the front, to the advantage of balance, direction and speed on landing.

Illusions. While, doubtless, a working knowledge of the mechanics of movement improves what has been called ‘the coaching eye’, action and reaction in the air are often difficult for even an experienced person to detect because of rotation or spin developed at take-off. To explain: when a pike dive is well executed it appears that throughout the upward part of the flight the diver brings his upper body down towards fixed legs and then, keeping the arms, head and trunk still, throws the legs back on a fixed trunk to assume a straight position for entry. Trunk and leg motions seem isolated and without reaction. This is because the diver leaves the springboard with forward rota- tion. In piking, as we have seen, the legs are in fact pulled towards the trunk but, in a good dive, their angular velocity is approximately equal to the body’s rotational speed in an opposite direction, thus the legs remain almost vertical as the trunk rotates down and forward.

Conversely, the trunk movement is the more pronounced during the upward flight because it blends with the body’s overall forward rotation off the board. Later, in straightening out, the trunk rotates backward and the legs forward in an approximate ratio of 1:3, but the body’s take- off rotation cancels out the trunk movement while greatly speeding up the movement of the legs. Thus, the diver’s total momentum is first concentrated in the trunk and then, subsequently, in the legs.

The following are examples from track and field athletics. The pole vaulter’s eccentric arm thrust at the top of the vault imparts a rotation (clockwise in our illustration) to his whole body. But as he simultaneously ‘arches’ or ‘jacks’, pulling the trunk and legs in towards each other, the leg movement is the more apparent to an onlooker because it blends with the vaulter’s overall clockwise rotation. But when, on releasing the pole, he ‘unjacks’ his body overall rotation favours the lifting of the arms, head and chest, and the counter-clockwise leg reaction is hardly noticeable, or, perhaps, is completely nullified.

Similarly, the muscles pulling the hurdler’s leading leg towards the ground also straighten up his trunk; at this stage of clearance he, too, ‘unjacks’. But because of taking off with an overall forward rotation his trunk maintains a forward lean for the landing and the leading leg sweeps down and back even more quickly. Likewise, in long- and high-jumping, action and reaction in the air are sometimes difficult to discern and analyse through being ‘superimposed’ upon rotations of the whole body.

We have seen already that an arm or leg can be moving relative to other parts of the body without force acting simultaneously. Theoretically, therefore, force can be applied to a limb while an athlete is in contact with the ground and, relative to the trunk, the limb will go on moving in the air without apparent reaction, for the reaction will have taken place on the ground. However, usually with body movement, where there is motion there is force also, because of the internal resistances which must be overcome.

Exploitation of relative moments of inertia. We know that movement originating in the air cannot change an athlete’s total angular momentum about his axis of momentum because the action of one part of his body is ‘cancelled out’ by the reaction of another. Hence, it might appear that it is impossible to turn the whole body in the air without a point d’appui. However, the following examples will prove to the contrary: (i) If a man, freely falling through space or standing on a frictionless turntable, extends his arms and then swings them horizontally in, say, a clockwise direction, the lower parts of his body will then turn in a counter-clockwise direction. And when he lowers his arms and takes the twist out of his abdominal muscles by turning his shoulders counter-clockwise, a clockwise reaction is produced in his hips and legs. However, now that the moment of inertia of the shoulders and arms is so much smaller than when the arms were extended, the clockwise reaction of his lower body is slight. In consequence, the whole of his body has turned counter-clockwise. (ii) This process—sometimes used at the completion of the twist, in a twisting somersault dive, to bring the hips and legs into line with the upper body—is similar to that which enables a cat or rabbit, dropped from an upside-down position, to twist in the air and land on its feet. During the first stage of its fall the animal ‘pikes’ or bends in the middle and stretches out its hind legs almost perpendicular to an axis passing through its trunk. It then twists the fore part of its body through 180 deg.; the head, fore legs and upper trunk are now ready for the landing. The hind parts react by being displaced through a much smaller angle, because of their much greater moment of inertia about this axis.

During the second stage of the animal’s fall twisting takes place about a new axis parallel to its hind legs , the twist being in the same direction as that of the head and trunk during the first phase. Therefore the hind legs now turn through the larger angle, for the moment of inertia of its upper body about this new axis is much greater. At the completion of this series of movements the animal’s whole body is free from deformation and has turned through 180 deg.. In our illustration the rabbit turns about these two axes of displacement at different times. However, the movements can be—and often are— made simultaneously.

Likewise, when a diver assumes a front-arch position, ‘breaking’ at the hips , extended legs and hips can be used to rotate the trunk in an opposite direction. But to originate twists in the air by this method (and 540 deg. twists can be achieved) the body must be neither too straight nor too ‘piked’; and considerable spinal flexibility is required to continue the movement forward, laterally and backward. (iii) An outstanding example of the use of this principle in sport is seen in the half-twisting somersault, piked, dive. The diver leaves the board with forward rotation and then adopts a pike position. Now he twists his trunk through 90 deg. and then extends his arms sideways.

With the moment of inertia of the head, arms and trunk now greatly increased about the body’s long axis, the legs are momentarily brought into line and twist taken out of the abdominal muscles ; their twist continues until a pike position has again been assumed, but with the legs now on the side of the trunk farthest from take-off.

These leg movements produce no noticeable reaction or change of position in the rest of the diver’s body because of (a) the latter’s greater moment of inertia about the trunk’s long axis, and (b) the illusion created by his forward rotation off the springboard—a rotation in the sagittal plane which tends to conceal his upper body’s reaction to leg movement in that plane.

The diver has again assumed a pike position, but with his body still twisted. He then brings his arms back to their first position, stretched above his head, afterwards completing a 90-deg. trunk turn. Finally, now facing the springboard, he straightens out and enters the water.

Just as animals employ these movements without conscious analysis of the mechanics involved so, too, can athletes. And just as animals do not all conform precisely to the same movement pattern when faced with such problems of balance and safety so, too, in diving and other forms of athletic technique do the methods of exploitation vary from one person to another. In originating twisting movement in the air, for example, there are many variations of head, shoulder, trunk, arm, hip and leg movement available to the flexible, physically-clever performer. (Diving experiments appear to demonstrate that about three-quarters of twisting originating in the air is developed through the use of the spine; only about a quarter through hip movement.)

To re-emphasise: these movements of the whole body in space always take place about axes of displacement which pass through the body’s Centre of Gravity, but which, in most cases, are distinct from its axes of rotation and momentum. They cannot change total angular momentum

Secondary axes. Another type of movement involving the exploitation of relative moments of inertia is concerned with motion of a part of an athlete about an axis at a distance from his Centre of Gravity—a ‘local’ or ‘secondary’ axis.

From Newton’s Third Law it follows that when a man falling freely through space, or standing on a frictionless turntable, twirls an arm in a circle above his head so that its axis of movement corresponds with the body’s longitudinal axis, the angular momentum thus developed is simultaneously compensated for by turning the whole body mass in an opposite direction. In such a circumstance the arm action possesses a constant turning effect on the rest of the body.

Note. The arm’s radius of gyration is the horizontal distance between its axis and a point which represents the ‘mean’ of all the separate moments of inertia of its many parts; a point where, for the purpose of rotation only, we can consider the mass of the arm to be concentrated. It is not the arm’s Centre of Gravity. The circle traced out by this radius will be referred to as the circle of gyration.

When a similar movement is made slightly to the side, however, the turning effect is in proportion to the arm’s distance from the axis of displacement. In reaction, greatest angular displacement is produced when the arm is farthest from this main axis, and least when nearest, but the total rotational effect on the whole body is the same as before.

Should the axis of arm action (now a secondary axis) be moved so far to the side that the arm’s circle of gyration no longer ‘embraces’ the axis of displacement , then part of the arm movement produces a turning effect on the rest of the body contrary to that hitherto; for, relative to the axis of displacement, the arm moves through a sector of the circle of gyration in an opposite direction. However, when the arm is farthest from the body’s main axis its turning effect exceeds that of our two previous examples and, through a 360-deg. arm movement, the angular displacement of the whole body remains the same.

The sector producing a contrary rotation is enlarged. By virtue of the greater distance between these axes it is also more powerful, but so, too, is the arm’s turning effect in an opposite direction. Therefore, through 360 deg. of arm movement, the angular displacement of the whole body is the same as before.

It should be noted that the secondary axis of each rotating arm passes through the shoulder, the main axis passing through the man’s Centre of Gravity.

Here it should be stressed that (i) while the arm movement continues the body can be turned horizontally through any required angle; but when the arm stops, the body stops also; (ii) although the foregoing examples show motion in a horizontal plane, the phenomenon applies to motion in any plane; (iii) such movement cannot change the body’s total angular momentum about its axis of momentum.

In effect, therefore, it is the arm’s momentum about the body’s main axis (passing through the Centre of Gravity) which is significant.

Balance in the air. Movements which exploit relative moments of inertia are of particular importance to the maintenance of an athlete’s balance in the air, for they can be used to counteract embarrassing rotations either originated at take-off or caused by air and wind resistance after contact with the ground has been broken. The following turntable experiment exemplifies this.

IT angular momentum in the same clockwise or counter-clockwise direction is given to the man and turntable before he makes any of the arm movements described under Secondary axes , the reaction to them will create an illusion of reducing total angular momentum or even of reversing its direction, depending upon the efficiency of the arm movements; but when the arm stops, the original angular momentum is again apparent.

Three examples taken from sport follow: (i) A ski jumper with forward rotation in the air (movement hardly conducive to a safe landing!) can move his arms about his shoulders, in the same vertical plane and in the same direction as the embarrassing body rotation, and thus take up some of it, causing his body rotation to slow down. However, when the arm movement stops, the original body rotation again reveals itself.

Factors limiting the value of this action to the ski jumper are (a) working together in a sagittal plane, the arms cannot be swung with full range behind the shoulders and (b) as they move past the trunk they could possibly add to the body’s forward rotation (for the reason given under Secondary axes). (ii) In the hitch-kick long-jumping technique the forward rotation of the arms and the movement of the legs have a forward angular momentum. This may be (a) less than, (b) equal to, or (c) greater than the forward angular momentum with which the jumper left the board, depending upon the take-off angular momentum and the efficiency of his arm and leg movements in the air.

In the first case (a) the arms and legs will take up some of his angular momentum, causing his forward rotation to slow down. In (b) they will take up all his angular momentum, and so his forward rotation will cease temporarily; and in (c) the trunk will automatically rotate backward about the body’s Centre of Gravity, otherwise the jumper will have generated angular momentum in the air, which is impossible. (iii) The long-jumper corrects lateral balance (i.e. his movement in a frontal plane) by the use of his arms. When they are moved about his head in that plane their influence on the rotation of his whole body is the more powerful because of the distance between his shoulders (secondary axes) and his Centre of Gravity (through which must pass his axis of displacement). Depending upon its efficiency, the arm action slows down, stops or even reverses body rotation, yet without changing his total angular momentum; when the arms stop, the rotation of his whole body ‘takes over’ again.

It is sometimes convenient to ‘break down’ movement originating in the air into simultaneous rotation in several planes. Thus, a combination of rear leg recovery and front leg downward drive in hurdling may be considered as simultaneous body rotations and reactions in the three main planes, i.e. horizontal, frontal and sagittal.

We have already seen that, in a horizontal plane, the action of the hurdler’s lower limbs twists his upper body towards the trailing leg ; in a frontal plane it also tilts it down in the direction of this leg ; and in a sagittal plane the reaction rotates him backward, as in a hitch-kick. In all three planes, clockwise motion produces simultaneous counter-clockwise reaction, and vice versa; in each case angular momenta are equal but opposite.

These principles of balance and reaction also apply to certain kinds of movement when an athlete is in contact with the ground. If a man balancing on a beam feels he is falling to one side, he might regain balance by rotating his extended arms in (he some direction, as with the maintenance of lateral balance of the long-jumper in the air , with the difference, however, that when in contact with a beam the movement could, at the same time, displace his Centre of Gravity.

In running, too, the alignment of the body (both on and off the ground) depends to some extent upon such principles.

Gyration and the trading of momentum. We have seen how in many activities the axis of momentum is present in only a theoretical sense, with the athlete turning in the air separately but simultaneously about his body’s longest and shortest axes.

It has been convenient to assume that the axes of momentum and rotation always coincide, but in fact this can be the case only where there has been an angular impulse imparted about either the axis of greatest (i.e. medial) or least (i.e. longitudinal) moment of inertia. Where the athlete develops rotation about his transverse axis or more than one of his principal axes simultaneously , his motion can best be described in terms of a twisting about his longitudinal axis and a gyrating of

Thus the high jumper twists about his longitudinal axis which, at the same time, describes part of a closed conical path about his axis of momentum; and if he maintains the posture shown, this path will not vary and there will be no interchange of momentum between the two motions.

However, through action-reaction in the air an athlete can alter his position relative to his axis of momentum, and so change the direction of his principal axes that he (a) rids himself of the complexities of multi-axial rotation by stabilizing his rotation about only one such axis or, conversely, (b) deliberately brings about gyration.

For example, a gymnast somersaulting forward as he leaves the trampoline (i.e. rotating forward about a horizontal axis of momentum: can originate movement in the air which will change the direction of his principal axes. In consequence, his body absorbs some angular momentum by twisting about its longitudinal axis, simultaneously changing the initial rotation about the transverse axis into the motion illustrated. He ‘trades’ some somersaulting for twisting, although his total angular momentum remains unchanged.

In this case, the initiation of a twisting motion will not alter the rate at which the somersaulting takes place; but, of course, if the twisting is rapid and the cone-angle is therefore much reduced, the motion will not look so much like a somersault. Likewise, by originating movement in the air, our straddle-jumper could bring his longitudinal axis more into line with his axis of momentum and so twist more rapidly away from the bar.

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