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Tuesday, December 14, 2010

Angular Momentum

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The angular momentum possessed by a rotating body is equal to the product of its moment of inertia and its angular velocity:
Angular momentum = la (57)
Thus, if the diver in Fig. 85(a) is rotating at 10 rad/sec about an axis through his center of gravity, his angular momentum is
2.58 x 10 = 25.8 slug-ft2/sec
(Note again the parallel between corresponding quantities in linear and angular motion: momentum = mv angular momentum = la.)
Analogues of Newton's Laws of Motion
Just as most of the quantities of linear kinematics and linear kinetics have equivalent (or analogous) forms in angular motion, so too do Newton's laws of motion. Although perhaps not so widely known as the laws applying to linear motion, these analogous forms are nonetheless of considerable importance in obtaining an understanding of many sports techniques.
The First Law. For present purposes, the angular analogue of Newton's first law can be stated in the following form:
A rotating body will continue to turn about its axis of rotation with constant angular momentum, unless an external couple or eccentric force is exerted upon it.
This statement, perhaps better known as the principle of conservation of angular momentum, means that a body spinning will continue spinning indefinitely (and with the same angular momentum) unless some other body exerts a couple or an eccentric force on it that causes it to modify its angular motion.
This fact is of particular significance to divers, gymnasts, jumpers, and other athletes who become airborne during the course of their events. Consider the example of a diver performing a tucked backward one-and-one-half somersault (Fig. 86). As he leaves the board, his extended body has certain amounts of linear and angular momentum—linear momentum to project him high into the air and to give him time to complete his mid-air actions, and angular momentum to rotate him through the required one-and-one-half somersaults. Shortly after takeoff the diver moves into a tucked position, thereby moving his body mass much closer to the axis of rotation and decreasing his moment of inertia. Because his angular momentum can be altered only by an external couple or eccentric force (principle of conservation of angular momentum), the internal forces involved in this tucking process have no effect upon it—it is the same when he is in the tuck as it was when he left the board in an extended position. A quick glance at the equation for angular momentum, viz.,

Fig. 86. A tucked backward one-and-one-half dive illustrates well the interplay between angular momentum, moment of inertia, and angular velocity.
Angular momentum = Iw
shows that the only way the diver can decrease his moment of inertia and yet still retain the same angular momentum is to increase his angular velocity by a corresponding amount. And this is exactly what happens. At takeoff he has a relatively large moment of inertia and a relatively small angular velocity. When he tucks, he decreases his moment of inertia and increases his angular velocity—and, of course, it is this latter effect that he is trying to produce. He wants to rotate quickly enough to complete his one-and-one-half somersaults in time to prepare himself for a controlled entry. As the diver comes out of his tucked position and extends his body in preparation for entry, this whole process is reversed—his angular velocity decreases as his body extension causes his moment of inertia to be increased.
A trampolinist performing the stunt known as "swivel hips" makes use of the same interdependence of angular velocity and moment of inertia as he brings his body mass close to the axis of rotation during the flight (Fig. 87). The beginner gymnast learning how to do a handspring also makes use of the same principle when he tucks in preparation for landing (Fig. 88). Instinctively he knows that he has insufficient angular momentum to enable him to complete the stunt in the approved manner, so he tucks and increases his angular velocity so that he can at least, avoid crashing on his back.

Fig. 88. Good form in the handspring, contrasted with the poor form used by a beginner to "salvage" the stunt

Fig. 87. Swivel hips, another example of the conservation-of-angular-momentum principle

The Second Law. The angular analogue of Newton's second law can be stated as
The rate of change of angular momentum of a body is proportional to the torque causing it and has the same direction as the torque.
Or it can be expressed algebraically in the form
RUMUS
where T = the applied torque, I1 and I2 = the initial and final moments of inertia, and w1 and w2; = the initial and final angular velocities. If I1 = I2, this relationship reduces to
RUMUS
where a = the angular acceleration. With the methods explained on pp. 61-62, this equation can be converted to the form
T=Ia (58)
The diver in Fig. 56 provides a useful example of the application of Eq. 58. At the instant depicted, his body is being angularly accelerated about ac axis through his feet. The torque causing this angular acceleration is equal to the product of the diver's weight and the horizontal distance x. If the diver is executing a dive requiring a large amount of rotation (say, a forward two-and-a-half), he will want this torque to be relatively large so that his body can leave the board with sufficient angular velocity to enable him to complete the dive. On the other hand, if he is executing a dive that involves very little rotation (say, a plain forward dive), he will have little need for angular acceleration during the takeoff and will want to keep the applied torque relatively small. Because his weight is constant, and the only other factor that influences the magnitude of the torque is the distance x, the only way in which he can control this torque is by making alterations in the magnitude of x. And this is exactly what he does. By varying the amount of forward body lean that he has at takeoff, he controls the magnitude of x and hence too the magnitude of the torque applied at this time.
The Third Law. The angular analogue of Newton's third law can be stated as
For every torque that is exerted by one body on another there is an equal and opposite torque exerted by the second on the first.
Sports offer countless examples to illustrate this angular equivalent of Newton's third law. Probably the most common is that in which an athlete applies a torque to one part of his body by contracting a muscle (or group of muscles), thereby causing that part to rotate. The equal and opposite reaction to this applied torque causes some other part of his body to rotate, or tend to rotate, in the opposite direction.
When a long jumper swings his legs forward ready for landing, a torque equal and opposite to that exerted on hi; legs is applied to the remainder of his body. The net effect is that as the jumper swings his legs forward and upward, in say a clockwise direction, the remainder of his body moves forward and downward in a counterclockwise direction (Fig. 89).
It should be noted here that since angular accelerations obtained when equal torques are applied to two different bodies depend on their respective moments of inertia (Eq. 58), the effect on one body is rarely equal to that produced on the other—of. the corresponding linear case, pp. 65-66. This is evident in a basketball jump shot, for example. When a player propels the ball toward the basket by extending his elbow and flexing his wrist, the remainder of his body is acted upon by torques equal in magnitude and opposite in direction to those causing these movements. Because the moment of inertia of the rest of the body is much greater than that of either the forearm or hand, the effects of these equal torques appear to be quite different. Whereas the forearm and hand sweep quickly through a relatively large angle, the remainder of the body exhibits only a slight tendency to rotate in the opposite direction.

Fig. 89. Angular action and reaction in a long-jump landing
The angular "action-reaction" effects are by no means limited to a forward-backward (or sagittal) plane. In an across-the-body (or frontal) plane these effects are very clearly seen in the instinctive responses of a gymnast who senses she is about to topple off a balance beam (Fig. 90). As soon as she feels she is starting to overbalance, she rotates her arms (and perhaps her non supporting leg) in the direction in which she is falling. The effect of these actions is to cause the rest of the body to rotate in the opposite direction. If this maneuver is successfully carried out, her tendency to overbalance is arrested.
The tennis player in Fig. 91 exerts torques to produce the clockwise backhand stroke depicted. These torques are accompanied by equal and opposite torques, which tend to turn the rest of his body in a counterclockwise direction. However, because his feet are in firm contact with the ground, this contrary tendency is transmitted to the ground. Now, the moment of inertia of the tennis player about the axis in question is relatively small and the angular acceleration he experiences is thus clearly apparent—one can readily see how he is affected by the torques produced by his muscles. In contrast, the moment of inertia of the earth is enormous, and the angular acceleration it experiences as a result of the torques transmitted via the player's feet is quite imperceptible. Players are often exhorted, therefore, to keep both feet in contact with the ground so that the reactions that accompany the strokes they make can be "absorbed" in this way.

Fig. 90. The gymnast's instinctive actions "create an angular reaction tending to restore her balance.

Fig. 91. The reaction to the tennis player's backhand drive is "absorbed" by the ground.

Fig. 92. The angular momentum of a-diver performing a piked front dive is first localized in his upper body and then in his legs.

Transfer of Momentum
When a body is airborne and the angular momentum of one part of the body is decreased, some part (or all) of the rest of the body must experience an increase in angular momentum if the total angular momentum is to be conserved (or held constant). For example, consider the case of a diver performing a piked front dive (Fig. 92). As he goes into his piked position, the angular momentum of his legs is reduced to zero or near-zero—they appear to be stationary in the air—and the angular momentum of his arms and trunk is markedly increased. Then, as he assumes his position for entry, this process is reversed. The angular momentum of the arms and trunk is reduced to zero, or near-zero—and they appear to remain stationary—while the legs swing upward and into line with the rest of the body. This process whereby momentum is redistributed within a body is commonly referred to as a transfer of momentum.

Fig. 93. With appropriate actions to rotate him sideways, a diver can "fade" somersaulting angular momentum for twisting angular momentum.
A similar process occurs in diving, gymnastics, and a few other activities where performers execute airborne movements involving both somersaulting and twisting. Such movements generally involve a transfer (or "trading") of part of the somersaulting angular momentum, initiated during the takeoff, for the twisting angular momentum needed during the flight.
Consider the case of a diver who leaves the board to execute a front dive [Fig. 93(a)j and then, while airborne, vigorously abducts one arm to bring it down alongside his body. The reaction to this motion is a contrary rotation of the rest of his body about his frontal axis—a rotation that tilts his body in the opposite direction [Fig. 93(b)]. Although virtually impossible in practice, suppose that the diver abducts his arm with such vigor that in reaction his body is brought into a horizontal position [Fig. 93(c)].
Now when the diver left the board he possessed a certain amount of angular momentum about a horizontal axis perpendicular to the line of the board (an axis sometimes referred to as the axis of momentum) and, because the transverse axis of his body coincided with this horizontal axis, he experienced a somersaulting rotation. When his body was subsequently rotated into a horizontal position, his longitudinal axis was brought into line with the axis of momentum. Thus, in keeping with the conservation-of-angular-momentum principle, his body acquired the same amount of angular momentum about its longitudinal axis as it had previously possessed about its transverse axis. In other words, by readjusting the position of his body relative to the axis of momentum, the diver was able to "trade" all his somersaulting momentum for twisting momentum.
In practice, the actions used to make the body tilt—generally both arms moved vigorously in the same angular direction—are insufficient to produce more than a very limited amount of side-somersaulting rotation. As a result, only part of the diver's somersaulting angular momentum is "traded" for twisting angular momentum, and the dive is executed with both somersaulting and twisting proceeding simultaneously. Finally, because the trading of somersaulting for twisting angular momentum requires the diver to move his body out of alignment with the vertical plane of his flight path, his body is not correctly positioned for entry once he has completed the prescribed number of twists and somersaults. To correct for this the good diver reverses the direction of his earlier arm action, thereby imparting sufficient side-somersaulting rotation to his body to bring it back into its original alignment for entry.
The concept of transferring momentum is most frequently used in explaining what takes place in situations other than those in which a body is airborne. For example, when a diver performs a backward dive, he swings his arms upward and backward prior to leaving the board. Then, as they near the limit of their range of motion in this direction, they begin to slow down. The angular momentum that the arms lose at this time (or at least a large portion of it) is "matched" by a corresponding increase in the angular momentum of the rest of the body. New, although the angular momentum of a body is not necessarily conserved in situations like this (since the body is subject to external torques that tend to alter its angular momentum), the effect is qualitatively very similar to cases in which the angular momentum is conserved—one part of the body loses (or "gives up") angular momentum at the same time as another part experiences a gain in angular momentum.
Initiating Rotation in the Air
While the angular analogues of Newton's laws indicate quite clearly that a body cannot acquire angular momentum unless acted upon by an external torque, the possibility of initiating rotation in the air has been of considerable interest for some time. Sparking much of this interest has been the performance of cats, rabbits, guinea pigs, and other animals, which have the ability to right themselves when falling upside down. While initially some doubts were expressed that these animals initiated the turns in the air rather than at the time of takeoff, it is now widely accepted that this is the case.
How are these turns initiated in apparent defiance of Newton's laws? The answer most frequently put forward lies in the relationship between the moments of inertia of the body parts that interact when an angular action is initiated in the air. Consider the cat being dropped in Fig. 94. As he begins to fall, he bends (or pikes) in the middle [Fig. 94(a)], brings his front legs in close to his head, and rotates his upper body through 180° [Fig. 94(b)]. In reaction to this rotation, his lower trunk, hind legs, and tail, all of which are some considerable distance from the axis of rotation, rotate in the opposite direction.

Fig. 94. A falling cat initiates rotation in the air in the absence of an external torque and in apparent defiance of the law of conservation of angular momentum. (The /inures from which these diagrams were redrawn first appeared in New Scientist, the weekly international review of science and technology, 128 Long Acre. London WC2. and appear with the publisher's permission.)
However, because the moment of inertia of these body parts is much greater than that of his upper body, the angular distance through which they move is correspondingly small ("about 5°" according to McDonald 26). To complete the required 180° turn, the cat then brings his hind legs and tail into line with his lower trunk and rotates these body parts about an axis running longitudinally through his hindquarters [Fig. 94(c)]. The reaction is again very small, this time due to the disposition of the upper body relative to the axis. Finally, to make any minor adjustments necessary, the cat rotates his tail in a direction opposite to that in which it is desired to move the body. (Since Manx cats and cats completely without tails can right themselves if held upside down and dropped, these final movements of the tail are clearly not essential ingredients of the righting maneuver.)
It should be carefully noted that throughout this whole sequence of movements the angular momentum of one body part has always been "matched" with an equal and opposite angular momentum of some other part and that, as a consequence, the total angular momentum has been quite unaffected. In short, the cat has not defied Newton's laws but has merely appeared to do so. (Note: Research by Smith and Kane 27 and Kane and Scher 28 has suggested some alternative explanations of the process by which the cat rights itself when dropped in an inverted position.)

_____________
26 Donald McDonald, "How Does a Cat Fall on Its Feet? "The New Scientist, VII, June 30, 1960, p. 1647.
With all the interest in this question of how a falling cat rights itself, some interesting side results have been reported 29
1. If dropped upside down, a cat can turn over within its own standing height.
2. The cat's eyes and the mechanisms of its inner ear both play a part in sensing the need to initiate a turn. Of these, the eyes seem to be the more important—a blindfolded cat dropped from as low as 3 ft lands clumsily, while a cat without an intact inner ear mechanism can still right itself efficiently. However, a cat deprived of both sensory organs made no attempt to right itself when dropped upside down.
3. A blindfolded cat, rotated in a special apparatus in order to "confuse" the organs of its inner ear, was reported to have rotated over and landed on its back when it accidentally slipped feet first out of the apparatus!
The possibility of athletes' using similar techniques has been the source of some interest, and various writers have discussed the matter relative to diving, 30,31,32,33 trampolining, 34 and track and field. 35,36. The question has also interested those engaged in research related to man's ability to maneuver in a weightless state. 37,38,39
27. Preston G. Smith and Thomas R. Kane, The Reorientation of a Human Being in Free Fall, Technical Report No. 171, Division of Engineering Mechanics, Stanford University, May 1967.
28. R. Kane and M. P. Scher, "A Dynamical Explanation of the Falling Cat Phenomenon," International Journal of Solids and Structures, V, July 1969.
29. Donald McDonald, "How Does a Cat Fall on its Feet?" pp. 1647-48.
30. Geoffrey H. G. Dyson, The Mechanics of Athletics (London: University of London Press Ltd., 1973), pp. 105-9.
31. George Eaves, Diving, The Mechanics of Springboard and Firmboard Techniques (London: Kaye & Ward, Ltd., 1969).
32. F.R. Lanoue, "Mechanics of Fancy Diving" (M. Ed. thesis, Springfield College, 1936).
33. Donald McDonald, "How Does a Man Twist in the Air?" New Scientist, X, June 1, 1961, pp.501-3.
34. Dennis E. Home, Trampolining: A Complete Handbook (London: Faber & Faber, Ltd., 1968), p. 111.
35. H. A. L. Chapman, "Rotation—Its Problems and Effects" in International Track and Field Digest, ed. by Don Canham and Phil Diamond (Ann Arbor, Mich., "Champions on Film," 1957), p. 243.
36. Dyson, The Mechanics of Athletics, pp. 111-13.
37. R. Kane and M. P. Scher, "Human Self-Rotation by Means of Limb Movements," Journal of Biomechanics, III, January 1970, pp. 39-49.
38. Kulwicki, Schlei, and Vergamini, Weightless Man: Self-Rotation Techniques.
39. Smith and Kane, The Reorientation of a Human Being in Free Fall.

From all this research and speculation several conclusions have been reached:
1. A man can initiate rotation while he is in the air. A simple demonstration of this can be given by a trained gymnast who does a series of consecutive, pike-to-front-drop movements on a trampoline and executes a half twist to land in a flat back position (instead of in a front drop) when called upon to do so. A shouted command to the gymnast shortly after he leaves the bed—and at which time he has zero angular momentum because that is what is required to perform a series of consecutive front drops—enables him to convincingly demonstrate the point (Fig. 95).
2. A man's ability to initiate rotation in the air is a function of how much training or practice he has had—in a plain jump from a 1-m board, a trained diver can initiate a twist in the air and turn through as much as 450°, while an untrained man can rarely exceed 90°.
Fig. 95. A trampolinist can initiate a rotation in the air, as in this front-drop pike half-twist flat-back sequence.
3. The basic mechanism involved appears to be similar to that used by the cat, although there are variations from movement to movement and from one individual to another in performing the same movement.
4. Starting positions in which the body is arched or piked facilitate the initiating of rotation. In addition, it appears that while a man can, with relative ease, initiate a twisting rotation while in the air, initiating rotations about either of the other two principal axes does not appear to be quite so easy.
The extent to which divers, trampolinists, and others involved in "aerial activities" actually use such techniques is anything but easy to determine. It is clear though that a considerable number of the movements in such activities are executed in a manner consistent with the initiation of catlike rotations in the air.

Centripetal and Centrifugal Force
When a tennis player executes a forehand drive, muscles of his trunk exert forces on his arm to cause it and the racket (which serves as an extension of the arm) to swing through an arc. Suppose that the vector F (Fig. 96) represents the forces exerted by the trunk on the arm and R and T represent the components of F along and at right angles to the arm. If the axis of rotation is a vertical one through the shoulder join', T causes .an angular acceleration about this axis that increases the tangential velocity of the racket. R, on the other hand, is the force responsible for the radial acceleration (see p. 54) that changes the direction in which the racket is moving. Because this latter force acts toward the center of rotation, the axis, it is known as the centripetal (or "center-seeking") force.
Fig. 96. Centripetal and centrifugal forces in a forehand shot in tennis

The magnitude of this force is obtained by combining the equations for Newton's second law (F = ma) and for radial acceleration (a -= v}/r). Thus:
RUMUS (59)
Because Vr = w (Eq. 26), it is sometimes useful to substitute w2r2 for v) in Eq. 59. The right-hand side of this equation is then expressed in terms of the angular rather than the linear velocity:
Centripetal force = mrco1 (60)
Suppose now that the racket has a mass of 0.026 slug and that its center of gravity is 2.92 ft from the axis and is moving with a speed of 35 fps. Then, according to Eq. 59, the centripetal force applied to the racket must be

RUMUS


Further, since the man's hand is the only other body in contact with the racket, this 10.91-lb force must be applied to the racket by his hand. Now whenever one body exerts a force on another "there is an equal and opposite force exerted by the second body on the first" (Newton's third law). Thus, if the hand exerts a force of 10.91 Ib on the racket, the racket in turn exerts a force of 10.91 Ib in the opposite direction on the hand. This force, which always acts away from the center of rotation, is termed the centrifugal (or "center-fleeing") force.
The concept of a centrifugal force is frequently the source of confusion. The two principal reasons appear to be
1. A failure to recognize that centrifugal and centripetal forces do not both act on the same body. Apart from the fact that Newton's third law indicates that the "action" acts on one body and the "reaction" on another, it should be obvious that the resultant of two equal and opposite forces acting on a body is zero and that application of a zero force would not change the direction in which a body is moving.
2. A failure to realize that "actions" and "reactions" occur simultaneously. Thus, if a tennis player released his grip on his racket partway through a stroke, both the centripetal force exerted by his hand on the racked and the centrifugal force exerted by the racket on his hand would cease to exist at the same time. Under such circumstances, and in accord with Newton's first law, the racket would tend to continue traveling in the same direction as it was at the moment of release (i.e., tangent to the point on the arc at which it was located at that instant).
[Note: Contrary to what is often supposed in such cases, the racket does not have a tendency to travel radially outward under the action of a centrifugal force—(a) because the centrifugal force acts on the player's hand and not on the racket and (b) because, even if it did act on the racket, it would cease to exist at the moment the corresponding centripetal force was removed.]
Centripetal and centrifugal forces are exerted whenever a body moves on a curved path. In sports, however, there are times when these forces seem more important than others because athletes must make conscious adjustments in technique to allow for their existence. Probably the most striking examples are seen when track sprinters and cyclists negotiate a bend in the track. Here the only body that can exert the required centripetal force on them is the ground, the only body with which they are in contact. If the ground exerts an inward horizontal force against the foot of the runner (or against the wheels of the bike), this eccentric force will have the required effect of changing the direction of his motion. However, it will also have the undesirable effect of rotating him outward. (Remember, an eccentric force causes both translation and rotation—see p. 105). To combat this rotary effect the athlete leans inward so that the vertical component of the ground reaction will also act eccentrically and provide a moment in the opposite direction to that produced by the eccentric centripetal force. When the speed of athletes exceeds a certain limit (as it often does in cycling and motor sports) or the radius of the track is very small (as at many indoor track meets), the ground is no longer able to provide the necessary amount of centripetal force. This means that, unless some additional provision is made, the athletes will have to slow down going into the bends or risk failing to safely negotiate them. It is to avoid these problems that banked tracks are built for cycling velodromes and indoor track meets. In this way the component of the athlete's weight acting down the slope can contribute to the centripetal force required, and the need for an inward directed force exerted by the ground is at least reduced, and perhaps eliminated entirely.