Angular Motion in Sports

Most of the movement in track and field athletics is of an angular character because the human body, mechanically speaking, comprises a system of levers capable only of rotational motion. And what at first sight often appears to be linear movement is, on analysis, found to be the end product of a series of complex rotations of many parts of the body. It is therefore important, in analysing athletic technique, to be familiar with the principles governing rotational motion.

Moments

It is easier to hold a javelin in a horizontal position at its point of balance (i.e. at its Centre of Gravity) than farther down the shaft, and most difficult of all (if not impossible!) to keep it in this position while grasping the tail end.

And yet the javelin weighs no more, and the person holding it does not get progressively weaker! The increasing difficulty is due to the tendency for the javelin to rotate—a tendency which grows with an increase in horizontal distance between the force of its weight, acting through the Centre of Gravity, and the hand.

The following are dynamic examples of the same phenomenon; wheels, revolving doors and propellers are easier to turn when force is applied as far away as possible from their axes—the hub, hinge or shaft. A heavy door is progressively easier to close as one pushes farther and farther away from its hinges. (i) To turn an object, force must be exerted at a distance to its axis, and the greater the distance, the greater will be the rotational or spinning effect. But it is important to note that the distance from the line of action of the force to the axis—the lever-arm, as it is called —must be measured along a perpendicular, i.e. at right angles to the direction of the force. (ii) A larger force will produce a greater turning effect.

The product of the force and the lever-arm is called the moment of the force, or torque, and is used as a measure of this turning effect; and the product of torque and the time for which it acts is known as the impulse.

F

Horizontal bar exercises. The turning effect of the force of gravity on the swinging gymnast is greater when his body is horizontal, and diminishes progressively as he assumes a more vertical position. Obviously, a change in body position during the swing also increases or decreases the lever arm and influences swinging speed.

Moments in equilibrium. In both horizontal positions the javelin is in equilibrium, because, first, the resultant of all the forces acting on it is zero and, second, clockwise and counter-clockwise moments cancel each other out. It will help to study this in greater detail.

The hand merely exerts an upward force of 1 lb 12 oz— equal to the javelin’s weight. But in the other position this force by itself will not produce equilibrium, even though the sum of the forces is zero. We must refer to the second condition for equilibrium and take the moments into account.

In this position three forces act on the javelin: (1) the forefinger, acting as the axis or fulcrum, exerts an upward force; (2) the javelin’s weight, which has a clockwise moment about this finger; and (3) the downward force exerted by the thumb, which has a counter-clockwise moment. These two moments cancel out each other.

As the sum of all the forces must be zero, the two downward forces (1 lb 12 oz and 21 lb 3 oz) must equal the forefinger’s upward pressure of 22 lb 15£ oz. So the hand exerts two forces; an upward one of 22 lb 15£ oz and a downward one of 21 lb 3 oz. It is hardly surprising that the javelin is more difficult to hold in this position!

From the first position in which the javelin was held it can be seen that an object’s or athlete’s Centre of Gravity is actually a point where the combined moments of the many separate gravitational pulls on all the particles of mass on one side, acting clockwise, equal the combined counter-clockwise moments on the other.

Whenever, at rest, a plumb-line can be dropped from the Centre of

Gravity to fall within the base the various turning effects automatically cancel out—as with the bricks and the balanced athletes.

Balance in acceleration can be maintained by altering the position of the body, thus shifting the Centre of Gravity so as to lengthen or shorten the lever-arms of the component forces applied. For example, the stick remains balanced during linear acceleration when the product of upward force-component and its horizontal distance to the Centre of Gravity is equal to that of the forward component and its distance vertically to the Centre of Gravity.

When the stick leans in the direction of the acceleration its Centre of Gravity is lowered, reducing the lever-arm of the horizontal component of drive and the backward-rotating effect it encourages. Simultaneously, the lever-arm of the vertical component is increased, as is the tendency to rotate the stick forward.

It follows that a sprinter who leans forward too much for the effective force of his leg drive totters, unbalanced, from his marks. Or if he is too upright his legs will seem to move ahead of him and lose their driving effect; this can also happen when a distance runner, surprised by an unexpected challenge, responds without making the necessary adjustment to his body lean. [n efficient running the ratio of forward to upward component of the reaction to leg drive influences the angle of the trunk. At the start, where it is easy to push forcefully backward against the ground, the sprinter needs a pronounced forward lean, but as his speed increases so will the effective force of his drive be reduced, necessitating a progressive modification towards a more erect position. Again, a bounding type of runner, using a comparatively greater vertical drive component may need a more upright carriage to shorten the lever-arm of the force and so reduce the tendency to rotate forward.

It is quite possible for athletes who enjoy exceptional hip mobility and who possess certain postural characteristics to make the necessary adjustments to the position of the Centre of Gravity without employing the more orthodox forward trunk lean—hence their comparatively upright running positions even during acceleration. The great Jesse Owens and E. McDonald Bailey spring readily to mind as good examples of this.

Problems of balance in running and jumping during those phases in which athletes are in contact with the ground are best considered, as in our previous examples, by taking moments about the Centre of Gravity, not the point of support. For, in dealing with the body as a whole, movement on the ground in these events is the combination of a series of linear accelerations whereby clockwise and counter- clockwise turning moments about the point of support cannot be equal.

But if an athlete is balanced in motion, in the sense that there is no turning of the whole body mass, moments about his Centre of Gravity may be considered equal. As a general principle it is usually more convenient to consider an axis passing through the Centre of Gravity in all events where, subsequently, the athlete leaves the ground.

However, in analysing balance in pole vaulting and throwing positions (where the athlete has contact with the ground) it is sometimes convenient to take moments about the base or support, thinking in terms of turning couples. An example follows.

A hammer thrower’s equilibrium in a sagittal plane is the product of two pairs of parallel, equal, forces acting in opposite directions. The hammer’s centrifugal pull and the equal force of the ground thrusting at an angle against the thrower’s feet form one couple tending to rotate him counter-clockwise; the distance at right angles to the line of these parallel forces is the arm of the couple.

In a clockwise direction, however, the force of the thrower’s weight, acting downward through his Centre of Gravity, plus the equal but upward force of the ground under his feet, form a second couple, the arm of which is the horizontal distance between the lines of these two vertically directed forces.

The turning moment of a couple is found by multiplying one of its forces (it should be noted that they are always equal in size) by its arm, and the thrower is balanced when the moments of the two couples are equal.

But as, in spinning, the athlete develops greater velocity in the hammer-head, he automatically increases the centrifugal force and the ground’s horizontal thrust against his feet, both of which tend to rotate him counter-clockwise in a sagittal plane. So he maintains balance by bending his legs and lowering his seat, thereby reducing the arm of this counter-clockwise couple , and increasing the arm of the clockwise one.

Moments in rotation and spin. Just as, with linear acceleration, moments about the support cannot be equal, so it is with angular acceleration, whether the axis is considered to be a point of contact with the ground or the Centre of Gravity.

In analysing a diver’s movements we take moments about his Centre of Gravity. When the reaction to his leg thrust (along a straight line drawn between foot and hip) passes directly through his Centre of Gravity, pure translatory motion occurs, i.e. he leaves the board without rotation. But as, in successive dives he exerts a force at increasing distances to his Centre of Gravity, the effect is as though more is used progressively in imparting a rotation, while less is available to project his body.

In fact, a given reaction (i.e. vertical impulse) to leg-thrust will impart the same vertical speed, and therefore the same height, to a diver’s body, whether directed through his Centre of Gravity or not. But here the difficulty is to exert the same impulse (i.e. force x time) as previously against a rotating body.

To achieve the same height the diver would have to develop a greater thrust from the board; greater still for the rotation. . In both cases the leg-thrust would need to be faster, reducing the time of operation and, consequently, requiring greater thrust.

Eccentric thrusts, are wasteful of energy. Assuming, for the moment, that the diver projects his

Centre of Gravity to the same height in all three dives; the point of application of the thrust moves through the same distance and acquires the same velocity as his Centre of Gravity; but when the same take-off impulse is directed eccentrically this point moves through a greater distance than his Centre of Gravity.

Thus, an impulse exerted through the Centre of Gravity changes a body’s linear kinetic energy while an equal impulse exerted ‘off centre’ changes the body’s linear kinetic energy by the same amount, but also imparts rotational kinetic energy. Hence, unless rotation is required, eccentric thrust is wasteful of energy.

The diver’s rotation off the board is brought about, in fact, by a turning couple; the reaction to his leg thrust, i.e. the force of the springboard, acts vertically upward and the force of his weight, acting through his Centre of Gravity, pulls vertically downward. He obtains this eccentric or sideways thrust about the Centre of Gravity by bringing his head and arms (i.e. just under a fifth of his total body mass) forward beforehand and, in bad diving, by bending forward markedly from his hips.

We have seen that the forces of a couple must be equal and yet, in diving from the springboard,the force of the upward vertical thrust actually exceeds the downward pull on the man’s body. The forces turning him are indeed equal, but it is the excess of force from the board which projects him into the air. If his Centre of Gravity moved downward, i.e. he fell off the board, then the reverse would apply: the force of his body weight would then be greater than that which passed through his feet.

As Orner has written, ‘If a competitor performs a well-executed dive, it will be good because the direction, height and the force of rotation imparted to his body through the diving-board are a close approximation to his exact needs for the correct number of somersaults in the particular position intended and his control in the air will be quite marginal. ‘The picture of a diver jumping from the board steeply without rotation then circling deliberately for a spin, finally halting his rotation completely to dart vertically into the water is pure illusion—and the diver’s art! The illusion is most effectively shattered when a diver errs on take-off by taking too much turning force, and is unable to maintain the impression of ‘stopped’ rotation near the water. ‘Unfortunately, many divers and coaches believe in the possibility of controlling dives fully in the air and are taken in by the illusion. The inevitable result is that they concern themselves mainly with the dive; that is the effects, rather than with the take-off in which errors showing in the dive originate in a proportion of at least 90 per cent.’

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