Chapter 12. The Conditions of Linear Motion

At the conclusion of this chapter, the student should be able to:

• 1. Name, define, and use the following terms properly as they apply to linear motion: force, inertia, mass, weight, momentum, and impulse.
• 2. Explain what is meant by the terms magnitude, direction, and point of application of force, and use these terms properly as they apply to internal and external forces.
• 3. Explain the effect of specified changes in the magnitude, direction, and point of application of force on the motion state of a body.
• 4. Define and give examples of the terms linear forces, concurrent forces, and parallel forces.
• 5. Determine the magnitude, direction, and point of application of muscle forces in hypothetical situations in which specific muscles are considered in isolation.
• 6. State Newton’s laws as they apply to linear motion.
• 7. Explain the cause and effect relationship between the forces responsible for linear motion and the objects experiencing the motion.
• 8. Name and define the basic external forces responsible for modifying motion: weight, normal reaction, friction, elasticity, buoyancy, drag, and lift.
• 9. Draw and analyze simple two-dimensional free-body diagrams in which all applicable external forces are properly accounted for.
• 10. Explain the work–energy relationship as it applies to a body experiencing linear motion.
• 11. Define and use properly the terms work, power, kinetic energy, and potential energy.
• 12. Using the concepts that govern linear motion, perform a mechanical analysis of a selected motor skill.

Objects start moving when they are pushed or pulled, that is, when some type of force acts on them. Forces produce motion, stop motion, and prevent motion. They may increase speed, decrease speed, or cause objects to change direction. They may push or pull to cause motion or produce a net effect so that bodies remain stationary. Force is defined as that which pushes or pulls through direct mechanical contact or through the force of gravity to alter the motion of an object. It is the effect that one body has on another. The identification of forces that act to produce motion of the body or of an object is an important element of a kinesiological analysis.

The action of a force may be internal or external. Internal forces are defined as forces exerted by bodies on other bodies within a defined system, whereas external forces are those exerted by bodies within an arbitrarily specified system on bodies outside the specified system. Internal forces cause differences in body shape, and external forces cause displacement of the body. In kinesiology, the system under consideration is the human body. Internal forces are therefore usually classified as muscle forces that act on the various structures of the body, and external forces are those outside the body. The best-known external force is weight, or gravitational force. Wind or water resistance forces, friction, or forces due to other objects acting on the body are also external forces.

Aspects of Force

Force is a vector quantity. Force possesses both magnitude and direction, as do all vector quantities. Force also has an exact point of application on the object. To describe a force fully, all three of these characteristics must be identified and taken into account. A change in any one of them alters the nature of the motion. Applying a force of insufficient magnitude, for instance, might produce no motion. A weight lifter trying to lift a 250 N barbell with 150 N of force would be unsuccessful. If the force applied by the weight lifter were increased to 260 N, the barbell would move if the force were applied in the proper direction and at the proper point of application. If the direction of the force is downward, the barbell will not be lifted upward. If the force is applied to only one end of the barbell, the bar will rotate rather than move upward in a linear manner. To successfully lift the 250 N barbell, the lifter must apply a force greater than 250 N in an upward direction, through the center of gravity of the barbell.

Magnitude, direction, and point of application as they relate to both external and internal forces are explained in the following sections.

Magnitude

The amount or size of the force that is being applied makes up the magnitude of the force vector. The basic unit of force in the U.S. system is the pound, and in the metric system it is the newton. In the weight-lifting example just used, the magnitude of the muscle force used in the first unsuccessful attempt was 150 N. The force being exerted by the barbell had a magnitude of 250 N. This 250 N force was the result of the pull of gravity acting on the mass of the barbell. This force is commonly referred to as weight. Weight can be expressed as mass times the acceleration due to gravity, or

When one holds a ball in the hand, the pull of gravity is felt as the weight of the ball. The ball stays in the hand as long as an equal and opposite force acting between the hand and the ball balances the downward gravitational force. In this example the equal and opposite force is muscular. When the opposing force is removed, the ball drops and gravity’s pull is apparent in the downward motion of the ball. The weight of the ball is the magnitude of the force of gravity acting on the ball.

The magnitude of muscular force is in direct proportion to the number and size of the fibers in the muscle that is contracting. If muscles contracted individually, it would be a relatively easy matter to measure the force exerted by each one in a given movement. Because they normally act in groups, however, their force or strength is measured collectively. It is customary to measure maximum muscular strength by performing a simple movement against the resistance of a dynamometer or similar instrument. The instrument thus serves as the resistance to an anatomical lever whose force is provided by a group of muscles that act as a functional team to produce the movement of the lever. Among the muscle groups that are frequently measured by this method are the finger flexors (grip strength), elbow flexors, and knee extensors.

The magnitudes of external forces other than gravity depend on other factors. Most of these external forces will be discussed in more detail later in this text.

Point of Application

The point of application of a force is that point at which the force is applied to an object. Where gravity is concerned, this point is always through the center of gravity of an object. For practical purposes it may be assumed that the point of application of muscular force is the center of the muscle’s attachment to the bony lever. This usually corresponds to the muscle’s insertion or distal attachment. Technically, however, it is the point of intersection between the line of force and the mechanical axis of the bone or segment serving as the anatomical lever. This axis does not necessarily pass lengthwise through the shaft of the bony lever. If the bone bends or if the articulating process projects at an angle from the shaft, the greater part of the axis may lie completely outside the shaft, as in the case of the femur (see Figure 2.3). The mechanical axis of a bone or segment is a straight line that connects the midpoint of the joint at one end with the midpoint of the joint at the other end or, in the case of a terminal segment, with its distal end.

Direction

The direction of a force is along its action line. Because the force of gravity pulls all objects toward the earth’s center, the direction of gravitational forces is vertically downward. The force of gravity acting on an object would be represented as a downward-directed vector starting at the center of gravity of the object.

The direction of a muscular forcevector is in the direction of the muscle line of pull. This direction is identified by the muscle angle of pull, which is defined as the angle between the line of pull and that portion of the mechanical axis of bone that lies between the point of application of the muscle force and the joint, which acts as the fulcrum (Figure 12.1).

Resolution of Forces

Because force has the qualities of both magnitude and direction, it is a vector quantity. Graphically, the magnitude of the force is represented by the length of the vector line. The point of application of the force is the point where the force vector starts. The vector line represents the line of force, and the arrowhead indicates the direction of the force application (Figure 12.2).

Angle of Pull

As with any vector, a force may be resolved into a vertical and a horizontal component, and the relative size of these component vectors depends on the angle at which the force is applied. Where muscles are involved, the size of a muscle’s angle of pull changes with every degree of joint motion, and consequently so do the sizes of the horizontal and vertical components. These changes have a direct bearing on the effectiveness of the muscle’s pulling force in moving the bony lever. The larger the angle between 0 degrees and 90 degrees, the greater the vertical component and the less the horizontal component (see Figure 12.1).

The vertical component of muscle pull is always perpendicular to the lever and is called the rotary component. It is that part of the force that moves the lever. The horizontal component is parallel to the lever and is the nonrotary component. It does not contribute to the lever’s movement. The angle of pull of most muscles in the resting position is less than a right angle, and it usually remains so throughout the movement (see Figure 12.1a). This means that the nonrotary component of force is directed toward the fulcrum, which gives it a stabilizing effect. By pulling the bone lengthwise toward the joint, it helps maintain the integrity of the joint. Under most circumstances, therefore, muscular force has two simultaneous functions—movement and stabilization. In the latter capacity it supplements the ligaments, an excellent example of the body’s efficiency because muscles perform this stabilizing function only during the period when the segment is moving, at which time the integrity of the joint may be threatened.

Occasionally the angle of pull becomes greater than a right angle, which means that the non-rotary component of force is directed away from the fulcrum and is therefore a dislocating component. This does not happen in many instances, however, when it does, the muscle is close to the limit of its shortening range and is therefore not exerting much force (see Figure12.1c).

When the angle of pull is 90 degrees, the force is completely rotary. When it is 45 degrees the rotary and stabilizing (nonrotary) components are equal. Because the angle of pull usually remains less than 45 degrees, more of the muscle’s force serves to stabilize the joint than to move the lever. In fact, there are some muscles whose angles of pull are always so small that their contributions to motion would seem to be negligible. This appears to be true of the coracobrachialis and the subclavius muscles. It is interesting to note that the upper extremity is frequently called upon to perform violent, powerful movements, as well as to support the body weight in suspension. The joints that bear the brunt of this violence and strain are the shoulder and sternoclavicular joints. They might well become dislocated more easily than they do, were it not for the coracobrachialis and the subclavius muscles, which pull the bones lengthwise toward their proximal joints and thus serve to stabilize these joints.

The effect the angle of pull has upon the rotary force of a muscle for a given angle is demonstrated in the following problem. The solution clearly shows an increase in rotary force with each increase in angle size.

Anatomical Pulley

The small angle of pull that most muscles have when in their normal resting position has already been noted. Were it not for anatomical devices that serve as fixed single pulleys, some muscles would probably be unable to effect any movement whatsoever. The anatomical pulley serves the same purpose as the mechanical fixed single pulley, that of changing the direction of force by changing the angle of pull of the muscle providing the force.

An anatomical pulley can alter the way a muscle acts on a joint in a number of ways. First, a pulley may act to improve the muscle angle of pull and increase the rotary component, such as the way in which the patella moves the quadriceps tendon away from the knee joint (Figure 12.4). Another example of this might be the condyles of the femur and tibia, effectively pushing the gracilis away from the joint (Figure 12.5). Second, a pulley may change the direction of the muscle action. A good example would be the peroneus longus, passing behind the lateral malleolus before it turns under the foot (Figure 12.6). If the peroneus longus did not do this, it would produce dorsiflexion. Another configuration with a pulley action is the pronator teres, which rotates the radius over the ulna. Tendons and ligaments may also act as pulleys. A good example of this is the pulley action of the finger and thumb flexor sheaths.

Resolution of External Force

The resolution of a single external forcevector into component force vectors acting at right angles to each other is accomplished in the same manner as that explained for muscular forces and would be used when the force is applied at an oblique angle. If, instead of giving a piece of furniture a horizontal push, one pushes it by standing close to it with the arms held at a forward-downward slant, only part of the force will push the table forward (Figure 12.7). The force is applied to the table at an oblique angle and thus has both a vertical and a horizontal component. Whether or not the table moves forward depends on the amount of the total force applied in the horizontal direction. If sufficient, only the horizontal component of force will overcome the table’s resistance to move. The downward component merely pushes the table against the floor and increases the friction between the floor and the table. The closer one can come to applying all the available force in the desired direction of movement, the more efficient will be the action. Another example of this principle is demonstrated in pulling actions. If a child’s sled or wagon is drawn by too short a rope, there will be a relatively large lifting component and a small forward pulling component. Because the purpose is to pull the wagon horizontally, it is more efficient to use a long rope because this gives a relatively greater horizontal or pulling component.

Composite Effects of Two or More Forces

Frequently two or more forces are applied to the same object. A canoe may be acted on by both the wind and the paddler. One force tends to send the canoe north and the other east. While in flight a punted ball’s path is the result of the force imparted to it by the kicker, the downward force of gravity, and the force of the wind, if any. In the body it is rarely, if ever, the case that an individual muscle acts by itself. For example, at least four muscles may act in flexion of the forearm, and more than five contribute to flexion of the leg at the knee joint. The effect that composite forces have on the human body to cause or modify motion may be classified according to their direction and application as linear, concurrent, or parallel.

Linear Forces

Forces applied in the same direction along the same action line are called linear forces. If a horizontal push is applied to a piece of furniture and the push is applied in line with the object’s center of gravity, the object will move forward in a horizontal direction, if, of course, there is no additional conflicting force or resistance. If another force is applied to the furniture in line with and in the same direction as the first force, the resultant of the two forces has a value equal to the sum of the two forces (a + b = c).

Similarly, if the forces act in opposite directions, the resultant still equals the algebraic sum of the two forces (a + (−b) = c). This might be the case in a tug of war.

Examples of more than one force applied at the same point and acting in identical directions in the body are rare. Possibly there are two such examples: the gastrocnemius and the soleus acting at the ankle joint, and the psoas and the iliacus acting at the hip joint. In each of these examples, the two muscles have a common tendon for their distal attachments.

Concurrent Forces

Forces acting at the same point of application but at different angles are called concurrent forces. Several football players opposing each other in a blocking situation furnish opposing forces that may be considered as acting at one point (Figure 12.8). The resultant outcome of the blocking is found using the method known as the combination of vectors described in Chapter 10. This method assumes a common point of application. Although this situation is common externally, it is not true of the majority of muscles that act on the same bone. Very few muscles have a common point of attachment. Nevertheless, the principle of finding the composite effect of two or more forces, with respect to magnitude and direction, is as true for forces acting on body segments as for forces acting on external objects. The important thing to remember is that the resultant magnitude of two or more concurrent forces is not their arithmetic sum, and the resultant direction of two concurrent forces is not halfway between them unless the two forces are of equal magnitude. The resultant of two or more concurrent forces depends on both the magnitude of each force and the angle of application—that is, the direction of each force. The techniques of vector resolution and combination must be used when doing force calculations.

Parallel Forces

In addition to linear forces, in which all forces occur along the same action line, and concurrent forces, in which forces acting at different angles are applied at the same point, another situation exists in which forces not in the same action line but parallel to each other act at different points on a body. An example of this is holding a 10-N weight in the hand when the forearm is flexed so that the angle of pull of the biceps is 90 degrees. The force of gravity may be represented as acting at two different points to push the forearm and the weight down while the force of the biceps acting in the opposite direction at another point pulls the forearm up (Figure 12.9).

The effect parallel forces have on the object they act on depends on the magnitude, direction, and application point of each force. Parallel forces may act in the same or opposite directions. They may be balanced and cause no motion, or they may cause linear or rotary motion. When parallel forces act on an object, their relationship to the object’s fixed axis or to its center of gravity, if it can move freely, determines the resultant action. Another example of parallel forces is shown in Figure 12.10.

The fact that motion is related to force in a precise manner was observed by Sir Isaac Newton in the seventeenth century. He formulated three laws of motion that explain why objects move as they do. Although all of these laws cannot be proved on earth, even in ideal experimental situations, they are accepted as universal truths to explain the effects of force.

Law of Inertia

Newton’s first law of motion states that a body continues in its state of rest or of uniform motion unless an unbalanced force acts on it. This Law of Inertia means that an object at rest remains at rest, and one in motion will continue at a constant speed in a straight line unless acted on by a force. In effect, the law identifies the conditions under which there is no external force. The fact that objects at rest need a force to move them seems obvious, but the need for force to slow or stop an object in motion is not always as apparent. We have all seen moving objects come to rest of their own accord, or what seems to be their own accord, but, in fact, other forces in the form of friction or air resistance caused the change in velocity. The tendency for a body to stay in motion for long periods of time is possible to observe more clearly in laboratory conditions where the effects of friction and air resistance can be minimized. Nevertheless, numerous examples of Newton’s first law can still be observed in sport and everyday experiences. Base runners in baseball know that it takes force to change velocity suddenly to avoid overrunning a base. Skiers continue into space if they traverse a hill or mogul at high speed (Figure 12.11). And nearly everyone at some time or another has experienced the frightening situation of continuing to move forward when the vehicle in which one is riding stops suddenly.

The property of an object that causes it to remain in its state of either rest or motion is called inertia. Because of inertia, force is needed to change the velocity of an object. The amount of force needed to alter the object’s velocity is directly related to the amount of inertia it has. The measure of inertia in a body is its mass, that is, the quantity of matter it possesses. The greater the mass of an object, the greater its inertia. A bowling ball obviously has greater inertia than does a volleyball. A tennis racket has greater inertia than a badminton racket, and a textbook more than a magazine.

Law of Acceleration

Newton’s second law of motion, the Law of Acceleration, concerns acceleration and momentum, and it tells how the quantities of force, mass, and acceleration are related and how to measure force when it exists. The law states that the acceleration of an object is directly proportional to the force causing it, is in the same direction as the force, and is inversely proportional to the mass of the object. It is quite easy to show that change in velocity (acceleration) of an object is proportional to the force and in the direction of the force. The velocity of a pitched ball reflects the force with which it is thrown, and it moves in the direction of the line of force at the moment of release. Similarly, it takes less braking force to stop a baseball moving at 5 meters per second than it does to stop a pitch of 50 meters per second. In both instances, the change in velocity is directly proportional to the amount of force and is in the direction of the force. The mass of the ball is a measure of its inertia, and the greater the inertia, the more force it takes to change the object’s velocity. Thus, a bowling ball requires more force to put it in motion than a playground ball, and the same is true for stopping it. Acceleration is inversely proportional to mass.

Newton’s second law may be expressed in equation form as:

For this to be possible, the force unit is the pound when acceleration is in feet/second/second and mass is in slugs. When acceleration is in meters/second/second and mass is in kilograms, force is expressed in newtons (see Table 10.1).

With this equation it is possible to determine the force needed to produce a given linear acceleration of a body if the weight of the body is known. Because we know that the equation for force, F = ma, can be written as , the force needed to accelerate a 300-N object 2 m/sec2 is 61 N, or

Impulse

The Law of Acceleration dictates that when sufficient force is applied to a mass, an acceleration will occur according to the equation

Because acceleration is the rate of change in velocity, this equation can be rewritten as

substituting for a. Multiplying both sides by time produces a new equation:

In other words, the product of a force and the time over which this force acts will produce a change in the velocity of the mass. This product of force and time is called impulse. From the impulse equation, Ft = m(vf − vi), it should be apparent that the force required to produce a given change of velocity in a given time period is proportional to the mass. It is also easy to see that as any change in velocity of an object of a given mass increases, so must the impulse increase proportionately. Conversely, if either force or time is increased, the velocity of the mass must also increase. Doubling either the force or the time over which a force is applied will double the velocity change.

The importance of creating as large an impulse as possible is evident in the skillful execution of many sports techniques. A baseball pitcher uses a form that allows the longest time over which to apply the force to the ball before releasing it. This same long windup is seen in the technique used by hammer throwers, discus throwers, and shot-putters. In each instance the performer accelerates the object to be thrown as much as possible by generating as much force as strength permits and by using body segment adjustments that increase the time over which the force can be applied (Figure 12.12).

Consider a pitcher throwing a 5-oz (0.142-kg) 90 mph baseball (40 m/s). If we know that the time of the forward motion of a throw is approximately 0.03 seconds, then using the impulse momentum relationship, the force required to bring about the change in velocity from 0 m/s (at the beginning of the forward motion) to 40 m/s at ball release can be determined.

To change the velocity of a baseball from rest to 40 m/s takes a force of 192.26 N (44.12 lbs).

Next, consider taking part in an egg toss game. When watching this activity, it is easy to see how the catchers are using the impulsemomentum relationship to catch the egg without breaking it. In this situation, the catcher wants to increase the time over which the force is being exerted to diminish the amount of force being applied. To increase the amount of time over which the force is being exerted, the catcher will “give” with egg.

The right side of the equation will remain the same and the velocity of the egg will be reduced to zero. The catcher can control only the left side of the equation. The egg can be brought to rest quickly (a small time and a large force), or it can be brought to rest over a longer time period (longer time and smaller force).

The impulse relationship also shows that force cannot be generated to cause a change in velocity unless time is available over which the force is applied. This helps explain the value of the follow-through in throwing, kicking, and striking movements. Although the follow-through does not affect the flight of the object once the object has left the throwing or striking implement, it does help ensure that the missile will stay in contact with the implement that is imparting force to it for as long as possible. The ball is actually carried along by the foot, golf club, or tennis racket, and the longer the time the force of the implement can be applied to the ball, the greater will be the change in velocity in the ball.

Momentum

The impulse equation is also written as

The product of mass and velocity (mvi or mvf) is momentum, and any change in momentum is equal to the impulse that produces it. Momentum is a quantity of motion that may be increased or decreased by increasing or decreasing either the mass or the velocity. The shot-putter who is able to push the shot with a greater speed than the opponent will cause the shot to have greater momentum at the moment of release. And, even though the mass is less, a small child may topple an adult if the child’s faster speed is sufficient to produce a larger momentum than the person the child collides with. Similarly, heavier bowling balls released with the same speed as lighter balls have greater momentum, and a heavier tennis racket will strike a tennis ball with greater momentum and cause a greater change in momentum in the tennis ball than will a lighter racket.

An increase in momentum occurs when the force is applied in the direction of the motion. Force applied in the opposite direction produces a slowing down or decrease in momentum. This is what happens when one catches a fast ball or lands from a jump or a fall. In both instances relatively large momentums are reduced to zero and large impulses occur. The momentum of the ball is large because of its large velocity, and the momentum of the person falling is great because of a large mass. Looking again at the impulse equation, Ft = mvf − mvi, it can be seen that a short stopping time will require a larger stopping force and that an increase in stopping time will reduce the amount of stopping force needed to change the momentum of the object to zero. This is why it is necessary to increase the stopping time by “giving” when catching the ball or when landing from a jump or fall. Without the “give” the impulse will not be enough, the momentum will not decrease to zero, and the ball will not be caught, or the momentum will reach zero but the force will be so great that injury in the form of damaged bones, joints, and soft tissue may result. A 20-N force falling for 5 seconds has the same impulse as a 100-N force falling for 1 second.

Law of Reaction

Newton’s third law of motion considers the way forces act against each other. A book lying on a table (Figure 12.13) exerts a downward force on the table, and because the book is stationary, another equal and opposite force must be acting on the book. Newton’s first law states that unbalanced forces produce motion. Because we have no motion here, we must have a balanced force system. The downward force of the book on the table is balanced by an upward force of the table on the book. The same is true when a person walks across a floor. The feet push back against the floor with the same magnitude as the floor pushes forward against the feet. Without the forward push of the floor against the feet, forward progress would not be possible. Notice how the front part of the foot in the footprint in Figure 12.14 makes a deeper impression than the heel. To accelerate forward, the walker has to push backward. For this reason, runners push against starting blocks at the beginning of a race so that a strong forward reaction push can be received from the blocks. In each of these instances, forces of equal magnitude are exerted in opposite directions. One force is called the action force and the other is the reaction force.

Newton’s Law of Reaction states that for every action there is an equal and opposite reaction. Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first.

In locomotion on the surface of the earth (ground or floor), this reaction force is referred to as ground reaction force. The ground reaction forcevector is equal in magnitude and opposite in direction to the force vector that is being applied to the ground. In walking, as the foot pushes down and back against the ground, the ground pushes up and forward against the foot. Again, notice how the front part of the footprint in Figure 12.14 makes a deeper impression in the front than the heel does in the back. Notice also how the sand had to be compressed by the walker before it was solid enough to produce an equal and opposite reaction. This is one reason that walking on soft sand or snow is more difficult than walking on a firm surface.

Although ground reaction force is necessary for locomotion, such forces do place stress on the body. It sometimes becomes necessary to minimize ground reaction force to prevent injury. A pole vaulter, for example, uses ground reaction force in the run and the takeoff but needs to minimize the ground reaction force at landing. The use of a landing pit or crash pad allows the vaulter to alter momentum by increasing the time over which velocity is reduced. In the impulse equation, Ft = m(vf − vi), if time (t) is increased, force (F) must decrease. A decrease in the force component of impulse is a decrease in the action force. With the Law of Reaction, a decrease in the action force will produce an equal and opposite decrease in the ground reaction force.

Action and reaction show up in countless other ways when objects are in motion. When a boat is rowed, the oars exert a force against the water and the water pushes against the oars with an equal and opposite reaction, causing the boat to be pushed forward as the force is transferred from the oar to the boat through the oarlock. The force of a volleyball can be felt pushing back against the hand as it is served with a forward force. A similar force is observed when a gun is shot or an arrow is released from a bow. The recoil of the gun or bow is due to the reaction force of the bullet or arrow to the action force of the gun or bow.

Conservation of Momentum

When an object is set in motion by a force, the momentum (mv) of the object is changed. Because the force that causes this change in momentum must have an equal and opposite force, another equal and opposite momentum change must occur in the object producing the reactive force. The time the objects are in contact with each other would also be equal, and therefore the impulses (Ft) would be equal. This means that m1v1f − m1v1i = m2v2f − m2v2i. This principle is summarized in the Law of Conservation of Momentum, which states that in any system where forces act on each other the momentum is constant. Thus, if an impact or action between objects occurs, the momentums before impact must equal the momentums after the impact, provided, of course, that no momentum is lost through friction or other forces. The momentum of a golf ball changes from zero to a larger quantity after it is struck by a club (Figure 12.15). Neglecting air resistance and friction, the momentum of the club will also change so that the momentum of ball and club after impact will equal the momentum of ball and club before impact.

This is proved by using the impulse equation

where the subscript b refers to the ball and the subscript c to the club. Rearranged, this equation becomes

That is, the combined momentums before impact equal the combined momentums after impact. Momentum is conserved; none is lost.

The conservation of momentum may be easily apparent in some instances, whereas in others it is harder to visualize. When one steps out of a canoe onto a dock, the canoe is pushed back by the passenger as the passenger is pushed forward by the canoe. The change in momentum of the canoe backward (m1v1f − m1v1i) will equal the change in momentum of the passenger forward (m2v2f − m2v2i). Here the action–reaction relationship is evident in the movements of both the canoe and the person. But if the same person steps off an ocean liner, the change in momentum of the ship is imperceptible, yet it will still equal the change in momentum of the passenger. Because the mass of the ship is so great, its velocity is not apparent. In both instances, the change in the mass–velocity product of the passenger equals the change in the mass–velocity product of the boat (canoe or ship), and momentum is conserved. Incidentally, if the passenger disembarking from the canoe does not account for the backward acceleration of the canoe, the step toward the dock may be a wet one.

Summation of Forces

The forces generated by the muscle structure of the body may be summated from one segment to another. In generating momentum, these forces, applied over a period of time, are used to build momentum, which is then transferred from segment to segment. In a typical throwing pattern, for instance, force is first generated in the muscles of the lower extremity as the legs extend to push against the ground. The momentum generated is transferred to the trunk, where the application of further muscular force acts to increase momentum, primarily through an increase in velocity. This momentum is then transferred to the upper arm. The great decrease in mass when moving from trunk to upper arm produces an increase in velocity, in accord with the momentum equation (mv) and the fact that momentum within a system must be conserved. If the muscles of the shoulder contract to provide further impulse, velocity will increase further. This sequential transfer of momentum continues with mass decreasing and velocity increasing until momentum is finally transferred to the thrown ball.

In addition to the forces that produce motion, a number of forces act to modify motion in some manner. These modifying forces must be considered when studying the kinetics of linear motion. When the kinesiological analysis model presented in Chapter 1 is applied to a motor skill, the forces that modify motion must be included in any discussion of the nature of the forces involved. Six of these forces are commonly categorized as follows:

• 1. Weight
• 2. Contact forces
  • a. Ground reaction force
  • b. Friction
• 3. Fluid forces
  • a. Buoyancy
  • b. Drag
  • c. Lift

For purposes of discussion in this chapter, an additional category of elasticity and rebound has been added.

Weight

In what came to be known as the Law of Gravitation, Newton (remember the apple) was the first to point out that all bodies are attracted to each other in direct proportion to their masses and in inverse proportion to the square of the distance between them. The amount of this attraction between bodies on earth is negligible except for the attraction between the earth itself and the bodies on it. Because of the earth’s huge mass, this attraction is quite noticeable. The force is called gravity, and it is measured as the weight of the body applied through the center of gravity of the body and directed toward the earth’s axis (Figure 12.16). The closer a body is to the earth’s center, the greater is the gravitational pull and, therefore, the more it weighs. When the body moves far enough away from the earth’s center, such as to the moon, the decrease in the gravitational pull is apparent, as in the ease of giant leaps made by astronauts. The mass of the body is the same as on earth, but the weight has decreased in proportion to the gravitational pull. The relation between weight, mass, and gravity is represented in equation form as w = mg. The force of weight must be considered in all motion analyses.

Contact Forces

Ground Reaction Force

As Newton’s third law states, for every action there is an equal and opposite reaction. Anytime a force is exerted on an object, that object will exert a force back. As you sit and read this section, you are exerting a force on a chair and the chair is exerting an equal and opposite force on you. If the chair did not exert an equal and opposite force, you would be falling toward the floor. If you are sitting still, the ground reaction force is equal to your body weight. Likewise, when you walk, run, or jump, the ground pushes back on you with the same force with which you exert on the floor. In Figure 12.17 the high jumper exerts a force on the ground (action), and the ground exerts a force on the jumper’s foot (reaction force).

Without the reactive force there would be no motion. You cannot walk without the ground reaction force. When the heel strikes the ground, the ground brings the foot to rest so the body can pass over the planted foot. Then the ground reaction force pushes back against the toe at push-off. Ground reaction force is usually measured with a force platform and during jogging can be as high as two to three times body weight. If a book (4.45 N) is sitting on the floor, the ground reaction force is equal to the weight of the book (4.45 N). If you were to push directly down on the book, the ground reaction force would be the sum of the weight of the book plus the pushing-down force. If the force that you exerted on the book was not directly down, but at an angle, the ground reaction force would be the sum of only that part of the force acting downward plus the weight of thebook.

Friction

Friction is the force that opposes efforts to slide or roll one body over another. Without friction it would be impossible to walk or run or do much of any kind of moving. On the other hand, friction increases the difficulty of moving objects about with its deterrent effect. There are numerous examples in which we attempt to increase friction for more effective performance. The use of rubber-soled shoes on hardwood floors or wet decks, spikes on golf shoes, and rubber knobs on hiking boots improve friction with the supporting surface. Chalk on the gymnast’s hands, on work gloves, and on cloth grips on tennis rackets are all used to decrease slippage. Even the surfaces of balls are designed to increase friction through irregularities such as the fuzz on the tennis ball or the dimples on a golf ball. Attempts to decrease friction are also evident in physical activity. The sole of one of a bowler’s shoes is made to have little friction so that it can slide more easily in the approach. Sharp ice skates apply pressure on the ice and cause slight melting, thus making it easier for the skates to move across the ice. Ice covered with a slight film of water is more slippery than colder, drier ice. Skis are waxed, bicycles are greased, and in-line skates have ball bearings, each for the purpose of minimizing the retardant effect of friction.

The amount of friction between one surface and another depends on the nature of the surfaces and the forces pressing them together. Generally speaking, smooth surfaces have less friction than rough surfaces. The area of surfaces in contact with each other does not affect friction. A footlocker pulled along on one end would take as much force to pull as one on its side. An empty footlocker, however, would take less force than one full of books. Friction is proportional to the force pressing two surfaces together.

In Figure 12.18 the book is pressing down on the table with a force (W) equal to its weight. The reactive force of the table pushing up against the book is called the normal force (N). The normal force acts perpendicular to the support surface and in this case is equal and opposite to the weight of the book. The maximum force that can be applied to the book before it begins to slide is called the force of static friction (Fmax). The ratio of the force of static friction to the normal force is a quantity called the coefficient of static friction (μs).

The symbol μs (mu) represents the coefficient of static friction and is an experimentally determined quantity that is a constant for any two surfaces. The larger the coefficient, the more the surfaces cling together and the more difficult it is for the two surfaces to slide over each other. Police officers use this information every day in investigating automobile accidents. To determine how fast a car was going, they look to the length of the skid marks and determine how much friction (the coefficient of friction) was between the tires and the road. This is achieved experimentally through use of a drag sled. A portion of a tire filled with concrete to approximate the weight on one tire of a car is pulled along the road by a spring scale. The reading of the spring scale just before the sled moves is recorded as Fmax. The total weight of the car divided by four is the N, and the coefficient of friction is Fmax/N. From this information the police officers can tell how fast the car was going and if it was traveling over the posted speed limit.

From an exercise science perspective, friction is a very important concept and is taken into consideration when looking at shoe to floor interface. It is used to answer such questions as which court shoes give better traction on hardwood floors or clay tennis courts.

Up to this point the discussion has centered on static friction, which is the amount of attraction between two surfaces that must be overcome for one of those surfaces to begin to move along the other. Once motion has been achieved, other types of friction take over such as sliding friction and rolling friction. Static friction will always be greater than sliding friction or rolling friction for a given set of surface conditions. It is easier to keep something moving than to get it moving in the first place. As the name implies, sliding friction is a measure of the resistance of one surface to continue to slide along another and can be used to explain how far from a base a player needs to begin a slide. Rolling friction considers how easily one surface can continue to roll over another surface. Rolling friction helps soccer players and golfers determine how a ball will react on short grass versus high thick grass, and helps people determine which surface is better for in-line skating. The smaller the coefficient, the easier it is for the two surfaces to begin sliding or rolling over each other. A coefficient of 0.0 would indicate completely frictionless surfaces. The equation also shows that the coefficient of friction is totally dependent on the force holding the surfaces together (N) and the force needed to slide one surface over the (Fmax). The coefficient will decrease as Fmax decreases.

The coefficient of friction between two objects may also be found by placing one object on the second and tilting the second until the first starts to slide. The tangent of the angle of the second object with the horizontal is the coefficient of friction (Figure 12.19). An interesting application of the use of μ can be seen with respect to the gripping power of basketball shoes. The amount of lean a player may safely take is equal to the angle θ that the player makes with the vertical, whose tangent = is the amount of horizontal force needed to cause the feet to start sliding horizontally, and W is the weight of the player (Figure 12.20). Shoes allowing a greater lean would certainly afford their wearers an advantage in the game. For this information to be of use to shoe manufacturers, however, a standard type of floor surface would have to be determined.

Elasticity and Rebound

Objects that rebound from each other do so in a fairly predictable manner. The nature of a rebound is governed by the elasticity, mass, and velocity of the rebounding surfaces, the friction between the surfaces, and the angle with which one object contacts the second.

Anytime two or more objects come into contact with each other, some distortion or deformation occurs. Whether or not the distortion is permanent depends on the elasticity of the interacting objects. Elasticity is the ability of an object to resist distorting influences and to return to its original size and shape when the distorting forces are removed. The force that acts on an object to distort it is called stress. The distortion that occurs is called strain and is proportional to the stress causing it. Stress may take the form of tension, as in the stretching of a spring; compression, as in the squeezing of a tennis ball (Figure 12.21); bending, such as the bending of a fencing foil; or torsion, as in the twisting of the spring. In all cases the object tends to resume its original shape when the stress is removed. If the stress is too large, the elastic limit of the object is exceeded and permanent distortion occurs.

Coefficient of Elasticity

Substances vary in their resistance to distorting forces and in their ability to regain their original shapes after being deformed. One usually thinks of a material such as rubber as being highly elastic because it yields easily to a distorting force and returns to its original shape. Actually, substances that are hard to distort and return perfectly to their original shapes are more elastic. Gasses, liquids, highly tempered steel, and brass are examples. In comparing the elasticity of different substances, coefficients of elasticity are used. A coefficient of elasticity or restitution is defined as the stress divided by the strain.

The coefficient of elasticity most commonly determined in sports activities is that caused in the compression of balls. If one drops a ball onto a hard surface like a floor, the coefficient of restitution may be determined by comparing the drop height with the bounce height in the equation

where e = coefficient of restitution or elasticity. The closer the coefficient approaches 1.0, the more perfect the elasticity. Rules require that a basketball should be inflated to rebound to a height of 49 to 54 inches at its top when its bottom is dropped from a height of 72 inches. For the maximum bounce height, this is a coefficient of .781. In comparison, a volleyball dropped from the same height and inflated to 6 psi (pounds per square inch) rebounds to 51 inches and has a coefficient of .84. A tennis ball has a coefficient of .73, and a leather-covered softball one of .46.

The coefficient of elasticity may also be found in another way. Because the Law of Conservation of Momentum states that the total momentum in any impact between two objects must remain the same, the momentum of one object may be reduced, so long as the momentum of the other object increases proportionately. If the objects are numbered 1 and 2 and it is assumed that mass of the objects remains constant, it is possible to find the coefficient of elasticity by using the change in velocity of the two objects:

where vf2 and vf1 are the velocities after impact and vi1 and vi2 are the velocities before impact.

Angle of Rebound

An elastic object dropped vertically onto a rebounding surface will compress uniformly on its underside and rebound vertically upward. An elastic object that strikes a rebounding surface obliquely will compress unevenly on the bottom and rebound at an oblique angle. The size of the rebound angle compared with that of the striking angle depends on the elasticity of the striking object and the friction between the two surfaces. The rebound of a perfectly elastic object is similar to the reflection of light: The angle of incidence (striking) is equal to the angle of reflection (rebound) (Figure 12.22). Variations from this ideal are to be expected as the coefficient of restitution varies. Low coefficients will generally produce angles of reflection greater than angles of incidence.

For example, an underinflated volleyball will not have a great coefficient of restitution. It will therefore rebound at an angle much less, or much closer to the floor, than the striking angle. The coefficient of restitution acts to affect the vertical component of rebound. In rebound at an oblique angle, friction affects the horizontal component of the rebound. An increase in friction will produce a decrease in the angle of rebound. It is possible to produce an angle of rebound equal to the angle of incidence by decreasing both the horizontal component (increased friction) and the vertical component (elasticity) proportionately. If the proportions between the two components are constant, the angle of rebound will equal the angle of incidence, but the resultantvelocity of the object after impact will be decreased.

Effects of Spin on Bounce

Spin also influences rebounding angles. Balls thrown with topspin will rebound from horizontal surfaces lower and with more horizontal velocity than that with which they struck the surface. They will tend to roll farther also, an action often desirable on long golf drives. Balls hitting a horizontal surface with backspin rebound at a higher bounce and are slower. Balls with backspin also roll for shorter distances than those with topspin or no spin. Because of the compression of the ball and the friction between it and the surface, a ball with no spin hitting the surface at an angle will develop some topspin on the rebound, and a ball hitting with topspin will develop greater topspin upon rebounding. With backspin, however, the spin may be completely stopped or reversed. A ball with sidespin will rebound in the direction of the spin. A ball spinning or curving to the right will “kick” to the right upon rebounding, and a left-spinning ball will do the reverse.

When spinning balls hit vertical surfaces, such as a basketball backboard or a racquetball court wall, they respond in relation to the surface in the same manner as with horizontal surfaces. When the vertical surface is struck from below, a ball with no spin or topspin will have added spin on rebound, and the angle of reflection will be greater than the angle of incidence. With backspin the spin will stop or reverse and the angle of reflection will be less than the incidence angle. For this to be true, a backspinning ball approaching a vertical wall from above must be regarded as moving with topspin in relation to the rebounding surface. Similarly, a topspinning ball hitting the vertical surface from above is in actuality spinning backward with respect to the rebounding surface.

When a spinning ball meets a forward-moving object, as in tennis or table tennis, the ball will rebound in a direction that results from the forces acting on it at impact. For example, a ball with topspin striking a stationary vertical surface head-on will rebound upward. That same ball (with topspin) striking a moving vertical surface, like a tennis racket, will also rebound upward, but not as much, because the resultant rebound has a horizontal force component from the forward-moving tennis racket. Nevertheless, the upward direction may be sufficient to send the ball out of bounds. To counteract the upward direction, the racket face can be turned obliquely downward, thus adding a downward compensating force component as well as changing the spin of the ball.

The resultantforce path of a rebounding ball is controlled by several factors. When one attempts to predict the direction of the rebound, the momentums of both the ball and the striking implement must be considered as well as the elasticity of the two objects, the spin of the ball, and the angle of impact. Awareness of these factors and the ways in which they affect play have numerous and valuable applications in sports such as handball, squash, racquetball, table tennis, paddleball, and tennis.

Fluid Forces

Water and air are both fluids and as such are subject to many of the same laws and principles. The fluid forces of buoyancy, drag, and lift apply in both mediums and have considerable effect on the movements of the human body in many circumstances. A discus sails, a baseball curves, a volleyball “wobbles,” and a shuttlecock drops because of contact with air currents. Sky divers and hang gliders control their flight paths by interacting with the air currents, whereas downhill racers, swimmers, and underwater divers streamline their bodies to minimize the effect of fluid resistance.

Buoyancy

If an adult female stands in shoulder-deep water and abducts her arms as she lays her head back on the water’s surface, most likely her feet will leave the bottom as her legs begin to rise. At some point between the vertical and horizontal, the swimmer will come to rest, and she will be in a motionless back float. For this to occur, an upward force must counterbalance the weight (force) of her body, acting vertically downward at her center of gravity. This upward force is called buoyancy and, according to Archimedes’ Principle, the magnitude of this force is equal to the weight of water displaced by the floating body. Specifically, Archimedes’ Principle states that a solid body immersed in a liquid is buoyed up by a force equal to the weight of the liquid displaced. This principle explains why some objects float and others do not, why some individuals float motionless like bobbing corks, and why others struggle to keep their noses above water while attempting a back float. When a body is immersed in water, it will sink until the weight of the water it displaces equals the weight of the body. Sinking objects never do displace enough water to equal their body weight and eventually settle to the bottom. If such objects are weighed underwater, they will be found to weigh less than when weighed in air. That difference in weight equals the weight of the water displaced and is the buoyant force acting on the immersed objects. Even a body that floats has some part of its volume beneath the surface and thus displaces a volume of water. The weight of the water it displaces equals the total weight of the floating object. Any object stops sinking when the weight of the water it displaces equals its weight.

The ratio between the weight of an object and its volume is referred to as density. The more weight per volume, the greater the density. An inflated volleyball, for example, has a slightly greater volume than a bowling ball. The bowling ball, however, weighs much more. Because the bowling ball has more weight for its volume than does the volleyball, it is said to have greater density.

The ratio between the density of any given object and that of water is called specific gravity. If an object or a body has the same density as water, or the same weight and volume ratio, it possesses a specific gravity of 1.0. Those objects with greater density will have a specific gravity greater than 1.0 and will sink. The bowling ball, being very dense, will sink very quickly. Objects with a density less than that of water will have a specific gravity less than 1.0 and will float with some part of the object or body exposed. An example is the inflated volleyball, which is composed primarily of air.

Human beings differ in specific gravity depending on individual body composition. Human bone and muscle tissue both have a specific gravity greater than water and as a result tend to sink. Fat and air, on the other hand, have specific gravities much less than that of water and will float. The ease with which one floats and the floating position assumed are therefore determined by the distribution of muscle, bone, fat, and air within the body. The specific gravity of the various body parts differs accordingly.

Usually the legs have a high specific gravity and consequently are the part of the body that most often sinks during the back float. The thoracic region is the most buoyant part, having the lowest weight for its volume. A person can increase the buoyancy of this region by keeping the lungs inflated with air, thus increasing the ease of floating. Some individuals have overall specific gravities greater than 1.0. It is impossible for these people to do a motionless float because they are “sinkers.” A practical way to determine whether a person is a floater or a sinker is to have the person assume the tucked jellyfish float position with lungs inflated. If any portion of the individual’s back is on or above the surface, the individual can learn to maintain a motionless floating position that, even though it may be more nearly vertical than supine, is still called a back float.

A floater has to be concerned with two forces, the downward force of the body’s weight and the upward buoyancy of the water. When these forces act on the body so that their resultant is zero, the forces will be in equilibrium and the body will be in a motionless float. The downward force acts at the center of gravity of the body, a point somewhere in the pelvis. The buoyant force acts at the center of buoyancy of the body, a point that varies with individuals but is usually closer to the head than the center of gravity. If the body were of uniform density, the center of gravity and the center of buoyancy would coincide, but because the body has less mass toward the head, the center of buoyancy is usually higher in the body than the center of gravity. The center of buoyancy is the point where the center of gravity of the volume of displaced water would be if the water were placed in a vessel the shape and size of the floater’s body. Because the water is of uniform density, its center of gravity will be in the direction of the greater volume, that is, near the chest region. If the center of gravity and center of buoyancy are not in the same force line with each other, as shown in Figure 12.23a, the body will rotate in the direction of the forces until the forces are equal and opposite in line, direction, and magnitude. At this point the floater will be in a balanced float (Figure 12.23b). Individuals who float horizontally have the center of gravity and center of buoyancy in the same vertical line while in the horizontal position. These are usually individuals whose bodies contain a high percentage of fat. Those floaters whose legs tend to drop when attempting the back float will have a balanced position somewhere between the horizontal and vertical at that point where the center of gravity and center of buoyancy are in the same vertical line.

The angle of the floating position with the horizontal may be decreased by making adjustments in the position of the body segments that move the center of gravity in closer alignment with the center of buoyancy. Raising the arms over the head, bending the knees, and flexing the wrists to bring the hands out of the water all contribute to moving the center of gravity closer to the head and thus closer to the center of buoyancy (Figure 12.23b).

Lift and Drag

The fluid resistance to movement through air or water consists of two forces: drag and lift. The fluid flow that produces these two resistance forces is the result of either fluid or object velocity, which acts to produce pressure. Putting one’s hand into a calm swimming pool will produce no fluid flow and no feeling of pressure because there is no velocity. Moving the hand back and forth in the pool (object velocity) or placing the hand in a fast-running- stream (fluid velocity) produces definite pressure sensations because of fluid flow. Similarly, a softball thrown at 30 mph on a calm day will produce an airflow around the ball in the opposite direction of 30 mph. If the ball were thrown into a 5-mph headwind, the airflow velocity around the ball would be 35 mph. The magnitude of the fluid flow is directly proportional to the fluid and object velocities.

The resistance to forward motion experienced by objects moving through a fluid is called drag. Drag force is the result of fluid pressure on the leading edge of the object and the amount of backward pull produced by turbulence on the trailing edge. In fluid flow, the layer of fluid next to the object is called the boundary layer. There will usually be some amount of friction between the boundary layer and the object, which will cause resistance to forward motion. This friction is referred to as surface drag. If the flow of fluid around an object is smooth and unbroken, it is referred to as laminar flow. Laminar flow usually occurs in fluids passing around a smooth surface at relatively slow speeds. In laminar flow, the boundary layer is slowed down slightly by surface drag but continues in a smooth, unbroken flow around the object (Figure 12.24a). A smooth surface will produce much less surface drag than a rough surface. New, smoother swimsuit fabrics and the practice of shaving body hair before competition are both examples of attempts to reduce surface drag in competitive swimming.

If the surface area presented by the object is great or if the flow velocity is high, there will be greater flow pressure on the leading edge of the object than on the trailing edge. In this situation, the boundary layer does not flow completely around the object. The boundary layer separates, creating a vacuum behind the trailing edge. Fluid rushes in to fill this vacuum, causing a back flow of fluid that is called turbulence. Turbulence produces a suction force backward, against the direction of motion. This turbulence is the result of form drag. In form drag the shape of the object is such that the fluid, moving at a given velocity, cannot follow the contours of the object (Figure 12.24b). Drivers on interstate highways often experience turbulence caused by form drag. When approaching a large truck from behind, the driver of a small car may experience an agitated airflow that tends to make the car harder to control. The driver may also feel a pull toward the truck. Both of these are the effects of turbulence. This turbulence might also increase as the velocity of the truck increases.

Form drag often can be reduced by streamlining. Examples of streamlining are common any time one desires to move easily and quickly through air or water. The crouch position on a racing bicycle, the shape of a race car, and the sharp bow of a boat are all examples of streamlining. The shape of a discus is also meant to take advantage of the effects of streamlining. The discus is tapered at both leading and trailing edges, so the airflow does not have to make any abrupt changes in direction. This streamlining principle will work only if the discus is thrown so that air flows past the tapered edges. If the discus is thrown with the flat side leading, air must make a very abrupt change of direction. There will be increased pressure- on the leading edge and turbulence on the trailing edge, creating a large form drag. For this reason, streamlining makes the angle between the long axis of the object and the direction of fluid flow very important. This angle is often referred to as the angle of attack.

The angle of attack is also critical in producing and utilizing the fluid force known as lift. Lift is the result of changes in fluid pressure as the result of differences in airflow velocities. An extremely important principle in understanding the mechanism of lift is Bernoulli’s Principle. Simply stated, Bernoulli determined that the pressure in a moving fluid decreases as the speed increases. The design of an airplane wing is based on Bernoulli’s principle of lift. The top side of an airplane wing has a higher curve than the bottom. To avoid creating a vacuum, airflow over the top will be at a higher velocity than the flow across the bottom. Pressure will therefore be lower on top of the wing. Lift will act upward, from the area of high pressure toward the area of low pressure, causing the wing to move upward. The swimmer uses lift in much the same way. Creating lift toward the forward surface of the hand is one force that aids in propelling the body forward. Lift, therefore, is the result of fluid on one side of an object having to travel farther in the same amount of time to avoid creating a vacuum. Lift always acts perpendicular to the fluid flow and therefore to the drag force (Figure 12.24c).

Ball Spin

Bernoulli’s Principle applies for moving objects passing through a stationary fluid as well as for fluids passing stationary objects and, in addition to explaining why airplanes can fly, it also explains why balls with spin follow a curved path. This latter application of Bernoulli’s Principle is called the Magnus effect after the German physicist who first explained the phenomenon. A ball moving through the air will also move in the direction of least air pressure. As shown in Figure 12.25a, the ball spinning in a clockwise direction drags around a boundary layer of air. At the bottom of the ball this air current is moving in the same direction as the oncoming air. At the top, the boundary air is moving in the opposite direction. The air at the bottom is moving faster, and therefore the pressure is reduced. The air at the top moves more slowly and the pressure is increased. Thus the ball will move in a downward curve in the direction of least pressure. Viewed from the side, this would be the behavior of a ball with topspin imparted to it. Balls with topspin drop sooner than balls with no spin. A ball with a counterclockwise or backspin will move in an upward curve and thus stay aloft longer than a ball with no spin (Figure 12.25b). Balls spinning about a vertical axis have sidespin. Right spin causes the ball to curve to the right and occurs when the forward edge of the ball moves to the right. Left spin is the opposite.

The amount of air a ball drags around with it when spinning depends on the surface of the ball and the speed of the spin. Rough or large surfaces, small mass, and a fast spin speed all produce a more noticeable spin and curve deflection. The small mass of a table tennis ball, the fuzz on a tennis ball, and the seams on a baseball all enhance spin, an important element in each game’s strategy. The deflection will also be more pronounced if the forward velocity is slow. This may occur because of little force imparted to the ball or a strong headwind. Spin on a ball may also smooth its flight by acting as a stabilizer. Like a gyroscope, a football or discus spinning around one axis resists spinning about another axis and therefore is less likely to tumble through the air.

In analyzing any technique, the student should consider all the external forces by methodically accounting for the effect of each one on the body. To do this it is helpful to look at the body as if it were isolated from its surroundings. The isolated body is then considered to be a separate mechanical system whose boundaries have been defined. The isolation makes it easier to identify the forces and to represent them as vectors in a diagram of the body. This type of representation is called a free-body diagram and can help settle any doubts about the application and direction of the various forces acting on the body in any given time frame. The direction of each of the external forces and the point through which each acts are summarized in Table 12.1.

In Figure 12.26 the magnitude of each external force acting on the body is represented by the arrow length. The direction of the force is represented by the arrowhead, and the point of application is located at the arrow tail. The body is acted on by its weight (W) applied through the center of gravity, the reactive force (R) and the friction force (F), both applied at the point of contact with the vault, and the horizontal force (H) directed through the center of gravity. In Figure 12.27, the forces acting on the body are weight (W), buoyancy (B), and drag (D). The importance of visualizing all the forces acting on the body relates to the fact that the state of motion or rest of a body depends on the vector sum of all these forces. With a diagram, the forces that can be manipulated to alter the vector sum and produce the desired outcome are more readily apparent. If the swimmer in Figure 12.27, for instance, wants to stress horizontal speed, attention should be paid to those factors that will maximize the propulsive force and minimize the opposing drag force, little can be done to alter the weight and buoyancy forces.

Free-body diagrams are also used to show the forces acting on an isolated body segment. In Figure 12.28, the thigh has been isolated from the rest of the lower limb, and the forces acting on it are shown. The forces acting on the limb include the weight of the thigh (W), the resultant muscle force acting across the hip joint (Mh), the resultant muscle force acting across the knee joint (Mk), and the reactive force components acting across the hip joint (Hx and Hy) and knee joint (Kx and Ky).

Work

Simple machines such as the lever are devices designed to do work. In each instance the machine aids in the use of a force to overcome a resistance. When the resistance is overcome for a given distance, work is done. Mechanically speaking, work is the product of the amount of force expended and the distance through which the force succeeds in overcoming a resistance it acts upon. For work done in making a body move in a linear fashion, this may be expressed as

where W stands for work, F for force, and s for the distance along which the force is applied. Units for expressing work are numerous because any force unit may be combined with any distance unit to form a work unit. In the U.S. system the foot-pound is the most common work unit; the joule is the most frequently used unit in the metric system. A joule is equivalent to 107 × 1 gram of force exerted through 1 centimeter.

In computing work, the distance s must always be measured in the direction in which the force acts, even though the object whose resistance is overcome may not be moving in the same direction. If one lifts a 20-N suitcase from the floor to place it on a shelf 2 meters above the floor, the work accomplished is 40 newton-meters (Nm). If, on the other hand, that same weight is lifted an equal distance upward but along a 4-meter inclined plane, the amount of work done is still 40 N · m (Figure 12.29). The work done along the incline is the same as the work done to lift the weight to the height of the incline. The horizontal distance the suitcase moves is not included.

Work done in the same direction that the body moves is called positive work, whereas work done in the opposite direction is called negative work. When an individual does a deep knee bend, the extensor muscles of the leg contract eccentrically to resist the effect of gravity on the body. The body moves in a direction opposite to the upward force of the muscles, and thus negative or resistive work is performed by the muscles. In the return to standing from the knee bend position, the body moves in the same direction as the concentrically contracting extensors, and the work of the muscles is positive. The net work accomplished during the down and up movement of the knee bend is the sum of the positive and negative work.

Consider the mechanical work done during a 444-N (50 lbs) bench press cycle. If the weight is moved through a distance of 0.33 m (13 in.), then the work done would be:

If the question was, “How much mechanical work is done in one complete bench press cycle?” the answer would be zero. Because the weight started and ended at the same point, no net motion occurred and, as such, no mechanical work was done.

When the exertion of effort produces no motion, such as might happen during a tug of war, no work is accomplished. Static contraction of muscles may be evident and considerable physiological effort may be expended but, in the strict mechanical sense, no work is being done. The physiological measure of such efforts is determined by obtaining the energy costs, which in turn are usually measured by computing the amount of oxygen consumed during the effort and converting it to calories per minute.

Theoretically, the mechanical work performed by an individual muscle can also be determined using the equation W = Fs. Suppose, for instance, that there is a rectangular muscle, 10 centimeters long and 3 centimeters wide, that exerts 240 newtons of force as it turns a bony lever. Because the average muscle fiber shortens to one-half its resting length and the fibers of a small rectangular muscle run the full length of the muscle, the force of the muscle in question is exerted over a distance of 5 cm (the amount of shortening equal to one-half the resting length). Therefore, F = 240 N and s = 5 cm, the muscle is performing 1200 newton-centimeters, or 120 newton-meters-, of work. In brief: W = F (240 N) × s (5/100 meters).

If the force of the muscle is not known, it is computed from the muscle’s cross section. Assuming the muscle in question to be 1 cm thick and the force per square inch of cross section to be 360 N, the following steps are taken:

• 1. Find the muscle’s cross section:
• Cross section = width × thickness.
• 3 cm × 1 cm = 3 sq cm (cross section).
• 2. Find the amount of force exerted by the muscle:
• Average force = 360 N per sq cm.
• Cross section = 3 sq cm.
• F = 360 × 3 = 1080 N.
• 3. Find the amount of work performed by this muscle, first in N · cm, then in N · m:
• W = Fs.
• W = 1080 N × 5 cm = 5400 N · cm.
• 5400/100 = 540 N · m.

For purposes of simplification, the internal structure of the hypothetical muscle used in these examples was rectangular. This means that a simple geometric cross-sectional measure could be used. For penniform and bipenniform muscles, however, it would be essential to determine the physiological cross section. It must also be remembered that s in the work equation represents one-half the length of the average fiber in the muscle and that this may or may not coincide with the total length of the fleshy part of the muscle, depending on its internal structure. The number of pounds of force per square inch exerted by the average human muscle must be selected arbitrarily, depending on whose research the student accepts.

A single equation for computing the work performed by a muscle whose average fiber length is known and whose physiological cross section (PCS) has been determined is as follows:

This gives work in terms of Newton-centimeters. Because it is customary to measure work in terms of Newton-meters, the result should be divided by 100. The following equation includes this step:

Any measure of work does not account for the time involved in performing the work. When one lifts 200 N a distance of 2 meters, 400 N · m of work is done, regardless of the time it takes to do it. Work per unit time or the rate at which work is done is called power.

Recall that W = Fs and by substituting Fs for work a different version of the power equation can be written:

Recall that is the equation for velocity, so a

different version of the power equation can be written:

Any form of the equation clearly points out that the machine or person who can perform more work in a given unit of time or who takes less time to do a specified amount of work is more powerful.

In the U.S. system, power is expressed as ft · lb/sec or as horsepower (550 ft · lb/sec = 1 horsepower). The metric system unit is the watt, which is equivalent to 1 joule/second.

Energy

Energy is defined as the capacity to do work. A body is said to possess energy when it can perform work, and the energy that the body possesses is measured as the work accomplished. Energy may take numerous forms, and it is common for it to be converted from one form to another, although according to the Law of Conservation of Energy it can neither be created nor destroyed. Some forms that energy may take are heat, sound, light, electric, chemical, nuclear, and mechanical. When a ball is hit with a bat, some of the mechanical energy is converted to sound and heat energy, but none of the energy is lost. The total amount of energy possessed by a body or an isolated system remains constant.

Potential energy and kinetic energy are two classifications of mechanical energy that have important implications in biomechanics. Potential energy is the capacity for doing work that a body has because of its position or configuration. A raised weight such as a diver standing on a platform, a bent bow, or a compressed spring all have potential energy. Measured in work units, potential energy is the product of the force an object has and the distance over which it can act. The potential energy of a 150-pound diver whose center of gravity is 20 feet above the water surface is 3000 foot-pounds. In this instance the appropriate form for the potential energy equation is

where m is the mass of the body, g is the acceleration of gravity, and h is the height between the diver’s center of gravity and the water surface. The product of m and g (mg) is the same as the diver’s weight.

The distance selected for h in the potential energy equation is established by choosing an arbitrary zero level. The potential energy of someone standing on a table 1 meter from the floor can be increased by boring a large hole in the floor directly in front of the table. The individual’s potential for dropping farther is now greater because h is greater.

The kinetic energy of a body is the energy due to its motion. The faster a body moves, the more kinetic energy it possesses. When a body stops moving, the kinetic energy is dissipated. This is readily seen in the equation for kinetic energy

where m is the mass of the object and v is its velocity. If v is zero, then KE is also zero.

Energy has been defined as the capacity to do work, and because energy can neither be created nor destroyed, the work done is equal to the kinetic energy acquired, or

This relationship is extremely helpful in explaining the value of “giving” when receiving the impetus of any moving object. In attempting to catch a fast-moving ball, one’s chances of success are improved by increasing the distance used for stopping the ball’s motion. The work of the kinetic energy against the hands depends on the kinetic energy the ball possesses when it hits the hand. Whether that work consists of a large F and a small s or a small F and a large s depends on the technique used by the catcher. As s increases in Fs = 1/2 mv2, the force of the impact F must decrease. The skillful performer knows this and will decrease the possibility of injury and increase the chances of holding onto the ball with the choice of a small F and a large s. Achieving a gradual loss of kinetic energy is likewise advantageous when landing from a jump or a fall (Figure 12.30).

Energy is frequently transformed from kinetic into potential energy or conversely from potential to kinetic energy. The diver who jumps from a platform above the water begins immediately to lose potential energy in proportion to the gain in kinetic energy. At platform height all of the diver’s energy is potential. At zero elevation all of the potential energy has been converted into an equivalent amount of kinetic energy. Barring no conversion of energy to other forms (heat, sound, and so forth) the potential energy at the top equals exactly the kinetic energy at the bottom.

The transformation from one energy state to another is interesting to note in a gymnast swinging on the flying rings. Like a pendulum, the swinger’s energy is transformed from kinetic to potential on every swing. As the suspended body moves upward, the kinetic energy is continuously changed to potential energy, and the higher the swinger goes, the greater the potential becomes for doing work on the downswing. At the top of the swing the conversion to potential energy is complete. Once more as the gymnast begins to fall, kinetic energy again develops and becomes greatest at the bottom of the swing, when the speed is greatest and potential energy is zero. Theoretically the swinging should continue uninterrupted, but because of friction some energy is converted to heat and eventually the “pendulum” will come to rest.

Establishing the mechanical principles that are used in the analysis of linear motion involves first identifying the nature of the forces involved in the motion of interest. As discussed in this chapter, several forces must be considered.

• 1. Weight—applied downward through the center of gravity.
• 2. Propulsive forces—forces generated by muscles, which, in turn, move segments to produce contact forces.
• 3. Ground reaction—applied equal and opposite to contact forces, perpendicular to contact surfaces.
• 4. Friction—applied perpendicular to normal reaction forces along contact surfaces opposite the direction of the propulsive force.
• 5. Buoyancy—acting in opposition to gravity to support the body in a fluid.
• 6. Drag—resistance to motion caused by fluid pressure and turbulence, acting in the direction of fluid flow.
• 7. Lift—acting perpendicular to the direction of fluid flow.

In the long jump example used throughout this text, weight is the resistance force that must be overcome in order for the jumper to leave the ground. The propulsive force is muscle force acting to produce forceful extension of the lower extremities and flexion of the upper extremities. The flexion of the upper extremities produces a downward reaction force. This, along with the downward-directed force produced by lower extremity extension, generates a downward-directed normal force and a backward-directed tangential force. It is the equal and opposite ground reaction force and the friction force that propel the body into the air as a projectile.

The principles that govern the mechanical aspects of a movement performance can be summarized by examining some of the basic concepts involved in the kinetics of linear motion. Some general movement principles based on these concepts might be as follows:

• Inertia An object will resist any change in motion state, based on mass or momentum (mass × velocity).
• Impulse Any change in the motion state depends on the applied force and the time over which it is applied. An impulse is required for any change in momentum.
• Work The amount of work done by the movement system is equal to the amount of force applied and the distance over which it is applied.
• Kinetic energy The amount of work done on a body or object is equal to the kinetic energy produced. Energy can be imparted to move objects or be dissipated to cause objects to stop motion.

To return to the long jump example, the inertia that must be overcome is the mass of the jumper’s body. The impulse generated in the propulsion phase must be one that will give maximum linear velocity to the body for its projectile flight. When this impulse is applied, work is done on the body. The result of this work is the production of the kinetic energy necessary to carry the body forward in flight. The amount of kinetic energy generated will be directly dependent on the impulse applied.

The student must understand that the linear forces that produce motion in the human body are often the result of the rotary motion of the body segments occurring at the joints. For this reason, an analysis of linear mechanics alone cannot fully describe the mechanical attributes of performance. Chapter 13 provides the student with the knowledge to analyze the rotary components of human motion.

• 1. Define the following key terms and give an example of each that is not used in the text:
  • Force
  • Inertia
  • Mass
  • Weight
  • Momentum
  • Impulse
• 2. In the diagram of the biceps muscle in Figure 12.31, find the size of the angle of pull.

• 3. Place a book on a table and stand a small bottle on top of it. Pull the book toward you with a quick jerk. Note the action of the bottle. Stand the bottle on the book again and pull the book across the table with a steady pull. How does the bottle move? Now pull the book and stop it suddenly. Explain the action of the bottle in all three instances.
• 4. Hit a softball with a bat from a tee, from a self-toss, and from a pitch. Which hit goes the farthest? Explain.
• 5. Lie on your back in the water with your arms over your head. Raise your legs by flexing your hips. What happens to your arms and trunk? Explain. Devise another experiment that demonstrates the same principle.
• 6. While both of you have on ice skates or roller skates, stand facing someone whose weight is the same as yours. Push against each other. Observe and compare the distance and velocity each of you moves. Repeat the procedure with someone who weighs considerably more or less than you do. Explain the difference between the two performances.
• 7. Place a long wide board on some rollers (pencils, dowels, or pipes) so that there is very little friction between the board and the floor. Carefully step on the board and attempt to walk normally along it. What happens to you? To the board? Take the board off the rollers and place it on the floor. Walk along it. Explain the reason for the differences in the behavior of the board and your ability to walk along it. The use of spotters is advised for this exercise.
• 8. Determine the coefficient of restitution for each of the following objects dropped from a height of 2 meters onto a wooden floor:
  • a. Tennis ball
  • b. Racquetball
  • c. Golf ball
  • d. Soccer ball
  • e. Baseball
  • Repeat the calculations for the same objects rebounding from concrete, from asphalt tile, from artificial turf, and from a tumbling mat.
• 9. Using a tennis racket or a paddle, impart topspin, backspin, no spin, and sidespin to a ball. Note the effect on the velocity and angle of reflection when the ball rebounds from a horizontal surface, from a vertical surface when struck from above, and from a vertical surface when struck from below.
• 10. Assume a back-lying position in the water with your arms at your sides. Note and explain any shift in body position. Slide your arms from your sides to a side horizontal position, keeping them close to and parallel to the surface of the water. What effect does this have on body position? Continue moving your arms until they are overhead. Next flex the wrists so the hands are out of the water. Finally, flex the legs at the knees. Explain the effect of these last three moves on your position in the water.
• 11. Record the body weight for each of five subjects. Using a stopwatch, record the time necessary for each subject to run up two flights of stairs. Determine the vertical distance from the bottom of the first flight to the top of the second flight. Calculate the work done by each subject and the power of each. Complete the chart below.
  • a. Which subjects did the most work? Why?
  • b. Did those who did the most work have the most power? Explain.
  • c. If power is an important part of an event, how should an athlete train for it?

• 12. Estimate the cross section of your own biceps muscle. Approximately how much force should it be able to exert?
• 13. Approximately how much force should your triceps be able to exert?
• 14. In each of the following movement patterns, identify the nature of the motion for each phase. State the underlying mechanical objectives and describe what must occur to achieve optimum results. Consider carefully the application of linear mechanical principles.
  • Front crawl stroke
  • Bench press
  • High jump