Chapter 13. The Conditions of Rotary Motion

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

• 1. Name, define, and use the following terms properly as they relate to rotary motion: eccentric force, torque, couple, lever, moment of inertia, and angular momentum.
• 2. Solve simple lever and torque problems involving the human body and the implements it uses.
• 3. Demonstrate an understanding of the effective selection of levers by relating speed, range of motion, and mechanical advantage to the properties of given lever systems.
• 4. Explain the analogous kinetic relationships that exist between linear and rotary motion.
• 5. State Newton’s laws of motion as they apply to rotary motion.
• 6. Explain the cause-and-effect relationship between the forces responsible for rotary motion and the objects experiencing the motion.
• 7. Define centripetal and centrifugal force, and explain the relationships that exist between these forces and the factors influencing them.
• 8. Identify the concepts of rotary motion that are critical elements in the successful performance of a selected motor skill.
• 9. Using the concepts that govern rotary motion, perform a mechanical analysis of a selected motor skill.

Eccentric Force

The effect forces have on an object depends on the magnitude, point of application, and direction of each force. When force is applied in line with a freely moving object’s center of gravity, linear motion occurs. When the direction of force is not in line, a combination of rotary and translatory motion is likely to occur. This relationship between force application and direction and the resulting motion are apparent when a book is pushed along a table. Linear motion occurs when sufficient force is applied in line with the book’s center of gravity, and a combination of linear and rotary motion results from a force directed left or right of center. Similarly, an object with a fixed axis, like a door or one of the body’s limbs, rotates when the force is applied off center but does not rotate when the force is in line with the axis of rotation. A force whose direction is not in line with the center of gravity of a freely moving object or the center of rotation of an object with a fixed axis of rotation is called an eccentric force. There must be an eccentric force for rotation to occur. Some examples of the application of eccentric force are shown in Figure 13.1.

Torque

The turning effect of an eccentric force is called torque, or moment of force. The torque about any point equals the product of the force magnitude and the perpendicular distance from the line of force to the axis of rotation. The perpendicular distance between the force vector and the axis is called the moment arm, or torque arm. In this text, the turning effect will be referred to as torque and the perpendicular distance as moment arm. An example of torque is illustrated in Figure 13.2. Neglecting the weight of the arm, the torque or turning effect caused by a 9-N weight held in the hand with the forearm flexed at the elbow to a horizontal position is the product of the weight times its perpendicular distance, or moment arm length, from the elbow. If the moment arm d is 0.3 meters, the amount of torque or downward turning force is 3 newton-meters (N · m). To keep the arm motionless in this position, the elbow flexors must exert an equal and opposite torque or upward-directed turning force of 3 N · m.

Because torque is the product of force and the length of the moment arm, it may be modified by changing either the force or the moment arm. Torque may be increased by increasing the magnitude of the force or by increasing the length of the moment arm. Decreasing either of these factors will produce a decrease in torque. In the preceding example, for instance, if the magnitude of the weight held in the hand were decreased to 6 N, the torque would also decrease, in this case to 2 N · m. Conversely, increasing the weight would increase the torque. If, however, the weight is increased but is moved closer to the axis (the elbow), the torque produced by the heavier weight would be reduced through reducing the length of the moment arm.

It is important to emphasize that the moment arm is the perpendicular distance from the direction of force to the axis of rotation. In the example just given, the moment arm length was the same as the length of the forearm because the horizontal forearm was perpendicular to the vertical direction of the force, and therefore the forearm length was equal to the perpendicular distance from the force direction to the axis. If the arm were in some position other than the horizontal, its length would no longer be equal to the moment arm distance between the force direction and axis. Suppose the forearm’s position is shifted from the horizontal to one of 45 degrees of flexion with the horizontal. Now the force line of the weight is not at right angles to the forearm (Figure 13.3). This means that the length of the moment arm is no longer the length of the forearm because, by definition, the moment arm is the perpendicular distance from the direction or line of force to the axis of motion. The moment arm now is considerably shorter. Using trigonometric relations (see Chapter 10), we find that the moment arm length is 0.2 meters. Consequently, the torque decreases from 3 N · m to 1.9 N · m, and the amount of muscular effort needed to counteract this downward force decreases proportionately.

In the human body, the mass or weight of a segment cannot be altered instantaneously. Therefore, the torque of a segment that is due to gravitational force can be changed only by changing the length of the moment arm in relation to the axis. This is done by moving a body segment so that the force line of the weight is closer to or farther from the axis. The effect of doing this is quite apparent in the stages of the sit-up. Compare the gravitational torque on the trunk when the trunk is just leaving the floor with the torque when the trunk is at a 30-degree angle from the floor (Figure 13.4).

Muscle forces also exert torque on rotating segments. The amount of torque depends on both the magnitude of the muscle force and the moment arm length. Unlike gravitational torque, each of these factors can be altered. The moment arm length depends on the point of insertion of the muscle and the position of the body segment at any point in a given motion. Figure 13.5 shows how the moment arm of the biceps changes for every change of arm position during elbow flexion. The magnitude of the force of the muscle contributing to the torque changes also as its length, tension, and angle of pull change.

With changes in muscle angle of pull, the magnitude of the force effectively producing torque also changes. When muscle force vectors are resolved into their two components, it can be seen that only the rotary component—that which is perpendicular to the mechanical axis of bone—is actually a factor in torque production. The stabilizing component of the muscle force acts along the mechanical axis of the bone through the axis of rotation; thus this component is not an eccentric, or off-center, force and cannot contribute to torque. It has a moment arm length equal to zero. Once a muscle force is resolved into a rotary and a stabilizing component, the moment arm is the distance between the axis of rotation and the point of application of the rotary component of the muscle force. Because the rotary component of any known force can be calculated trigonometrically, given the angle between the line of force and the mechanical axis, torque can be calculated as the product of the rotary component of the force (Fy) times the moment arm along the lever (d) (see Figure 13.9b). In the example used earlier, the angle between the line of force for the weight and the mechanical axis of the bone is 45 degrees. Using the angle and the 9-N magnitude of the force, the rotary component of the force is calculated to be 6.4 N. When multiplied by the 0.3-meter length of the moment arm, the torque is 1.9 N · m, the same value found when calculating the length of the true moment arm in Figure 13.3. Either method of finding torque is satisfactory.

The important thing to remember from this discussion is that the turning effect, or torque, produced on objects free to rotate depends on two factors: the magnitude of the applied force and the perpendicular distance from the force vector to the axis. Change in turning effect occurs when either of these factors is altered. Shortening the moment arm distance or decreasing the force magnitude will decrease the torque. Lengthening the moment arm or increasing the force magnitude will increase the turning effect and consequently the necessary effort to resist it.

Summation of Torques

The sum of two or more torques may result in no motion, linear motion, or rotary motion. When equal parallel eccentric forces are applied in the same direction on opposite sides of the center of rotation of an object, either no motion or linear motion will occur. An example of no motion is two children balanced on a seesaw. When the equal parallel forces are adequate to overcome the resistance of the object, linear motion will occur. This could be the situation with paddlers in a canoe, with one paddling on the port side and the other on the starboard side. When equal and opposite parallel forces are exerted on opposite sides of the axis of rotation, rotary motion will occur. The effect of equal parallel forces acting in opposite directions is called a couple, or force couple. An example of this type of movement is steering a car when both hands are used on opposite sides of the wheel (Figure 13.6). Another example is that of turning in a rowboat by simultaneously pulling on the handle of one oar while pushing on the handle of the other.

There are many examples in the human body in which two muscles rotate a bone by acting cooperatively as a force couple; for instance, trapezius II and the lower portion of serratus anterior are a force couple, rotating the scapula upward. Trapezius II and IV, acting on the two extremities of the spine of the scapula, similarly serve as a force couple to help rotate the scapula upward, as do the oblique abdominals in acting to rotate the trunk (Figure 13.7).

Principle of Torques

It is common for more than one torque to act on a body at any given time. The resultant effect of these torques is expressed in the principle of the summation of torques, which states that the resultant torques of a force system must be equal to the sum of the torques of the individual forces of the system about the same point. Because torques are vector quantities, the summation must consider both magnitude and direction. The direction of rotation is expressed as either clockwise or counterclockwise direction. Clockwise torques are usually labeled as negative, and counterclockwise torques as positive. Their signs must be accounted for when they are summed.

When the sum of counterclockwise torques equals the sum of clockwise torques, no turning will occur. This idea may also be expressed by stating that rotation will be absent when the sum of the torques of all forces about any point or axis equals zero. When the sum of clockwise torques does not equal the sum of the counterclockwise torques, the resultant turning effect (torque) will be the difference between the two opposing forces and in the direction of the larger.

In Figure 13.8 all the forces are applied perpendicular to the lever. Force A is 1.5 m from the axis and B is 3 m away. Both A and B are clockwise forces. Force C is a counterclockwise force that also is 3 m away from the axis. The sum of these torques produces a resultanttorque of 22.5 N · m in a clockwise direction about the center of rotation.

When an eccentric force is applied at an angle to the lever other than 90 degrees, the moment arm is not along the lever line. Before the torque can be found, the length of the moment arm must be determined. This can be done using trigonometric functions. An example of this procedure is shown in the following problem:

What muscular force F pulling at an angle of 25 degrees would be required to keep the abducted arm in a position of 20 degrees with the horizontal? The muscle inserts 10 cm from the shoulder joint. The arm weighs 50 N and its center of gravity is located 30 cm from the shoulder. A 45-N weight is held in the hand 60 cm from the shoulder joint. Note: For the arm to be held stationary, the sum of the counterclockwise torques must equal the clockwise torques (∑CCW = ∑CW). (See Figure 13.9 for solution.)

A simple machine that operates according to the principle of torques is the lever. A lever is a rigid bar that can rotate about a fixed point when a force is applied to overcome a resistance. When they move, levers serve two important functions. They are used either to overcome a resistance larger than the magnitude of the effort applied or to increase the speed and range of motion through which a resistance can be moved. When there is no motion, the torque produced by the effort and the torque produced by the resistance are equal and the lever system is said to be balanced.

External Levers

We use levers every day of our lives. In the kitchen the old-fashioned manual can opener, punch can opener, and bottle opener or lid pry are all examples of simple levers. Likewise, in the workshop or about the house and grounds the crowbar, pry bar, and wheelbarrow are levers. What do these implements have in common? Even though their shapes vary and their structures differ in complexity, each of these is a rigid bar. When a force is applied to one of them, it turns about a fixed point known as a fulcrum, and it overcomes a resistance that may, in some cases, be no more than its own weight. All the levers mentioned are designed to enable use of a relatively small force to overcome a relatively large resistance. In levers such as these, the range of movement of the resistance is relatively slight, whereas the range of movement of the effort is large. The crowbar, for instance, lifts a rock only a few centimeters, whereas the handle moves through a much larger distance. In other words, the force to overcome a considerable resistance is gained at the expense of range of motion.

The striking implements used in sports are levers that do the opposite of this. The golf club, for instance, is used to gain range of motion at the expense of force. The length of the shaft enables the club head to travel through a large arc of motion, but it is used to overcome the relatively slight resistance of the weight of the club itself. Tennis and squash rackets, baseball bats, hockey sticks, and fencing foils are other examples of levers used to gain distance at the expense of force. These levers do not save the strength of the user, as do the household levers mentioned, but they increase the user’s range and speed of movement. By striking a ball with a racket, for instance, the striker can impart more speed to it and send it a greater distance than could be done by striking it with the hand. This is because the head of the racket travels a greater distance, and therefore at a greater speed, than the hand alone is able to do.

A still different kind of lever is seen in the seesaw on the playground, the scales in the laboratory, and the mobile hanging in the art gallery. These levers gain neither force nor distance but provide for a balancing of weights. If the loads are equal, they will balance each other when they are equidistant from the fulcrum. If they are unequal, they will balance only if the heavier load is placed closer to the fulcrum. There is an exact relationship between the magnitude of the weights and their respective distances from the fulcrum so that when the torques are summed, the resultant is zero.

This kind of lever may also be used to balance a force and a load. The skier carrying skis over one shoulder, balanced by a hand holding the other end, is an example. The amount of effort exerted by the hand depends on what portion of the ski is in contact with the shoulder. If the weight of the skis is evenly distributed on each side of the shoulder, the hand needs to exert little or no effort.

Anatomical Levers

But where in the human body do we have anything even faintly resembling a punch can opener, hockey stick, or seesaw? When we recognize each of these levers as a rigid bar that turns about a fulcrum when force is applied to it, it is then apparent that nearly every bone in the skeleton can be looked upon as a lever. The bone itself serves as the rigid bar, the joint as the fulcrum, and the contracting muscles as the force. A large segment of the body, such as the trunk, the upper extremity, or the lower extremity, can likewise act as a single lever if it is used as a rigid unit. When the entire arm is raised sideward, for instance, it is acting as a simple lever. The center of motion in the shoulder joint serves as the fulcrum. The effort is supplied mainly by the deltoid muscle and the resistance in this instance is the weight of the arm itself. The point at which the effort is applied to the lever is approximately the point at which the deltoid inserts into the humerus, and the point at which the resistance is applied is the center of gravity of the extended arm. If a weight is held in the hand, the resistance point is then the center of gravity of the arm plus its load and is located closer to the hand than before. If a relatively heavy weight is lifted, for practical purposes the weight of the arm may be disregarded and the resistance point may be assumed to be the center of the object’s point of contact with the hand.

Anatomical levers do not necessarily resemble bars. The skull, shoulder blade, and vertebrae are notable exceptions to the definition. The resistance point, also, may be difficult to identify, especially in the seesaw type of lever. It is not always easy to tell whether the resistance is the weight of the lever itself or is the resistance afforded by antagonistic muscles and fasciae that are put on a steadily increasing stretch as the movement progresses. For instance, when the head is turned easily to the left, the resistance point may be regarded as the center of gravity of the head. We can only guess at the approximate location of this. If the turning of the head is resisted by the pressure of someone’s hand against the left side of the chin, the resistance point is the midpoint of the contact area. If the head is turned without external resistance but is forced to the limit of motion, resistance to the movement is afforded by the antagonistic rotators, and possibly by the ligaments and fasciae. The resistance point in such a case is the midpoint of the area over which these resisting forces act on the head. As in the first example, the location of this point can only be estimated.

Lever Arms

Lever arms are commonly defined as the portion of the lever between the fulcrum and the force points. The effort arm is the distance between the fulcrum and the effort point, and the resistance arm is the distance between the fulcrum and the resistance point. These definitions are valid, however, only when the effort and resistance are applied at right angles to the lever. When the effort and resistance are applied at some angle other than 90 degrees to the lever, these definitions are inaccurate. A better definition of a lever arm that applies regardless of the angle of force application is one synonymous to that of a moment arm. Indeed, a lever arm is a moment arm. In a lever the perpendicular distance between the fulcrum and line of force of the effort is the effort arm (EA). Similarly, the perpendicular distance between the fulcrum and the line of resistance force is the resistance arm (RA). In Figure 13.16, EA is the effort arm and RA is the resistance arm.

Classification of Levers

Three points on the lever have been identified: the point about which it turns, the point at which effort is applied to it, and the point at which the resistance to its movement is applied or concentrated. Because there are three points, there are three possible arrangements of these points. Any one of the three may be situated between the other two. The arrangement of these three points provides the basis for the classification of levers.

First-Class Levers

In a first-class lever, the axis lies between the effort and the resistance (Figure 13.10a). The mechanical advantage of the first-class lever is balance. The first-class lever also can be used to magnify the effects of the effort or to increase the speed and range of motion on the basis of the relative lengths of the effort moment arm (EA) and the resistance moment arm (RA). Examples of external first-class levers are the seesaw, balance scale, crowbar, scissors, and automobile jack (Figure 13.11).

The head, tipping forward and backward, is a good example of a first-class lever in the body (Figure 13.11). The effort is supplied by the extensors of the head, notably the splenius and upper portions of the semispinalis, and is applied to the head at the base of the skull. The resistance to the movement is furnished by the weight of the head itself, together with the tension of the antagonistic muscles and fasciae, as the limit of motion is approached.

Another first-class lever is seen in the foot when it is not being used for weight bearing, as when the knees are crossed in the sitting position. As the soleus pulls upward on the heel, the foot plantar flexes at the ankle joint where the fulcrum is situated. The resistance seems to be provided by the tonus of the dorsiflexor muscles.

The forearm is another example of a first-class lever when it is being extended by the triceps muscle against a resistance (Figure 13.12). The fulcrum is situated at the elbow joint, the effort is applied at the olecranon process, and the resistance point is located at the forearm’s center of gravity when no external resistance is present, and at the hand when the latter is pushing against an external resistance. Internal resistance does not appear to be a factor in this movement.

Second-Class Levers

In a second-class lever, the resistance lies between the axis and the effort (Figure 13.10b). In this lever, the EA is always longer than the RA. This arrangement has the advantage of magnifying the effects of the effort so that it takes less force to move the resistance. Its disadvantage is that range of motion is sacrificed. Examples of external second-class levers are the wheelbarrow, door (effort applied at the knob), and nutcracker. Whether or not there are any second-class levers in the body seems to be a controversial matter among anatomists and kinesiologists. Some claim that when the foot is being plantar flexed in a weight-bearing position, as when rising on the toes, it is a second-class lever. The fulcrum is said to be at the point of contact with the ground, the effort point at the heel where the tendon of Achilles attaches, and the resistance at the ankle joint where the weight of the body is transferred to the foot.

There are numerous second-class levers involving body segments if those actions where gravity supplies the effort for the movement and muscles control or resist the movement through eccentric contraction are included. The forearm in slow downward extension is an example. The fulcrum is at the elbow joint, the effort may be considered to be applied at the center of gravity of the forearm, and the resistance at the insertion of the elbow flexor, which for this example is considered to be the brachialis. The brachialis through eccentric contraction resists the downward-directed weight of the arm (Figure 13.13). Another example is the slow lowering of the leg at the knee from an extended to a flexed position. The fulcrum is at the knee joint, the effort is the lower leg applied at its center of gravity, and the resistance is the eccentric contraction of the quadriceps at its point of insertion on the tibial tuberosity.

Third-Class Levers

In a third-class lever, the effort lies between the axis and the resistance (see Figure 13.10c). In this lever, RA is always the longer of the two moment arms. This means that it will take greater effort to overcome a given resistance. However, a very small motion by the effort will produce a large motion in the resistance. The advantage of the third-class lever is speed and range of motion at the expense of force.

There are not many examples of external third-class levers. One example is a screen door with a spring closing. There are, however, many examples of anatomical third-class levers. Most body segments that are moved by muscular effort fall into this class. The forearm is a good example of a third-class lever when it is being flexed by the biceps and the brachialis (Figure 13.14). Another third-class lever is seen in the example cited earlier—namely, the arm as it is abducted by the deltoid muscle (Figure 13.15).

The Principle of Levers

A lever of any class will balance when the product of the effort and the effort arm equals the product of the resistance and the resistance arm. This is known as the principle of levers. It enables us to calculate the amount of effort needed to balance a known resistance by means of a known lever or to calculate the point at which to place the fulcrum to balance a known resistance with a given effort. If any three of the four values are known, the remaining one can be calculated by using the following equation:

This equation restates a principle already studied under the summation of torques. When the torques in one direction equal the torques in the opposite direction, equilibrium exists. E × EA is a torque, as is R × RA. When applied to Figure 13.16, this means that it will be possible to balance the weight in the hand when the product of the weight and the perpendicular distance from its vector line of action to the fulcrum (R × RA) equals the product of the muscle effort and the perpendicular distance from its vector line of action to the fulcrum (E × EA).

The lever equation also shows the importance of lever arm lengths in determining the amount of effort needed to balance a given resistance. If EA is lengthened while R × RA remains constant, the amount of effort needed to balance the lever must decrease. If, on the other hand, RA is increased, the amount of effort must increase. For this reason, second-class levers require less effort to balance or move heavier resistances, whereas third-class levers require more effort than the resistance they must balance or move (see Figures 13.14 and 13.15). The first-class lever may have either a longer EA or RA, depending on the placement of the fulcrum (see Figures 13.11 and 13.12).

A lever whose effort arm is the longer of the two moment arms, whether it be a first- or second-class lever, is said to favor force. Less effort is required to overcome a resistance with this kind of lever than it would take to overcome the same resistance without the aid of the lever. This advantage is gained at the expense of speed and range of movement. The heavier resistance will always move through a smaller distance than the effort (Figure 13.17). Conversely, a lever whose resistance arm is longer, whether it be a first- or third-class lever, is said to favor speed and distance. However, more effort is required to move it than would be the case if the relative lengths of the effort and resistance arms were reversed. Furthermore, an object of negligible weight can be moved a greater distance and more rapidly by this kind of lever than it could without the aid of the lever.

Relation of Speed to Range in Movements of Levers

The reader will have noticed that in the foregoing discussion of levers, the terms speed and range were usually linked together. There is a reason for this. In angular movements, speed and range are interdependent. For instance, if two third-class levers of different lengths each move through a 40-degree angle at the same angular velocity, the tip of the longer lever will be traveling a greater distance or range than the tip of the shorter lever. Because it covers this distance in the same amount of time it takes the tip of the shorter lever to travel the shorter distance, the former must be moving faster than the latter. This is easily seen if the shorter lever is superimposed on the longer, as in Figure 13.18. Here the shorter lever AB has been superimposed on the longer lever AC. The levers are moving from the horizontal position to the diagonal one. Because point C travels to its new position C′ in the same time it takes B to travel to B′, point C must obviously be moving farther and faster than point B. This is the same relationship that was discussed in Chapter 11. The linear kinematics at the end of a lever vary with the length of the radius. Thus, the greater the length of the lever, the greater the linear displacement, velocity, and acceleration that occur at the distal end point of the lever.

The effort arms in the skeletal lever systems of the human body are determined by the point of muscle attachment, usually close to the proximal end of the lever, and the muscle angle of pull. Because of these two factors, with few exceptions the effort arm in skeletal levers is shorter than the resistance arm. Thus anatomical levers tend to favor speed and range of movement at the expense of effort. Examples of this preference are the throwing of a baseball or the kicking of a soccer ball. The hand in the one case and the foot in the other travel through a relatively long distance at considerable speed. Both movements require strong muscular action, even though the balls are relatively light in weight. This type of leverage is the reverse of the kind usually seen in mechanical implements such as the crowbar and the automobile jack, both of which are used to move heavy weights a relatively short distance. Sports implements that may serve as levers in themselves or as artificial extensions of the human arm also favor speed and range of movement. Frequently the sport implement, arm, and a large part of the rest of the body act together as a system of levers. In hitting a baseball, for instance, the trunk forms one lever, the upper arms another, the forearms another, and the hands and bat still another. This use of multiple leverage builds up speed at the tip of the bat because the greater this speed, the greater the force that can be imparted to the ball.

Selection of Levers

Skill in motor performance depends on the effective selection and use of levers, both internal and external. Long golf clubs are selected for distance, and shorter clubs for accuracy at close range. Heavy baseball bats are chosen by those with the strength to swing them, whereas children are often taught tennis with short-handled rackets. In most instances external levers are designed for a specific purpose and are selected accordingly. The levers of the human body, on the other hand, are not designed for one action or purpose. Body parts or segments may be held in numerous positions for any given joint action and thus provide a great variety of lever arrangements. Consequently, skill depends on the right choice of joint axis, joint action, and moment arm length. In all three instances in Figure 13.19, the ball in the hand is carried forward by virtue of the counterclockwise (as seen from above) rotation of the pelvis that takes place at the left hip joint. The pelvis carries the trunk with it, but no movement occurs in any joint other than the left hip. In all three positions the resistance moment arm for the action is the perpendicular distance between the hip axis and the line of action of the resistance (the weight of the ball). However, the resistance moment arm is a different length in each situation. It is smallest in (a) and largest in (c). Consequently, when the angular velocity is the same in each instance, the linear velocity of the ball is largest in (c) and smallest in (a) (see Figure 13.19). Similarly, if flexion of the shoulder is the joint action, the position of the arm that will provide the greatest linear velocity to the object in the hand at the moment of release is one in which the elbow is extended and the arm is perpendicular to the desired direction of flight (Figure 13.20).

It is not always desirable to choose the longest lever, however. The positioning of body parts to form short levers enhances the angular velocity of the levers while sacrificing linear speed and range of motion at the end of the lever. And although forming the longest possible lever can increase speed and range of motion at the end of the lever, the strength needed to maintain the desired angular velocity increases as the lever lengthens. This is why the tennis player may flex the elbow in the forehand drive or the batter may “choke up” when swinging a heavy bat. As the magnitude of the resistance to be moved increases, the more the performer may want to shorten the lever. Moving very large weights produces very high levels of external torque. This torque can be reduced by decreasing the RA. One should not attempt to move heavy loads by using a long lever and high movement speeds.

Mechanical Advantage of Levers

Machines are judged good if they are efficient, poor if they are inefficient. How is the efficiency of a machine measured? Because the machines used in industry and in the workshop are usually for the purpose of magnifying force, it is customary to measure their efficiency in terms of their mechanical advantage—in other words, their ability to magnify force. Another way to express this ability is to state the “output” of the machine relative to its “input.” In levers this is the ratio between the effort applied to the lever and the resistance overcome by the lever. It may be expressed in terms of the following equation: Mechanical advantage equals the ratio of the resistance overcome to the effort applied, or, simply,

Because the balanced lever equation may also be expressed as

the mechanical advantage may be expressed in terms of the ratio of the effort arm to the resistance arm. Hence, if

it also holds that

When a muscle is said to have poor leverage, it means that it has poor mechanical advantage. In other words, the effort arm of the lever on which it is acting is short compared with the resistance arm.

There are four functions of a simple machine:

• Balance two or more forces
• Favor force production
• Favor speed and range of motion
• Change the direction of the applied force

Depending on the relative lengths of EA and RA, a lever can achieve one or more of these functions.

Identification and Analysis of Levers

Figures 13.11, 13.12, 13.13, 13.14, and 13.15 depict certain anatomical levers and their mechanical counterparts. These should enable the student to understand the principle of leverage as it applies in the human body and to see how the anatomical levers compare with the levers of everyday life. For each of these levers and for every lever that the student observes, these questions should be answered: (1) What are the locations of the fulcrum, effort point, and resistance point? (2) At what angle is the effort applied to the lever? (3) At what angle is the resistance applied to the lever? (4) What is the effort arm of the lever? (5) What is the resistance arm of the lever? (6) What are the relative lengths of the effort and resistance arms? (7) What kind of movement does this lever favor? (8) What is the mechanical advantage? (9) What class of lever is this?

When applied to rotary motion, Newton’s laws of motion may be stated as follows:

• 1. A body continues in a state of rest or uniform rotation about its axis unless acted upon by an external torque.
• 2. The acceleration of a rotating body is directly proportional to the torque causing it, is in the same direction as the torque, and is inversely proportional to the moment of inertia of the body.
• 3. When a torque is applied by one body to another, the second body will exert an equal and opposite torque on the first.

The laws themselves are analogous to those stated for linear motion. To comprehend their application to rotary motion, however, an understanding of the rotary equivalents of linear mass, momentum, and force is necessary. These parallel relationships between linear motion and rotary motion quantities become quite apparent when presented side by side, as in Table 13.1.

Moment of Inertia

In the preceding discussion of Newton’s laws, the ability of a body to resist change in its state of motion was identified as inertia. The inertia of a body with respect to linear motion was shown to be directly proportional to the mass of the object. The heavier an object is, the greater the force that is required to start it moving and to stop it. This is also true in rotary motion. Once they are set in motion, spinning bodies tend to keep spinning. As with linear motion, the amount of force needed to start or stop a spinning object is related to its mass: the heavier the object, the greater the effort that is required to start or stop its rotation. However, an additional factor needs to be considered. As an example, the hammer thrower knows that it takes more effort to start the weight moving in a circular fashion at the end of the wire than it does to start it moving in a linear fashion. It is also true that once the hammer is moving in a circular fashion, it is more difficult to slow it down than if it were moving in a straight line. Therefore, it seems that the hammer’s inertia is greater when moving in an angular pattern than when moving in a linear path. Because the mass of the hammer does not change, this increase in inertia must be due to some other element. In the same way that the torque causing rotary motion depends on the magnitude of force and the distance from the line of action of the force to the the axis, the inertia of a rotary body is affected by both the mass and the distance between the mass and the axis of rotation. A ball rotating on the end of a long string is harder to stop and start than one on a short string. As the distance between the axis and the mass increase, the inertia increases. Thus the size of the angular inertia, called the moment of inertia, I, depends on both the quantity of the rotating mass and its distribution around the axis of rotation. The exact nature of this relationship is shown in the equation for the moment of inertia:

where m is the mass of a particle in the rotating body and r is the perpendicular distance between the mass particle and the axis of rotation. Thus the moment of inertia of an object is the sum of the mass of each of the particles multiplied by the square of the distance to the axis (Figure 13.21). If the mass of an object is concentrated close to the axis of rotation, the object is easier to turn because the radius for each particle is less, thus making I less. Conversely, if the mass is concentrated farther away from the axis, I becomes greater and the rotating body will require more force to start or stop it.

In the human body the mass distribution may be altered by changing the body position, thereby changing the moment of inertia. A runner is able to move the recovery leg forward more rapidly when it is in a flexed position than when it is extended. The mass of the leg is the same in both instances, but it is closer to the axis when the leg is flexed and the moment of inertia is less.

Using a method similar to finding the center of gravity of the body by the segmental method (see Chapter 14, Segmental Method), Hay (1993) reported the moment of inertia of the human body in some common rotating positions. In the tuck position, I about the body’s center of gravity was 3.5 kg · m2; in the pike position it was 6.5 kg · m2 and in the layout, 15.0 kg · m2. The moment of inertia about the hands of the giant swing position was 83.0 kg · m2. These figures verify that inertia to rotation increases as the mass is distributed farther away from the axis.

When one is interested in the moment of inertia of a specific body segment, it is usually taken about the center of the joint serving as the axis, although other axes may be chosen. If another axis is selected, such as the center of gravity of the segment, I will change accordingly because values of r will be different in the equation I = ∑mr2.

Acceleration of Rotating Bodies

The equation for Newton’s second law of motion when applied to linear motion was shown to be F = ma. The rotational equivalent is T = Iα—that is, rotary force, or torque (T), is equal to the product of the moment of inertia (I) of the rotating object and the angular acceleration (α). The rotary equivalent of linear force is torque, the rotary equivalent of mass is the moment of inertia, and the rotary equivalent of linear acceleration is angular acceleration. With a little transposing of the equation it becomes apparent how the rotary equivalent of Newton’s second law of motion states that the change in rotary velocity (α) is directly proportional to the torque (T) and inversely proportional to the moment of inertia (I)—that is, α = T/I.

When a net unbalanced torque is applied to a mass, this mass will begin to accelerate angularly according to the equation

Where T is torque, I is the massmoment of inertia, and α is the angular acceleration. Because angular acceleration is the time rate of change of angular velocity (rad/sec), this equation can be rewritten as

In other words, the product of a torque and the time over which this torque acts will produce a change in angular velocity of the mass, provided that the distribution of the mass is constant. Called angular impulse, it is the effect of a torque, applied over a given time, on the change of angular velocity of an object with a certain mass and mass distribution. Consider the exercise where you lay on your back and your partner stands above you. You bring your feet upward, and your partner throws your feet downward. During this downward motion, your abdominals are resisting the anterior pelvic girdle rotation. If your partner throws your legs at the same rate each time, then we know the change in angular momentum, specifically, from that initial angular velocity to 0 rad/sec. To stop the downward rotation of your legs quickly will take a relatively large torque (very strong abdominals). However, if you use the whole range of motion to stop the legs before they hit the floor, less torque is required.

Angular Momentum

Momentum is a measure of the force needed to start or stop motion. Objects undergoing angular motion have momentum in the same fashion as objects engaged in linear motion. Linear momentum is the product of mass and velocity. Angular momentum is the product of the angular equivalents of mass and velocity—that is, the moment of inertia I and angular velocity ω.

The momentum of a skater skating in a straight line is mv. The momentum of the skater spinningis I ω.

Angular momentum can be increased or decreased by increasing or decreasing either the angular velocity or the moment of inertia. A heavy bat swung with the same velocity as a light bat has greater angular momentum. However, increasing the velocity of the lighter bat could produce an angular momentum equal to or greater than that of the heavier bat. Similarly, the angular momentum of a kicking leg can be increased by increasing its angular velocity. Conversely, it will decrease if the leg is flexed at the knee (thus decreasing this moment of inertia) while keeping the velocity constant.

Conservation of Angular Momentum

Newton’s first law as applied to rotary motion can also be stated in terms related to momentum. Worded thus, it states that the total angular momentum of a rotating body will remain constant unless acted upon by external torques. This is why the law is also known as the law of conservation of angular momentum. Like linear momentum, angular momentum is conserved. Once a spinning body is in motion, its angular momentum will not change, provided, of course, that no outside forces are introduced. Divers, gymnasts, dancers, and other rotating performers all make use of the conservation of angular momentum to control the speed of bodily spin. A skater starts to spin with the arms outstretched. After spinning slowly in this position, the skater brings the arms in close to the body and immediately begins to spin rapidly. The increase in angular velocity occurs with no effort on the skater’s part. By bringing the arms in closer to the axis of rotation, the moment of inertia is decreased about that axis. Because the angular momentum must remain constant, a decrease in I produces an increase in ω. Increasing I by stretching the arms and perhaps a leg out to the side will again slow down the skater. A diver performing a tuck somersault can control the speed of rotation by changing the tightness of the tuck, thus changing the moment of inertia. If the diver thinks the entry into the water will be spoiled because of too rapid a rotation, the I can be increased and the angular velocity decreased proportionately by “opening up” or loosening the tuck (Figure 13.22).

Action and Reaction

As previously explained, the force for angular motion is torque. For every torque exerted by one body on a second body, there is another torque equal in magnitude and opposite in direction exerted by the second body on the first. This is Newton’s third law of motion as applied to angular movement. A force that causes a change in angular momentum must have an equal and opposite force creating an equal and opposite momentum change. In other words, angular momentum is conserved. When one jumps straight up from a trampoline bed with arms stretched upward and then swings the legs forward and up, the body will jackknife at the hips. The torque acting on the legs results in an equal and opposite torque acting on the rest of the body. A change in angular velocity is observed in both body segments, although that in the trunk segment is less because I is greater. Another jump upward followed by a sharp flexion of the wrists only will also produce an equal and opposite torque on the rest of the body. However, in this case the change in angular velocity of the arms and trunk resulting in forward-upward motion will be barely visible. I for the arms and trunk is so large compared with I for the hands that the opposite rotation will be very slight. The angular velocity of the two moving parts is inversely proportional to their moments of inertia about the axis of motion (the wrist joint in the example just given). Axes for such actions may be in any plane, but the reactions must be in the same or parallel planes. The arms swinging horizontally across in front of the trunk will produce an opposite action of the rest of the body in the transverse plane (Figure 13.23). Movement of one arm downward in the frontal plane produces an equal and opposite action of the rest of the body in the frontal plane. This is the movement that may be used to regain balance on a beam or tightrope. If one falls to the left, a downward movement of the left arm produces a reaction of the rest of the body to the right and hopefully prevents a fall.

The equation describing this action–reaction relationship may be written as

Any changes in the moments of inertia of the two bodies (I1 and I2) or of their respective angular velocities (ω1 and ω2) will produce equal and opposite momentum changes so that I1(ω1f − ω1i) and I2(ω2f − ω2i) continue to be equal and opposite.

Sometimes action and reaction in specific body parts is not desired and should be controlled. This can be done by substitution or absorption of the undesired action. In walking and running, the rotation of the pelvis and legs about a vertical axis produces an undesired reaction of the upper trunk in the opposite direction if it is not compensated in some fashion. To prevent this, the arms move in opposition to the legs to “absorb” the reaction by producing a countertwist that cancels out the reaction of the body to the leg action. The faster one moves or the more massive the legs, the more vigorous must be the arm action. The arms are used in the same fashion to prevent undesired trunk reaction to leg action in hurdling, diving, and jumping.

Transfer of Momentum

Because of the law of conservation of momentum, angular momentum may be transferred from one body or body part to another as the total angular momentum remains unaltered. In the suspended balls shown in Figure 13.24, the sudden checking of motion in ball A produces a comparable action in ball E. Similarly, checking the simultaneous motion in A and B produces equal momentum in D and E. Examples of transfer of angular momentum in movement patterns are numerous. As a back diver leaves the board, the arms are swung upward until the motion is checked by the limitation of range of motion in the shoulder joint. The checked momentum transfers to the body and increases its angular momentum, thus helping turn the body in the air. In the racing dive, the angular momentum of the diver’s arms is also transferred to the body as the diver stops the upward movement of the arms in the forward reach, and in a jump twist the dancer transfers twisting momentum from the upper arms and trunk to the legs as the feet leave the ground. In each of these instances the body is in contact with the ground while the momentum is developing. Otherwise, an equal and opposite reaction, rather than a transfer, would occur. The effect that the transfer of angular momentum has on unsupported bodies is discussed in Chapter 21.

A very common example of the transfer of angular momentum is the series of sequential segmental rotation that occurs in throwing, kicking, and striking actions. In an efficient performer, a throwing motion involves a series of rotations of progressively smaller body segments. The initial angular momentum generated by the application of torque to the fairly massive leg and trunk segments is transferred to the much lighter upper arm. The decrease in moment of inertia between the trunk and the arm produces an increase in the angular velocity of the upper arm. By the time this angular momentum has been transferred to the hand, angular velocity will be at a maximum.

Angular momentum also can be transferred or converted into linear momentum, or the reverse can occur. The angular momentum of the hammer becomes linear momentum when it is released. Conversely, a ball that hits the ground with no spin but rebounds spinning has had some of its linear momentum converted to angular momentum. Similarly, a long jumper’s linear momentum most often is partially converted to angular momentum at the takeoff. Those who advanced the somersault technique for long jumping thought that as long as the angular momentum existed, it should be used to advantage. Unfortunately there were too many disadvantages, and the style is no longer legal. The hitch kick technique that most good jumpers currently use provides the necessary “opposite” reaction to minimize the undesired forward rotation “action” of the trunk while putting the feet in an optimum landing position.

A weight whirling around on the end of a string is engaged in linear motion of the circular type [see Translatory (Linear) Movement], while the string itself is undergoing rotary motion about an axis. If one lets go of the string, both the string and weight fly off and experience linear motion. This is expected because, according to Newton’s first law, a moving body left alone travels uniformly in a straight line. To make that body leave the straight path requires application of an additional force. In the case of the weight on the string, the force has to be applied in a manner that will cause the weight to change direction and move in a circular path about the axis. This force is called centripetal force. It is a constant center-seeking force that acts to move an object tangent (at right angles) to the direction in which it is moving at any instant, thus causing it to move in a circular path.

The centripetal force acting on the weight is an external force that is applied by the finger through the string to the weight. The finger, in turn, must have a force pulling on it because, according to Newton’s third law, if one body exerts a force on the second body, that body will reciprocate with an equal and opposite force on the first body. This outward-pulling force is called centrifugal (center-fleeing) force and is felt in the tension exerted on the finger by the string attached to the circling object. Centrifugal force is equal in magnitude to centripetal force. If centripetal force ceases, there is no longer an inward pull on the object and the latter then flies off at a tangent to the direction in which it was moving at the instant the force stopped. Without centripetal force, there will be no centrifugal reaction, and the object will once more travel uniformly in a straight line.

Even though the speed of a whirling object is constant, its direction changes continually—that is, there is a constant change in velocity. This, in effect, is the same as saying that the weight is uniformly accelerating because, by definition, acceleration is the rate of change of velocity. The amount of acceleration that a whirling object possesses increases with the speed with which it is orbiting and decreases with the distance from the axis to the object. This relationship is represented in equation form as a = v2/r. The substitution of this value for acceleration in the equation obtained from Newton’s second law of motion, F = ma, results in the equation for centripetal force:

The equation for centrifugal force is the same, because it is equal in magnitude and opposite in direction to centripetal force.

From the equation it can be seen that the amount of centripetal force necessary to keep an object moving in a circular path is proportional to its mass and the square of its velocity and inversely proportional to its radius. Doubling the mass doubles the centripetal force, whereas doubling the radius decreases it by half. Changes in velocity have an even greater impact and are more dramatic. Doubling the velocity along the curve increases the centripetal force fourfold, and decreasing the velocity by one-half decreases to one-fourth the force needed to keep the object in orbit.

When runners or bicyclists negotiate corners with any appreciable speed, they lean into the curve. If they did not, they would tend to topple outward. In the same manner that centripetal force was exerted by the finger on the weight at the end of the string, the ground exerts centripetal force on the body of the runner through the feet and on the body of the cyclist through the bicycle wheel.

Centrifugal force, acting equal and opposite, tends to pull the body outward, back into a linear path. This centrifugal force produces a torque on the body in an outward direction, with the foot or the bicycle wheel acting as the axis. To maintain equilibrium against this torque, the runner or the bicyclist must lean inward, using the body weight to produce an equal and opposite inward torque. In Figure 13.25a, this outward-rotating torque Fcy is shown as the product of the centripetal–centrifugal force and the distance between the center of gravity of the runner and the axis of rotation about the foot. When one leans in, an opposite reacting torque is created, also about the feet, and is equal to the product of the weight and the perpendicular distance from the direction of weight to the axis. In Figure 13.25b, the torque that is due to the lean (Wx) balances the opposite acting torque that is due to centripetal force, so Fcy = Wx. This relationship demonstrates that as Fc increases, the amount of lean will also have to increase. As this occurs, x increases, y decreases, and W, of course, remains the same. It also shows that the lower the center of gravity of a body negotiating a curve, the smaller will be the outward-rotating force because the moment arm y will be less.

Where speeds are too great or radii too small for participants to negotiate curves successfully, the track may be banked. The amount of banking is determined for the particular radius and the average expected velocity so that the performer may remain perpendicular to the surface while rounding the curve. As might be expected, the greater the anticipated velocity for negotiating the curve, or the smaller the radius of the turn, the more the track must be banked.

Centripetal–centrifugal forces are evident in many instances in which the body imparts force to an external object. Overarm, sidearm, or underarm throwing patterns all involve the application of force to the object to be thrown in order to keep it moving in a circular path for all or part of the time before it is released. As soon as the object is released, its inertia causes it to move in a straight line. In the hammer throw (Figure 13.26), the thrower transmits force to the head of the hammer along the flexible wire. This is centripetal force directed inward toward the axis of rotation. As the performer increases the speed of rotation, the velocity of the hammer increases, and the increased centripetal force the thrower must exert is proportional to the square of the velocity. The outward-pulling centrifugal force increases accordingly on the thrower, who must adjust by leaning back or be pulled off balance. A hammer thrower needs considerable strength to resist centrifugal force. This is also true with discus throwers who generate considerable centripetal force during the turn prior to delivery, or baseball pitchers whose elbows are subject to great forces because of the tremendous speed with which the forearm is whipped forward in the pitch. Another instance in which centripetal force is an important factor is in all swinging activities in gymnastics.

As most motions of the human body involve rotation of a segment about a joint, any mechanical analysis of movement requires an analysis of the nature of the rotary forces, or torques, involved. In examining and evaluating performance of any movement skill, both internal and external torques must be considered. Internal torques will be generated by applied muscle forces as identified in the anatomical portion of the skill analysis. External torques now must be identified as they are produced by the external forces identified in the analysis of linear motion.

The principles that govern rotary motion are based primarily on the production of torque and the results of torque production. Some general principles of rotary motion might be stated as follows:

• Torque The torque about any point equals the product of the force magnitude and its perpendicular distance from the line of action. The force rotating the body segments depends on the related joint torques, which are governed by the muscle forces and the locations of their attachments.
• Summation of Torques The resultant torque of a force system equals the sum of the individual torques. Therefore, if a maximum resultant is desired, all body segments that have the capability must contribute to their maximum. Torques may be summed either simultaneously or sequentially.
• Conservation of Angular Momentum In the absence of any outside torque, an object undergoing rotary motion will continue to rotate. Because angular momentum is conserved, alterations in the moment of inertia will produce changes in the angular velocity and vice versa. In other words, a decrease in the moment of inertia will produce an increase in angular velocity, with the opposite also being true.
• Principle of Levers Any lever will balance when the torque of the effort equals the torque produced by the resistance. The amount of effort required may be calculated using this principle if the resistance torque and the length of the effort arm are known.
• Transfer of Angular Momentum Angular momentum may be transferred from one body part to another when total angular momentum remains unchanged. In many motions, this transfer of momentum moves from more massive segments to less massive segments. When this occurs, the velocity of rotation increases with each transfer based on the conservation of angular momentum.

In the continuing example of the long jump, for instance, the force applied to generate the propulsive impulse will be the result of the joint torques generated. To be most effective, these joint torques must be summed, meaning that each body segment that has the capability must contribute to its maximum. Maximum thrust of the body in the jump depends on maximum resultant torques developed at the individual joints.

• 1. Define the following key terms and give an example of each that is not used in the text.
  • Torque
  • Moment arm
  • Lever
  • Mechanical advantage
  • Moment of inertia
  • Angular momentum
  • Centripetal force
  • Centrifugal force
• 2. For each of the following anatomical levers, estimate the approximate length of the effort arm, that is, the perpendicular distance. Refer to anatomical illustrations or to a muscle mannequin to help you estimate the angle of pull of the muscles. Unless otherwise stated, assume that the segments are in their normal resting position. (Hint: Use the mechanical axis of the segment for the lever.)
  • a. The forearm lever with the biceps providing the effort.
  • b. The upper arm lever with the middle deltoid providing the effort.
  • c. The upper arm lever in the side-horizontal position with the entire pectoralis major providing the effort. (Hint: Which view would show you the angle of pull for this movement, facing the subject or looking down from above?)
  • d. Same lever as in c, with the anterior deltoid providing the effort.
• 3. Referring to the diagrams in Figure 13.5, assume that a weight is suspended from the wrist. Find the resistance arm of the forearm lever in positions b, d, and e. Consider that the scale for these diagrams is 2 cm = 25 cm.
• 4. Assume that a muscle having a force of 300 N is pulling at an angle of 75 degrees. Find the components of the force (a) by the trigonometric method and (b) by the graphic method.
• 5. Assuming that the weight used in Experience 3 is 8 kg, determine the rotary and non-rotary components of the weight (a) by the trigonometric method and (b) by the graphic method.
• 6. Referring to Figure 7.21, make a tracing of the femur and the adductor longus muscle. Draw a straight line to represent the mechanical axis of the femur and another to represent the muscle’s line of pull (connecting the midpoints of the proximal and distal attachments).
  • a. Using a protractor, measure the angle of pull (i.e., the angle facing the hip joint) formed by the intersection of the muscle’s line of pull with the bone’s mechanical axis.
  • b. Measure the effort moment arm of the lever (i.e., the perpendicular distance from the fulcrum [center of hip joint] to the muscle’s line of pull). To convert this to a realistic figure use the scale 1 cm = 0.25 m.
  • c. Assuming the muscle’s total force to be 900 N, calculate the moment of force (i.e., the torque) of the lever.
  • d. Calculate the rotary and the stabilizing components of force.
• 7. Place a pole across the back of a chair, hang a pail or basket containing a 3-kg weight on one end, and hold the other end of the pole. Now adjust the pole so that it becomes increasingly difficult to balance the weight of the pail with one hand. Stop when you reach the point where you are barely able to lift the pail by pushing down on the opposite end of the pole. Have an assistant measure the effort arm and the resistance arm of the lever. Without shifting the position of your hand, draw the pole toward you until you can easily lift the pail by pushing down on the opposite end of the pole with one finger. Again, have an assistant measure the effort and resistance arms of the lever. What do you conclude concerning the relative length of the two arms? In which case does the pail move through the greater distance? What do you conclude about the relationship between the effort required to move an object by leverage and the distance that the object is moved?
• 8. Balance a pole across the back of a chair, hanging a 3-kg weight at each end. Let one of these represent the effort and the other the resistance. The effort arm and the resistance arm should be exactly equal if the pole is symmetrical. Now add 3 more kilograms at another point on the resistance end and adjust the pole until it balances. Measure the effort and resistance moment arms. What do you conclude about:
  • a. The relationship between the resistance and the resistance arm? Between the effort and the effort arm?
  • b. The relationship between the resistance and the resistance arm, given a changing resistance and a constant effort?
• 9. Compute the amount of effort necessary to lift a resistance of 20 N with a 2-meter, first-class lever whose fulcrum is located 0.5 meters from the point at which the weight is attached and 1.5 meters from the point at which the effort is applied. Assume that both the effort and the resistance are applied at right angles to the lever.
• 10. Where would the fulcrum have to be located in the above-mentioned 2-meter, first-class lever if only 5 N of effort were available to balance the 20-N weight? Determine this by experiment. Check your answer by the algebraic method.
• 11. For each of the anatomical levers listed, identify the class of lever represented; identify the fulcrum, effort point, and resistance point; and name the kind of movement favored by this type of lever.
  • a. Leg being flexed at knee by hamstrings
  • b. Leg being extended at knee by quadriceps femoris
  • c. Pelvis being tilted to right by left quadriceps lumborum
  • d. Clavicle being elevated by trapezius I
  • e. Lower extremity being abducted by gluteus medius
  • f. From supine lying position, lower extremities being raised by hip flexors
  • g. From supine lying position, trunk being raised by hip flexors
• 12. Perform the following paired movements, noting the difference in the effort needed. Diagram and explain the reasons for the difference.
  • a. A push-up from the toes and a push-up from the knees
  • b. A sit-up with hands on the thighs and a sit-up with a 3-kg weight held behind the head
  • c. A 3-kg weight held in the hand with the arm outstretched horizontally sideward and a 3-kg weight suspended from the elbow crotch when the arm is outstretched sideward
• 13. Explain the difference in effort needed to hold the body levers in Experience 12a, b, c, at a 10-degree angle with the horizontal compared with holding them at a 30-degree angle and a 45-degree angle.
• 14. What muscular effort E, pulling at an angle of 80 degrees, would be required to keep the lower leg in a position of 10 degrees with the horizontal? The muscle inserts 8 cm from the knee joint. The lower leg and foot weigh 35 N and the center of gravity is located 18 cm from the knee joint. A 50-N weight is hung from the ankle 33 cm from the knee joint.
• 15. A gymnast weighing 56 kg attempts a handstand on the balance beam. Her center of gravity is 1 meter above the beam. What is the moment of force tending to pull her over if her center of gravity moves 3 cm ahead of the center of her support? 15 cm ahead?
• 16. Perform a vertical jump with and without the use of your arms. Have someone compare the height of the two jumps by noting the level of the top of your head. A chart with numbered horizontal lines on the wall behind you will help in scoring. The observer’s eyes should be at the level of the top of your head at the peak of the jump. Repeat several times and compare your results with those of others doing the same experiment. Explain the results.
• 17. Stand on a frictionless turntable or sit on a swivel stool. Hold a 3-kg weight in each hand and hold your arms out to the side. Have a partner spin you around. Alternately bring your hands into your shoulders and move them out to the side. Explain the changes in your angular velocity.
• 18. Stand on a frictionless turntable or sit on a swivel stool. Determine how you can rotate yourself moving just your arms. Explain why your arm motions make the rotation possible.
• 19. Firmly tape equal-length chains of small rubber bands to a golf ball and a table tennis ball.
  • a. Holding the end of the elastic, swing the golf ball around in a circle. Repeat with the lighter ball, swinging it at the same speed as the heavier ball.
  • b. Swing the table tennis ball at different speeds.
  • c. Swing the table tennis ball with one-half the radius length; one-fourth the radius length. What does this experiment tell about the relationship of mass, velocity, and radius to centripetal force? How can you tell whether the centripetal force is increasing or decreasing?
• 20. 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 rotary mechanical principles.
  • Fencing lunge
  • Bowling delivery
  • Shot put
  • Tennis lob