- Identify the angular analogues of mass, force, momentum, and impulse.
- Explain why changes in the configuration of a rotating airborne body can produce changes in the body’s angular velocity.
- Identify and provide examples of the angular analogues of Newton’s laws of motion.
- Define centripetal force, and explain where and how it acts.
- Solve quantitative problems relating to the factors that cause or modify angular motion.

Why do sprinters run with more swing phase flexion at the knee than do distance runners? Why do dancers and ice skaters spin more rapidly when their arms are brought in close to the body? How do cats always land on their feet? In this chapter, we explore more concepts pertaining to angularkinetics, from the perspective of the similarities and differences between linear and angular kinetic quantities.

Inertia is a body’s tendency to resist acceleration (see Chapter 3). Although inertia itself is a concept rather than a quantity that can be measured in units, a body’s inertia is directly proportional to its mass (Figure 14-1). According to Newton’s second law, the greater a body’s mass, the greater its resistance to linear acceleration. Therefore, mass is a body’s inertial characteristic for considerations relative to linear motion.

Resistance to angular acceleration is also a function of a body’s mass. The greater the mass, the greater the resistance to angular acceleration. However, the relative ease or difficulty of initiating or halting angular motion depends on an additional factor: the distribution of mass with respect to the axis of rotation.

Consider the baseball bats shown in Figure 14-2. Suppose a player warming up in the on-deck circle adds a weight ring to the bat he is swinging. Will the relative ease of swinging the bat be greater with the weight positioned near the striking end of the bat or with the weight near the bat’s grip? Similarly, is it easier to swing a bat held by the grip (the normal hand position) or a bat turned around and held by the barrel?

Experimentation with a baseball bat or some similar object makes it apparent that the more closely concentrated the mass is to the axis of rotation, the easier it is to swing the object. Conversely, the more mass is positioned away from the axis of rotation, the more difficult it is to initiate (or stop) angular motion. Resistance to angular acceleration, therefore, depends not only on the amount of mass possessed by an object but also on the distribution of that mass with respect to the axis of rotation. The inertial property for angular motion must therefore incorporate both factors.

The inertial property for angular motion is **moment
of inertia**, represented as I. Every body is composed of particles
of mass, each with its own particular distance from a given axis
of rotation. The moment of inertia for a single particle of mass
may be represented as the following:

In this formula, m is the particle’s mass and r is the particle’s radius of rotation. The moment of inertia of an entire body is the sum of the moments of inertia of all the mass particles the object contains (Figure 14-3):

The distribution of mass with respect to the axis of rotation is more important than the total amount of body mass in determining resistance to angular acceleration, because r is squared. Since r is the distance between a given particle and an axis of rotation, values of r change as the axis of rotation changes. Thus, when a player grips a baseball bat, “choking up” on the bat reduces the bat’s moment of inertia with respect to the axis of rotation at the player’s wrists, and concomitantly increases the relative ease of swinging the bat. Little League baseball players often unknowingly make use of this concept when swinging bats that are longer and heavier than they can effectively handle. Interestingly, research shows that when baseball players warm up with a weighted bat (with a larger moment of inertia than a regular bat) post-warm-up swing velocity is actually reduced (21).

Within the human body, the distribution of mass with respect to an axis of rotation can dramatically influence the relative ease or difficulty of moving the body limbs. For example, during gait, the distribution of a given leg’s mass, and therefore its moment of inertia with respect to the primary axis of rotation at the hip, depends largely on the angle present at the knee. In sprinting, maximum angular acceleration of the legs is desired, and considerably more flexion at the knee is present during the swing phase than while running at slower speeds. This greatly reduces the moment of inertia of the leg with respect to the hip, thus reducing resistance to hip flexion. Runners who have leg morphology involving mass distribution closer to the hip, with more massive thighs and slimmer lower legs than others, have a smaller moment of inertia of the leg with respect to the hip. This is an anthropometric characteristic that contributes to running economy (3). During walking, in which minimal angular acceleration of the legs is required, flexion at the knee during the swing phase remains relatively small, and the leg’s moment of inertia with respect to the hip is relatively large.

Modern-day golf irons are commonly constructed with the heads bottom weighted, perimeter weighted, or heel and toe weighted. These manipulations of the amount of mass and the distribution of mass within the head of the club are designed to increase club head inertia, thus reducing the tendency of the club to rotate about the shaft during an off-center hit. Research findings indicate that perimeter-weighted club heads perform best for eccentric ball contacts outside the club head center of gravity (CG), with a simple blade head club superior for contacts below the club head CG (18). The most consistent performance, however, was exhibited by a toe-and-bottom-weighted club head, which was second best for all eccentric hits (18). A golfer’s individual preference, feel, and experience should ultimately determine selection of club type.

Assessing moment of inertia for a body with respect to an axis by measuring the distance of each particle of body mass from an axis of rotation and then applying the formula is obviously impractical. In practice, mathematical procedures are used to calculate moment of inertia for bodies of regular geometric shapes and known dimensions. Because the human body is composed of segments that are of irregular shapes and heterogeneous mass distributions, either experimental procedures or mathematical models are used to approximate moment-of-inertia values for individual body segments and for the body as a whole in different positions. Moment of inertia for the human body and its segments has been approximated by using average measurements from cadaver studies, measuring the acceleration of a swinging limb, employing photogrammetric methods, and applying mathematical modeling (11).

Once moment of inertia for a body of known mass has been assessed, the value may be characterized using the following formula:

In this formula, I is moment of inertia with respect to an axis,
m is total body mass, and k is a distance known as the **radius of gyration**. The radius of
gyration represents the object’s mass distribution with
respect to a given axis of rotation. It is the distance from the
axis of rotation to a point at which the mass of the body can theoretically
be concentrated without altering the inertial characteristics of
the rotating body. This point is *not* the
same as the segmental CG (Figure 14-4). Since the radius
of gyration is based on r2 for individual particles, it
is always longer than the radius of rotation, the distance to the
segmental CG.

The length of the radius of gyration changes as the axis of rotation changes. As mentioned earlier, it is easier to swing a baseball bat when the bat is grasped by the barrel end rather than by the bat’s grip. When the bat is held by the barrel, k is much shorter than when the bat is held properly, since more mass is positioned close to the axis of rotation. Likewise, the radius of gyration for a body segment such as the forearm is greater with respect to the wrist than with respect to the elbow.

The radius of gyration is a useful index of moment of inertia when a given body’s resistance to rotation with respect to different axes is discussed. Units of moment of inertia parallel the formula definition of the quantity, and therefore consist of units of mass multiplied by units of length squared(kg · m2)

Moment of inertia can only be defined with respect to a specific axis of rotation. The axis of rotation for a body segment in sagittal and frontal plane motions is typically an axis passing through the center of a body segment’s proximal joint. When a segment rotates around its own longitudinal axis, its moment of inertia is quite different from its moment of inertia during flexion and extension or abduction and adduction, because its mass distribution, and therefore its moment of inertia, is markedly different with respect to this axis of rotation. Figure 14-5 illustrates the difference in the lengths of the radii of gyration for the forearm with respect to the transverse and longitudinal axes of rotation.

The moment of inertia of the human body as a whole is also different
with respect to different axes. When the entire human body rotates
free of support, it moves around one of three **principal
axes**: the transverse (or frontal), the anteroposterior (or
sagittal), or the longitudinal (or vertical) axis, each of which
passes through the total body CG. Moment of inertia with respect
to one of these axes is known as a **principal
moment of inertia**. Figure 14-6 shows quantitative
estimates of principal moments of inertia for the human body in
several different positions. When the body assumes a tucked position
during a somersault, its principal moment of inertia (and resistance
to angular motion) about the transverse axis is clearly less than
when the body is in anatomical position. Divers performing a somersaulting
dive undergo changes in principal moment of inertia about the transverse
axis on the order of 15 kg · m2 to 6.5
kg · m2as the body goes from a layout
position to a pike position (9).

As children grow from childhood through adolescence and into adulthood, developmental changes result in changing proportions of body segment lengths, masses, and radii of gyration, all affecting segment moments of inertia (10). Segment moments of inertia affect resistance to angular rotation, and therefore performance capability, in sports such as gymnastics and diving. Because of smaller moments of inertia, smaller gymnasts have an advantage in performing skills involving whole-body rotations, despite the fact that larger gymnasts may have greater strength and be able to generate more power (1). Several prominent female gymnasts who achieved world-class status during early adolescence faded from the public view before reaching age 20 because of declines in their performance capabilities generally attributed to changes in body proportions with growth. Substantial changes in principal moments of inertia for the body segments occur with age, and large interindividual differences exist in the growth patterns of these principal moments of inertia (11). According to Jensen (10), the best predictor of moment of inertia values among children is the product of body mass and body height squared, (m) (ht)2, rather than age.

Since moment of inertia is the inertial property for rotational
movement, it is an important component of other angular kinetic
quantities. As discussed in Chapter 12, the quantity of
motion that an object possesses is referred to as its *momentum.* Linear momentum is the product
of the linear inertial property (mass) and linear velocity. The quantity
of angular motion that a body possesses is likewise known as **angular momentum**. Angular momentum,
represented as H, is the product of the angular inertial property
(moment of inertia) and angular velocity:

Three factors affect the magnitude of a body’s angular momentum: (a) its mass (m), (b) the distribution of that mass with respect to the axis of rotation (k), and (c) the angular velocity of the body (ω). If a body has no angular velocity, it has no angular momentum. As mass or angular velocity increases, angular momentum increases proportionally. The factor that most dramatically influences angular momentum is the distribution of mass with respect to the axis of rotation, because angular momentum is proportional to the square of the radius of gyration (see Sample Problem 14.1). Units of angular momentum result from multiplying units of mass, units of length squared, and units of angular velocity, which yields kg · m2/s.

For a multisegmented object such as the human body, angular momentum about a given axis of rotation is the sum of the angular momenta of the individual body segments. During an airborne somersault, the angular momentum of a single segment, such as the lower leg, with respect to the principal axis of rotation passing through the total body CG consists of two components: the local term and the remote term. The local term is based on the segment’s angular momentum about its own segmental CG, and the remote term represents the segment’s angular momentum about the total body CG. Angular momentum for this segment about a principal axis is the sum of the local term and the remote term:

In the local term, Is is the segment’s moment of inertia and ωs is the segment’s angular velocity, both with respect to a transverse axis through the segment’s own CG. In the remote term, m is the segment’s mass, r is the distance between the total body and segmental CGs, and ωg is the angular velocity of the segmental CG about the principal transverse axis (Figure 14-7). The sum of the angular momenta of all the body segments about a principal axis yields the total-body angular momentum about that axis.

During takeoff from a springboard or platform, a competitive diver must attain sufficient linear momentum to reach the necessary height (and safe distance from the board or platform) and sufficient angular momentum to perform the required number of rotations. For multiple-rotation, nontwisting platform dives, the angular momentum generated at takeoff increases as the rotational requirements of the dive increase (8). Angular momentum values as high as 66 kg · m2/s and 70 kg · m2/s have been reported for multiple gold medalist Greg Louganis during his back two-and-a-half and forward three-and-a-half springboard dives, respectively (15). When a twist is also incorporated into a somersaulting dive, the angular momentum required is further increased. Inclusion of a twist during forward one-and-a-half springboard dives is associated with increased angular momentum at takeoff of 6–19% (20). Adding a somersault while rotating in a tuck rather than a pike position also requires a small increase in angular momentum (16).

Whenever gravity is the only acting external force, angular momentum is conserved. For angular motion, the principle of conservation of momentum may be stated as follows:

- The total angular momentum of a given system remains constant in the absence of external torques.

Gravitational force acting at a body’s CG produces no torque because d⊥ equals zero and so it creates no change in angular momentum.

The principle of conservation of angular momentum is particularly useful in the mechanical analysis of diving, trampolining, and gymnastics events in which the human body undergoes controlled rotations while airborne. In a one-and-a-half front somersault dive, the diver leaves the springboard with a fixed amount of angular momentum. According to the principle of conservation of angular momentum, the amount of angular momentum present at the instant of takeoff remains constant throughout the dive. As the diver goes from an extended layout position into a tuck, the radius of gyration is decreased, thus reducing the body’s principal moment of inertia about the transverse axis. Because angular momentum remains constant, a compensatory increase in angular velocity must accompany the decrease in moment of inertia (Figure 14-8). The tighter the diver’s tuck, the greater the angular velocity. Once the somersault is completed, the diver extends to a full layout position, thereby increasing total-body moment of inertia with respect to the axis of rotation. Again, because angular momentum remains constant, an equivalent decrease in angular velocity occurs. For the diver to appear to enter the water perfectly vertically, minimal angular velocity is desirable. Sample Problem 14.2 quantitatively illustrates this example.

Other examples of conservation of angular momentum occur when an airborne performer has a total-body angular momentum of zero and a forceful movement such as a jump pass or volleyball spike is executed. When a volleyball player performs a spike, moving the hitting arm with a high angular velocity and a large angular momentum, there is a compensatory rotation of the lower body, producing an equal amount of angular momentum in the opposite direction (Figure 14-9). The moment of inertia of the two legs with respect to the hips is much greater than that of the spiking arm with respect to the shoulder. The angular velocity of the legs generated to counter the angular momentum of the swinging arm is therefore much less than the angular velocity of the spiking arm.

Consider a rotating 10 kg body for which k = 0.2 m and ω = 3 rad/s. What is the effect on the body’s angular momentum if the mass doubles? The radius of gyration doubles? The angular velocity doubles?

The body’s original angular momentum is the following:

With mass doubled:

With k doubled:

With ω doubled:

A 60 kg diver is positioned so that his radius of gyration is 0.5 m as he leaves the board with an angular velocity of 4 rad/s. What is the diver’s angular velocity when he assumes a tuck position, altering his radius of gyration to 0.25 m?

To find ω, calculate the amount of angular momentum that the diver possesses when he leaves the board, since angular momentum remains constant during the airborne phase of the dive:

Position 1:

Use this constant value for angular momentum to determine ω when k = 0.25 m:

Position 2:

Although angular momentum remains constant in the absence of external torques, transferring angular velocity at least partially from one principal axis of rotation to another is possible. This occurs when a diver changes from a primarily somersaulting rotation to one that is primarily twisting, and vice versa. An airborne performer’s angular velocity vector does not necessarily occur in the same direction as the angular momentum vector. It is possible for a body’s somersaulting angular momentum and its twisting angular momentum to be altered in midair, though the vector sum of the two (the total angular momentum) remains constant in magnitude and direction.

Researchers have observed several procedures for changing the
total-body axis of rotation. Asymmetrical arm movements and rotation
of the hips (termed *hula movement*)
can tilt the axis of rotation out of the original plane of motion
(Figure 14-10). The less-often-used hula movement can produce
tilting of the principal axis of rotation when the body is somersaulting
in a piked position. These asymmetrical movements can be used to
generate twist and to eliminate twist (24). The results
of a study on twisting somersault performances executed from a trampoline
indicate that less-skilled performers tend to rush their asymmetrical
movements, detracting from performance quality (19).

Even when total-body angular momentum is zero, generating a twist
in midair is possible using skillful manipulation of a body composed
of at least two segments. Prompted by the observation that a domestic
cat seems always to land on its feet no matter what position it
falls from, scientists have studied this apparent contradiction
of the principle of conservation of angular momentum (7).
Gymnasts and divers can use this procedure, referred to as *cat rotation,* without violating the
conservation of angular momentum.

Cat rotation is basically a two-phase process. It is accomplished most effectively when the two body segments are in a 90˚ pike position, so that the radius of gyration of one segment is maximal with respect to the longitudinal axis of the other segment (Figure 14-11). The first phase consists of the internally generated rotation of Segment 1 around its longitudinal axis. Because angular momentum is conserved, there is a compensatory rotation of Segment 2 in the opposite direction around the longitudinal axis of Segment 1. However, the resulting rotation is of a relatively small velocity, because k for Segment 2 is relatively large with respect to Axis 1. The second phase of the process consists of rotation of Segment 2 around its longitudinal axis in the same direction originally taken by Segment 1. Accompanying this motion is a compensatory rotation of Segment 1 in the opposite direction around Axis 2. Again, angular velocity is relatively small, because k for Segment 1 is relatively large with respect to Axis 2. Using this procedure, a skilled diver can initiate a twist in midair and turn through as much as 450˚ (7). Cat rotation is performed around the longitudinal axes of the two major body segments. It is easier to initiate rotation about the longitudinal principal axis than about either the transverse or the anteroposterior principal axes, because total-body moment of inertia with respect to the longitudinal axis is much smaller than the total-body moments of inertia with respect to the other two axes.

When an external torque does act, it changes the amount of angular momentum present in a system predictably. Just as with changes in linear momentum, changes in angular momentum depend not only on the magnitude and direction of acting external torques but also on the length of the time interval over which each torque acts:

When an **angular impulse** acts on
a system, the result is a change in the total angular momentum of
the system. The impulse–momentum relationship for angular
quantities may be expressed as the following:

As before, the symbols T, t, H, I, and ω represent torque, time, angular momentum, moment of inertia, and angular velocity, respectively, and subscripts 1 and 2 denote initial and second or final points in time. Because angular impulse is the product of torque and time, significant changes in an object’s angular momentum may result from the action of a large torque over a small time interval or from the action of a small torque over a large time interval. Since torque is the product of a force’s magnitude and the perpendicular distance to the axis of rotation, both of these factors affect angular impulse. The effect of angular impulse on angular momentum is shown in Sample Problem 14.3.

In the throwing events in track and field, the object is to maximize
the angular impulse exerted on an implement before release, to maximize
its momentum and the ultimate horizontal displacement following
release. As discussed in Chapter 11, linear velocity is
directly related to angular velocity, with the radius of rotation
serving as the factor of proportionality. As long as the moment
of inertia (mk2) of a rotating body remains constant, increased
angular momentum translates directly to increased linear momentum
when the body is projected. This concept is particularly evident
in the hammer throw, in which the athlete first swings the hammer
two or three times around the body with the feet planted, and then
executes the next three or four whole-body turns while facing the
hammer before release. Some hammer throwers perform the first one
or two of the whole body turns with the trunk in slight flexion
(called *countering with the hips*),
thereby enabling a farther reach with the hands (Figure 14-12).
This tactic increases the radius of rotation, and thus the moment
of inertia of the hammer with respect to the axis of rotation, so
that if angular velocity is not reduced, the angular momentum of the
thrower/hammer system is increased. For this strategy,
the final turns are completed with the entire body leaning away
from the hammer, or *countering with the
shoulders.* Researchers have suggested that although the ability
to lean forward throughout the turns should increase the angular
momentum imparted to the hammer, a natural tendency to protect against
excessive spinal stresses or shoulder strength limitations may prevent
the thrower from accomplishing this technique modification (6).

The angular momentum required for the total body rotations executed during aerial skills is primarily derived from the angular impulse created by the reaction force of the support surface during takeoff. During back dives performed from a platform, the major angularimpulse is produced during the final weighting of the platform, when the diver comes out of a crouched position through extension at the hip, knee, and ankle joints and executes a vigorous arm swing simultaneously (17). The vertical component of the platform reaction force, acting in front of the diver’s CG, creates most of the backward angular momentum required (Figure 14-13).

On a springboard, the position of the fulcrum with respect to the tip of the board can usually be adjusted and can influence performance. Setting the fulcrum farther back from the tip of the board results in greater downward board tip vertical velocity at the beginning of takeoff, which allows the diver more time in contact with the board to generate angular momentum and increased vertical velocity going into the dive (12). Concomitant disadvantages, however, include the requirement of increased hurdle flight duration and the necessity of reversing downward motion from a position of greater flexion at the knees (12). In an optimum reverse dive from a springboard, peak knee extension torque is generated just prior to maximum springboard depression, so that the diver exerts force against a stiffer board (22).

The motions of the body segments during takeoff determine the magnitude and direction of the reaction force generating linear and angular impulses. During both platform and springboard dives, the rotation of the arms at takeoff generally contributes more to angular momentum than the motion of any other segment (8, 15). Highly skilled divers perform the arm swing with the arms fully extended, thus maximizing the moment of inertia of the arms and the angular momentum generated. Less-skilled divers often must use flexion at the elbow to reduce the moment of inertia of the arms about the shoulders so that arm swing can be completed during the time available (15). In contrast to the takeoff during a dive, during the takeoff for aerial somersaults performed from the floor in gymnastics, it is forceful extension of the legs that contributes the most to angular momentum (5). Optimizing performance of aerial somersaults requires generating high linear and angular velocities during the approach, as well as precise timings of body segment motions (13). During performance of the Hecht vault in gymnastics, the angular momentum produced by the forceful push-off from the horse is one of the key variables found by researchers to influence judges’ scores (23).

Angular impulse produced through the support surface reaction
force is also essential for performance of the *tour
jeté,* a dance movement that consists of a jump accompanied
by a 180˚ turn, with the dancer landing on the foot opposite
the takeoff foot. When the movement is performed properly, the dancer
appears to rise straight up and then rotate about the principal
vertical axis in the air. In reality, the jump must be executed
so that a reaction torque around the dancer’s vertical
axis is generated by the floor. The extended leg at the initiation
of the jump creates a relatively large moment of inertia relative
to the axis of rotation, thereby resulting in a relatively low total-body
angular velocity. At the peak of the jump, the dancer’s
legs simultaneously cross the axis of rotation and the arms simultaneously
come together overhead, close to the axis of rotation. These movements
dramatically reduce moment of inertia, thus increasing angular velocity
(14).

Similarly, when a skater performs a double or triple axel in figure skating, angular momentum is generated by the skater’s movements and changes in total-body moment of inertia prior to takeoff. Over half of the angular momentum for a double axel is generated during the preparatory glide on one skate going into the jump (2). Most of this angular momentum is contributed by motion of the free leg, which is extended somewhat horizontally to increase total-body moment of inertia around the skater’s vertical axis (2). As the skater becomes airborne, both legs are extended vertically, and the arms are tightly crossed to minimize moment of inertia around the vertical axis and thereby maximize rotational velocity.

In the performance of the tennis serve, although the server does not typically become airborne, the angular momentum generated plays an important role. Angular momentum is produced by the movements of the trunk, arms, and legs, with a transfer of momentum from the extending lower extremity and rotating trunk to the racket arm, and finally to the racket (3).

What average amount of force must be applied by the elbow flexors inserting at an average perpendicular distance of 1.5 cm from the axis of rotation at the elbow over a period of 0.3 s to stop the motion of the 3.5 kg arm swinging with an angular velocity of 5 rad/s when k = 20 cm?

The impulse–momentum relationship for angular motion can be used.

Table 14-1 presents linear and angular kinetic quantities in a parallel format. With the many parallels between linear and angular motion, it is not surprising that Newton’s laws of motion may also be stated in terms of angular motion. It is necessary to remember that torque and moment of inertia are the angular equivalents of force and mass in substituting terms.

Linear | Angular |
---|---|

mass (m) | moment of inertia (I) |

force (f) | torque (T) |

momentum (m) | angular momentum (H) |

impulse (Ft) | angular impulse (Tt) |

The angular version of the first law of motion may be stated as follows:

- A rotating body will maintain a state of constant rotational motion unless acted on by an external torque.

In the analysis of human movement in which mass remains constant throughout, this angular analogue forms the underlying basis for the principle of conservation of angular momentum. Because angular velocity may change to compensate for changes in moment of inertia resulting from alterations in the radius of gyration, the quantity that remains constant in the absence of external torque is angular momentum.

In angular terms, Newton’s second law may be stated algebraically and in words as the following:

- A net torque produces angular acceleration of a body that is directly proportional to the magnitude of the torque, in the same direction as the torque, and inversely proportional to the body’s moment of inertia.

In accordance with Newton’s second law for angular motion, the angular acceleration of the forearm is directly proportional to the magnitude of the net torque at the elbow and in the direction (flexion) of the net torque at the elbow. The greater the moment of inertia is with respect to the axis of rotation at the elbow, the smaller is the resulting angular acceleration (see Sample Problem 14.4).

The knee extensors insert on the tibia at an angle of 308 at a distance of 3 cm from the axis of rotation at the knee. How much force must the knee extensors exert to produce an angular acceleration at the knee of 1 rad/s2, given a mass of the lower leg and foot of 4.5 kg and k = 23 cm?

The angular analogue of Newton’s second law of motion may be used to solve the problem:

The law of reaction may be stated in angular form as the following:

- For every torque exerted by one body on another, there is an equal and opposite torque exerted by the second body on the first.

When a baseball player forcefully swings a bat, rotating the mass of the upper body, a torque is created around the player’s longitudinal axis. If the batter’s feet are not firmly planted, the lower body tends to rotate around the longitudinal axis in the opposite direction. However, since the feet usually are planted, the torque generated by the upper body is translated to the ground, where the earth generates a torque of equal magnitude and opposite direction on the cleats of the batter’s shoes.

Bodies undergoing rotary motion around a fixed axis are also
subject to a linearforce. When an object attached to a line is
whirled around in a circular path and then released, the object
flies off on a path that forms a tangent to the circular path it
was following at the point at which it was released, since this
is the direction it was traveling in at the point of release (Figure
14-14). **Centripetal force** prevents
the rotating body from leaving its circular path while rotation
occurs around a fixed axis. The direction of a centripetal force
is always toward the center of rotation; this is the reason it is also
known as *center-seeking force.* Centripetal
force produces the radial component of the acceleration of a body
traveling on a curved path (see Chapter 11). The following
formula quantifies the magnitude of a centripetal force in terms
of the tangential linear velocity of the rotating body:

In this formula, Fc is centripetal force, m is mass, v is the tangential linear velocity of the rotating body at a given point in time, and r is the radius of rotation. Centripetal force may also be defined in terms of angular velocity:

As is evident from both equations, the speed of rotation is the most influential factor on the magnitude of centripetal force, because centripetal force is proportional to the square of velocity or angular velocity.

When a cyclist rounds a curve, the ground exerts centripetal force on the tires of the cycle. The forces acting on the cycle/cyclist system are weight, friction, and the ground reaction force (Figure 14-15). The horizontal component of the ground reaction force and laterally directed friction provide the centripetal force, which also creates a torque about the cycle/cyclist CG. To prevent rotation toward the outside of the curve, the cyclist must lean to the inside of the curve so that the moment arm of the system’s weight relative to the contact point with the ground is large enough to produce an oppositely directed torque of equal magnitude. In the absence of leaning into the curve, the cyclist would have to reduce speed to reduce the magnitude of the ground reaction force, in order to prevent loss of balance.

When rounding a corner in an automobile, there is a sensation
of being pushed in the direction of the outside of the curve. What
is felt has been referred to as *centrifugal
force*. What is actually occurring, however, is that in accordance
with Newton’s first law, the body’s inertia tends
to cause it to continue traveling on a straight, rather than a curved,
path. The car seat, the seat belt, and possibly the car door provide
a reaction force that changes the direction of body motion. “Centrifugal
force,” then, is a fictitious force that might more appropriately
be described as the absence of centripetal force acting on an object.

Whereas a body’s resistance to linear acceleration is
proportional to its mass, resistance to angular acceleration is
related to both mass and the distribution of mass with respect to
the axis of rotation. Resistance to angular acceleration is known
as *moment of inertia,* a quantity that
incorporates both the amount of mass and its distribution relative
to the center of rotation.

Just as linear momentum is the product of the linear inertial property (mass) and linear velocity, angular momentum is the product of moment of inertia and angular velocity. In the absence of external torques, angular momentum is conserved. An airborne human performer can alter total-body angular velocity by manipulating moment of inertia through changes in body configuration relative to the principal axis around which rotation is occurring. Skilled performers can also alter the axis of rotation and initiate rotation when no angular momentum is present while airborne. The principle of conservation of angular momentum is based on the angular version of Newton’s first law of motion. The second and third laws of motion may also be expressed in angular terms by substituting moment of inertia for mass, torque for force, and angular acceleration for linear acceleration.

1. If you had to design a model of the human body composed entirely of regular geometric solids, which solid shapes would you choose? Using a straightedge, sketch a model of the human body that incorporates the solid shapes you have selected.

2. Construct a table displaying common units of measure for both linear and angular quantities of the inertial property, momentum, and impulse.

3. Skilled performance of a number of sport skills is characterized by “follow-through.” Explain the value of “follow-through” in terms of the concepts discussed in this chapter.

4. Explain the reason the product of body mass and body height squared is a good predictor of body moment of inertia in children.

5. A 1.1 kg racquet has a moment of inertia about a grip axis of rotation of 0.4 kg · m2. What is its radius of gyration? (Answer: 0.6 m)

6. How much angular impulse must be supplied by the hamstrings to bring a leg swinging at 8 rad/s to a stop, given that the leg’s moment of inertia is 0.7 kg · m2? (Answer: 5.6 mg · m2/s).

7. Given the following principal transverse axis moments of inertia and angular velocities, calculate the angular momentum of each of the following gymnasts. What body configurations do these moments of inertia represent?

(Answers: A = 70 kg · m2/s; B = 70 kg · m2/s; C = 70 kg · m2/s)

8. A volleyball player’s 3.7 kg arm moves at an averageangular velocity of 15 rad/s during execution of a spike. If the average moment of inertia of the extending arm is 0.45 kg · m2, what is the average radius of gyration for the arm during the spike? (Answer: 0.35 m)

9. A 50 kg diver in a full layout position, with a total body radius of gyration with respect to her transverse principal axis equal to 0.45 m, leaves a springboard with an angular velocity of 6 rad/s. What is the diver’s angular velocity when she assumes a tuck position, reducing her radius of gyration to 0.25 m? (Answer: 19.4 rad/s)

10. If the centripetal force exerted on a swinging tennis racket by a player’s hand is 40 N, how much reaction force is exerted on the player by the racket? (Answer: 40 N)

1. The radius of gyration of the thigh with respect to the transverse axis at the hip is 54% of the segment length. The mass of the thigh is 10.5% of total body mass, and the length of the thigh is 23.2% of total body height. What is the moment of inertia of the thigh with respect to the hip for males of the following body masses and heights?

(Answers: A = 0.25 kg · m2, B = 0.32 kg · m2, C = 0.30 kg · m2, D = 0.37 kg · m2)

2. Select three sport or daily living implements, and explain the ways in which you might modify each implement’s moment of inertia with respect to the axis of rotation to adapt it for a person of impaired strength.

3. A 0.68 kg tennis ball is given an angular momentum of 2.72 · 10–3 · m2/s when struck by a racket. If its radius of gyration is 2 cm, what is its angular velocity? (Answer: 10 rad/s)

4. A 7.27 kg shot makes seven complete revolutions during its 2.5 s flight. If its radius of gyration is 2.54 cm, what is its angular momentum? (Answer: 0.0817 kg · m2/s)

5. What is the resulting angular acceleration of a 1.7 kg forearm and hand when the forearm flexors, attaching 3 cm from the center of rotation at the elbow, produce 10 N of tension, given a 90˚ angle at the elbow and a forearm and hand radius of gyration of 20 cm? (Answer: 4.41 rad/s2)

6. The patellar tendon attaches to the tibia at a 20˚ angle 3 cm from the axis of rotation at the knee. If the tension in the tendon is 400 N, what is the resulting acceleration of the 4.2 kg lower leg and foot given a radius of gyration of 25 cm for the lower leg/foot with respect to the axis of rotation at the knee? (Answer: 15.6 rad/s2)

7. A cavewoman swings a 0.75 m sling of negligible weight around her head with a centripetal force of 220 N. What is the initial velocity of a 9 N stone released from the sling? (Answer: 13.4 m/s)

8. A 7.27 kg hammer on a 1 m wire is released with a linear velocity of 28 m/s. What reaction force is exerted on the thrower by the hammer at the instant before release? (Answer: 5.7 kN)

9. Discuss the effect of banking a curve on a racetrack. Construct a free body diagram to assist with your analysis.

10. Using the data in Appendix D, calculate the locations of the radii of gyration of all body segments with respect to the proximal joint center for a 1.7 m tall woman.

• The more closely mass is distributed to the axis of rotation, the easier it is to initiate or stop angular motion.

• The fact that bone, muscle, and fat have different densities and are distributed dissimilarly in individuals complicates efforts to calculate human body segment moments of inertia.

*Current research in sports biomechanics,*Basel, 1987, Karger.

*The biomechanics of sports techniques*(3rd ed), Englewood Cliffs, NJ, 1985, Prentice-Hall.

*Describes the way in which divers, gymnasts,
astronauts, and cats perform rotational maneuvers in midair that
seem to violate the conservation of angular momentum.*

*Discusses the linear and angular momentum
requirements for performing total body rotations at the world-class
level, and describes the methods by which these factors may be studied.*

*Includes a chapter on running, a chapter
on jumping, and sections on gymnastics and high-board diving that
discuss the principle of conservation of angular momentum with respect
to various applications in the identified sports.*

Yeadon MR and Mikulcik EC: Stability and control of aerial movements.
In Nigg BM, Stefanyshyn D, and Denoth J: *Biomechanics
and biology of movement,* Champaign, IL, 2000, Human Kinetics.

- http://regentsprep.org/Regents/physics/phys06/bcentrif/centrif.htm
*Includes a description and simulation of a car going around a curve to illustrate the effect of inertia on the contents of the car.*

- http://schutz.ucsc.edu/~josh/5A/book/torque/node15.html
*Describes a demonstration, with diagrams and formulas, for conservation of angular momentum.*

- www.walter-fendt.de/ph11e/carousel.htm
*Displays an applet allowing user control of a rotating carousel to illustrate centripetal force.*