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
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).
of Angular Momentum
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.
When angular momentum is conserved, changes in body configuration
produce a trade-off between moment of inertia and angular velocity,
with a tuck position producing greater angular velocity.
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.
During the airborne execution of a spike in volleyball,
a compensatory rotation of the lower extremity offsets the forcefully
swinging arm so that total body angular momentum is conserved.
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:
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:
Use this constant value for angular momentum to determine ω when k = 0.25 m:
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).
Asymmetrical positioning of the arms with respect to
the axis of angular momentum can shift the axis of rotation.
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
A skillful human performer can rotate 180˚ or
more in the air with zero angular momentum because in a piked position
there is a large discrepancy between the radii of gyration for the
upper and lower extremities with respect to the longitudinal axes
of these two major body segments.
Change in Angular
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).
A hammer thrower must counter the centrifugal force of
the hammer to avoid being pulled out of the throwing ring. Countering
with the shoulders (A) results in
a smaller radius of rotation for the hammer than countering with
the hips (B).
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).
The product of the springboard reaction force (F) and
its moment arm with respect to the diver’s center of gravity
(d⊥) creates a torque, which generates the angular impulse
that produces the diver’s angular momentum at takeoff.
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
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).
The arm swing during takeoff
contributes significantly to the diver’s angular momentum.
The surface reaction force is
used by the dancer to generate angular momentum during the takeoff of
the tour jeté.
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.