## SUMMARY

Crocodilians, including the alligator (*Alligator
mississippiensis*), perform a spinning maneuver to subdue and dismember
prey. The spinning maneuver, which is referred to as the `death roll',
involves rapid rotation about the longitudinal axis of the body. High-speed
videos were taken of juvenile alligators (mean length=0.29 m) performing death
rolls in water after biting onto a pliable target. Spinning was initiated
after the fore- and hindlimbs were appressed against the body and the head and
tail were canted with respect to the longitudinal body axis. With respect to
the body axis, the head and tail bending averaged 49.2° and 103.3°,
respectively. The head, body and tail rotated smoothly and freely around their
individual axes of symmetry at 1.6 Hz. To understand the dynamics of the death
roll, we mathematically modeled the system. The maneuver results purely from
conservation of angular momentum and is explained as a zero angular momentum
turn. The model permits the calculation of relevant dynamical parameters. From
the model, the shear force, which was generated at the snout by the juvenile
alligators, was 0.015 N. Shear force was calculated to scale with body length
to the 4.24 power and with mass to the 1.31 power. When scaled up to a 3 m
alligator, shear force was calculated at 138 N. The death roll appears to help
circumvent the feeding morphology of the alligator. Shear forces generated by
the spinning maneuver are predicted to increase disproportionately with
alligator size, allowing dismemberment of large prey.

## Introduction

Crocodilians, including the American alligator *Alligator
mississippiensis*, are large aquatic predators. These reptiles approach
their prey with stealth and forcefully grab the prey with their conical teeth
and large jaws (Davenport et al.,
1990; Cleuren and De Vree,
1992; Cleuren and De Vree,
2000; Erickson et al.,
2003). Although small prey are swallowed whole, large prey are
subdued and dismembered with a spinning maneuver
(McIlhenny, 1935;
Neill, 1971;
Guggisberg, 1972;
Pooley and Gans, 1976;
Ross, 1989). This maneuver is
dramatically termed the `death roll'. The death roll is an example of a
behavioral strategy referred to more generally as rotational feeding.

Body-rolling inertial feeding or rotational feeding is used by elongate
vertebrates that lack specialized cutting dentition
(Gans, 1974;
Helfman and Clark, 1986;
Davenport et al., 1990; Maesey
and Herrel, 2006). The inability to cut food into smaller portions requires
such species to use mechanisms to remove manageable pieces from prey that are
too large to consume whole. Large crocodiles and alligators will grab a limb
or lump of flesh with their jaws and then rotate around the longitudinal axis
of their body until the piece is torn free
(Guggisberg, 1972;
Cleuren and De Vree, 2000).
While there have been numerous observations of the spinning behavior for prey
reduction, there is only one description of the gross motions of the body
components for the alligator (McIlhenny,
1935). McIlhenny reported that an alligator would immediately roll
when it caught an animal that was too large to be instantly killed. The
alligator would initiate the roll by throwing its tail up and sideways. The
body and tail would turn simultaneously in the same direction. The feet were
not used as they were folded against the body. Observations from a second
crocodilian species, large (>3 m) Nile crocodiles, *Crocodylus
niloticus*, reported spin rates of 0.55–1.11 rotations
s^{–1} (Helfman and Clark,
1986).

The mechanics of the spinning maneuver in crocodilians have not been previously examined. The goal of this study was to understand how the alligator is able to initiate and sustain a spinning maneuver in an aquatic medium and to construct a model to describe the relevant dynamics. In this study, we were able to elicit juvenile alligators in the laboratory to spin in the manner of the death roll. By using high-speed video recordings of the rolling maneuver, we detailed the movements of body components and measured spinning performance. From this information, a mathematical model was produced that satisfactorily described the dynamics of the rolling maneuver, allowing the model to predict the torque and shear forces produced at the snout during this feeding behavior.

## Materials and methods

Nine juvenile alligators *Alligator mississippiensis* Daudin were
purchased from a commercial alligator farm (Everglades Outpost, Homestead, FL,
USA). Each alligator was weighed, measured, and sketched for identification.
Morphometrics of the alligators are provided in
Table 1. The total body length
(mean ± s.d.; tip of rostrum to tip of tail) and body mass were
299±9 mm and 66±8 g, respectively. One animal, which died, was
used to determine the relative proportions of mass for the head, body and
tail. The animals were housed together in a large aquarium (1.23 m×0.46
m×0.53 m) that was filled with water to a height of 50 mm. Bricks placed
in the aquarium acted as islands where the alligators could rest out of the
water. Alligators were maintained at 21–22°C with a light cycle of
12 h:12 h L:D. Animals were supplied with a diet of live earthworms
(*Lumbricus terrestris*) and strips of beef.

Experiments on spinning were conducted in a 38 l aquarium (0.51 m×0.26 m×0.32 m). Water depth was 100 mm, which was sufficient to keep the alligator from touching the bottom of the aquarium with any part of its body. Water temperature was 20–23°C. Alligators were placed singly in the test aquarium and allowed to acclimate for a minimum of 10 min. Immediately upon entry into the aquarium, the alligator would dive and swim. The alligator would eventually return to the water surface where it would float quiescently. The alligator would be presented with a small (approximately 50 mm) strip of meat held with tongs at the water surface. Once the animal grasped the meat, one to several small, sharp tugs were given to induce it to spin.

To determine if motions or orientation of the tail were associated with spinning, the tails of the alligators were restrained. Two test groups of four animals each were chosen. Strips of duct tape were used to bind a wooden stick (180 mm×6 mm×1 mm) to the dorsum of the alligators in one group (Fig. 1) and the venter of alligators in the other group. The neck and legs were free to move in all animals. The alligators were able to float at the water surface. As with unrestrained alligators, these animals were presented with a strip of meat to bite in order to initiate spinning. Restrained alligators were tested for no longer than 10 min.

The spinning maneuver was recorded with a high-speed video camera (Redlake
Imaging MotionMeter, Morgan Hill, CA, USA) at 250 frames s^{–1}
with a 6 mm lens (Cosmicar Television Lens, Japan). The camera was mounted on
a tripod 1.6 m above the aquarium. Video recordings from the camera were
played back at 60 frames s^{–1} and stored on videotape using a
Panasonic AG-7300 video recorder. Two 250 W halogen lamps supplied lighting at
water level.

Sequential frames of videotape were viewed using a Panasonic CT 2600 M monitor and Panasonic AG 7300 video recorder. Video records were chosen for analysis only if the animal displayed at least one full rotation, the animal was not pushing off the walls or floor of the aquarium, and the entire animal was in the field of view. Each spinning sequence was analyzed frame-by-frame. Data were collected on the duration of a complete spin, number of spins, and angular displacements of the head and tail relative to the longitudinal axis of the body. Angular displacements were measured using a protractor on the video frame at the initiation of the spin when the animal's dorsum was directed toward the camera and the animal's head, body and tail were parallel to the plane of the water surface. These angular data were combined with the morphometrics data to construct a mathematical model that allowed calculation of torques and shear forces, resulting from death roll behaviors.

## Results

### Live animals

A total of 52 sequences of spinning by unrestrained alligators was recorded. Spinning was induced by tugging on the meat, and spinning stopped when the alligator succeeded in tearing off a section of meat. Each sequence contained either one (73%), two (17%), or three (10%) complete spins. In every instance, the meat was proffered directly at the snout tip. Because the alligator did not move to approach the target meat, the animal did not initially possess any linear or angular momentum. This experimental condition of zero initial angular momentum will be important to understand the mechanics of the death roll.

Sequential images of spins are shown in Figs 2 and 3. The spin was observed after limbs and tail were moved (Fig. 2). The head, body and tail were bent into a C-shape. The fore- and hindlimbs were appressed against the sides and venter of the body. The head and tail could be flexed laterally, dorsally, or ventrally. Once the spin was initiated, the body remained relatively straight from the pectoral region to the pelvic region (Fig. 3). The alligator maintained this shape throughout the maneuver. The head, body and tail rotated around their individual longitudinal axes. The tail was rotated at its base, maintaining its position throughout the spin. However, the relative orientation of the body parts change with respect to each other. In Fig. 2, the tail starts bent to the left side of the alligator, but is bent to the right side of the animal later in the spin. At the end of the spin, the head, body and tail straighten out. In all cases, the legs abduct from the body and return to a typical sprawled posture, thus ending with zero angular momentum. With this condition, there are no external torques or forces operating during the spinning maneuver. Drag from the interaction of the animal and the fluid is thus negligible. In a few cases, after the animal straightens at the end the maneuver, a slow residual spin remains. This small amount of angular motion was attributed to an inadvertent external torque applied in the feeding.

The angle (θ) between the longitudinal axes of the head and body at
the start of each maneuver ranged between 20° and 75° with a mean of
49±10° (Fig. 4). The
angle (ϕ) between the body and tail at the same time ranged between
79° and 139° with a mean of 103±13°
(Fig. 4). There was no
significant correlation between head and tail angles (d.f.=50;
*R*=0.043). The mean rate of rotation was 1.5±0.5 rotations
s^{–1} or 560±170° s^{–1}. The rotation
rate ranged from 0.7 to 2.7 rotations s^{–1} (257–978°
s^{–1}) No significant correlations were found for head or tail
angle with rotation rate (head: d.f.=50; *R*=0.131; tail: d.f.=50;
*R*=0.184).

When the tail was restrained, alligators could not be induced to spin. In all cases, the legs were never tucked against the body.

### Model

Based upon the observations and kinematics of the spinning maneuver, a mathematical model was developed that was based on a spinning maneuver with a zero not angular momentum. Such zero angular momentum turns have been analyzed for some simple cases, such as a falling cat and aerial human maneuvers (Kane and Scher, 1970; Frohlich, 1979; Edwards, 1986; Galli, 1995). The dynamics of our model permit a calculation of the torque and shearing force produced at the snout.

The alligator was modeled as ellipsoidal head and body with a right
circular cone as a tail (Fig.
5). The head and body had circular cross sections. The joints at
the junctions of head and body and the body and tail can rotate freely without
slipping. As indicated above, the initial state is one of zero angular
momentum. The head and body sections each possess three principal moments of
inertia. For the model head with semi-major axis of *a*_{H} and
semi-minor axes each of length *b*_{H}, we denote the smallest
moment of inertia about the major axis as *i*_{H}. The moments
about the two equal minor axes are each denoted by *I*_{H} and
are larger than the moment about the major axis
(Table 1). The length of the
head *l*_{H} is 2*a*_{H} and the width and
thickness are each 2*b*_{H}. In this case,
(1)
and
(2)
where *m*_{H} is the mass of the head alone
(Gray, 1963). Similarly for
the model ellipsoidal body (or trunk) with axes of length
*a*_{B} and *b*_{B}, the principal moments of
inertia *i*_{B} and *I*_{B} are given by:
(3)
and
(4)
For the model right circular cone tail, the three principal moments are
*i*_{T} and *I*_{T} given by:
(5)
and
(6)
where *m*_{T} is the mass of the tail, *r* is its radius
at the base, and *l*_{T} is its length.

The model head, body and tail all roll without slipping with angular speedsω
_{H}=ω_{B}=ω_{T}=ω and
simultaneously revolve around the RR′-axis, the roll axis, with angular
speed ω_{rev} (Fig.
5).

The rotating head, body and tail each possess angular momentum. To
determine the moments of inertia of the body parts and the resulting angular
momenta about the RR′-axis, we adopt the coordinate system shown in
Fig. 5. The unit vectors for
each body part are described in Cartesian coordinates of
*x̂* and *ŷ*. The
*ŷ* axes lie along the spin axes of each body part and the
*x̂* axes are perpendicular to the
*ŷ* axes. The angular momentum of the head is:
(7)
Similarly, for the body and tail, respectively,
(8)
(9)
The parallel axis theorem was used to determine the moment of inertia of the
body revolving around the RR′-axis, which is a distance *d* away
from the longitudinal axis of the body.

In a zero angular momentum maneuver, the vector sum of these angular
momenta vanishes, that is,
.
For this case,
(10)
If the total angular momentum of a system is zero, it is zero about any axis.
The angular momentum projected onto the RR′-axis is therefore:
(11)
where we have used the fact that *d*=*l*_{H}sinθ
with *l*_{H} the length of the head. After rearranging terms to
form the ratio ω/ω_{rev}, we find:
(12)
However, for θ=45° and ϕ=90°, which are typical values for
these angles (Fig. 4), this
expression reduces to:
(13)
This expression is consistent with the observed characteristics of the death
roll (see below).

It is important to note that the ω_{rev} motion (i.e. the
motion of the animal revolving around the RR′-axis) is a reaction to the
rolling motions initiated by the animal after it fastens onto its prey. Before
the spin is initiated the angular momentum of the alligator is observed to be
zero, must remain zero during the spin, and is observed to be zero when the
spin terminates. The motion around the RR′-axis, which occurs at an
angular frequency approximately an order of magnitude slower than the rolling
motions, results purely from the conservation of angular momentum. This is
roughly analogous to how a figure skater controls spin rate
(Giancoli, 1985). By
voluntarily bringing both arms close to his or her body from an extended
position, a figure skater can increase angular speed to conserve angular
momentum. Rather than this one-dimensional case, the death roll is a
two-dimensional example.

## Discussion

### Significance of prey inertia to crocodilian spin feeding

Spinning is a maneuver to reduce large prey to small enough pieces that a
crocodilian can swallow (McIlhenny,
1935; Neill, 1971;
Guggisberg, 1972;
Pooley and Gans, 1976;
Ross, 1989). The conical teeth
of crocodilians are useful for grasping prey with a large bite force
(Erickson et al., 2003), but
not for tearing and cutting flesh
(Guggisberg, 1972). Spinning
is a mechanism that can tear apart large prey by subjecting the tissue to
torsional stresses. Animals and their tissues are weak in torsion
(Gordon, 1978;
Currey, 2002). The spinning
maneuver is used predominately by crocodilians with broad, short snouts, which
feed on large prey and on a more general diet
(Cleuren and De Vree, 2000).
This skull structure can resist the substantial forces associated with the
maneuver (Cleuren and De Vree,
1992). Inertia of the prey is required for the maneuver to be
effective. Spinning does not work with small prey animals, because as the
crocodile spins, the prey will also rotate. Thus, when groups of crocodilians
(e.g. *Crocodylus niloticus*) feed on a carcass at the same time
(Pooley and Gans, 1976;
Guggisberg, 1972;
Ross, 1989), the inertia added
by attached predators would facilitate the success of spin feeding by
individual crocodilians by helping to secure the prey.

We discovered that juvenile alligators are capable of performing the death
roll. Previous reports of spinning were associated with large crocodilians
subduing or dismembering large prey items
(McIlhenny, 1935;
Pooley and Gans, 1976).
Hatchling (50 g) and juvenile (100–550 g) salt-water crocodiles
(*Crocodylus porosus*) feeding on carrion were observed to use
side-to-side head shaking, rather than spinning, to detach small pieces
(Davenport et al., 1990).
Side-to-side head shaking was used to detach small pieces of the carrion.
However, the carrion was a large fish, which may not have offered resistance
to tearing (Davenport et al.,
1990). The toughness of the food presented to the alligators in
this study provided sufficient resistance to initiate the spinning
behavior.

### Conservation of angular momentum in crocodilian death rolls

The ferocity of the death roll of alligators and crocodiles is particularly enhanced by the rapid speed of the spinning motions. How can the animal generate these motions and still conserve angular momentum? From a configuration where the symmetry axes of the head, body and tail are all aligned, the animal quickly bends itself into a C-shape and commences spinning. Consequently, each body part possesses a vector angular momentum (Fig. 5). While the horizontal components of the angular momenta of the head and tail largely cancel, the vertical components add. This angular momentum vector, however, is canceled by a more subtle motion of the entire animal. As a reaction to the spinning motion, the animal also revolves around a roll axis roughly parallel to the animal's trunk (body). The roll axis runs through its snout, which is fastened onto meat, and a point approximately one-quarter of the distance from base of the tail to its tip. The revolution of the animal's head, body and tail about the roll axis also has an angular momentum, which is directly opposite to the vector sum of the angular momentums of each body segment. Thus, the initial angular momentum is zero, the total angular momentum during the roll is zero, and when the maneuver terminates by the alligator straightening, it remains zero.

The reason that the motion about the roll axis is less apparent than the spinning motions of the head, body and tail is because it takes place with an angular speed that is an order of magnitude smaller than the spinning motion (Fig. 6). When an animal executes a roll of one spin, it only completes a tenth of a revolution around the roll axis. This relatively small angular velocity is not measurable in this experiment. The relatively small magnitude of this compensatory rotation can ultimately be attributed to the large size of the moment of inertia of the alligator bent into a C-shape with the massive trunk relatively far from the roll axis.

The alligator is able to centralize its mass and its axis of rotation by keeping its legs in close to its body. This also effectively helps reduce drag and enables it to create a faster, more powerful spin. Similarly, human divers create a central axis as they somersault from a diving board (Frohlich, 1980). By drawing their arms and legs in close to their body, they can isolate their axis of rotation. The same principle applies when a person is spinning on ice skates. When their arms are extended the spin is slowed down, but when tucked in, the person is able to increase their speed of spinning. This can be seen in the spinning alligator when the legs are tucked in close to its body. Because the legs play no role in actually producing the torque of the spin, it appears that the alligator relies completely on the axial components of its body. The mechanics of the spinning behavior indicate that orientation between the body and tail and, to a lesser extent, the head are important in the maneuver of the alligator. The angular displacement between body parts changes the moment of inertia, which is necessary to conserve angular momentum during the spin. McIlhenny originally noted the reorientation of the tail and tucking of the legs during the roll (McIlhenny, 1935).

The angular momentum balance and lack of external torques to maintain the maneuver make spinning of the alligator a zero angular momentum maneuver. A similar maneuver is observed in an inverted cat during free-fall (Frohlich, 1980; Galli, 1995). The cat in an inverted position is able to twist its body in mid air to land on its feet. The cat begins its free fall with no initial angular momentum (Arabyan and Tsai, 1998). As it falls, the cat bends at the waist. The anterior and posterior body sections rotate in the same direction (Frohlich, 1980; Fredrickson, 1989). Each section has an angular momentum, whose vector sum gives a counter-rotation to the entire body (Edwards, 1986). This results in no net change in angular momentum for the cat. The legs are positioned close to the symmetry axis of each body section during the maneuver. This orientation reduces the moment of inertia and increases the spin rate of the body sections. When the cat has rotated 180°, it straightens its spine to stop rotating and can land on its feet (Fredrickson, 1989). The cat then terminates its maneuver with no angular momentum. The alligator and the cat both generate internal forces that enable these animals to spin.

### Generation of shear force in the death roll

To tear apart its food using the death roll, the alligator needs to
generate large shear forces. Although data on the magnitude of shear forces
required to dismember bodies have not been collected, the shear force in a
death roll can be calculated over a range of sizes for the alligator. To
illustrate this computation, the morphometric data
(Table 1) of a model juvenile
(0.3 m) and adult (3 m) specimen of *A. mississippiensis* are used. The
calculation estimates the total rotational kinetic energy
(*K*_{rot}) in the spinning maneuver. *K*_{rot}
equals the work needed to remove that energy and bring the roll to a halt.

*K*_{rot} of an alligator executing a spin possesses two
contributions: (1) the rotation at relatively high angular speed of each body
section about their individual symmetry axis, ω, and (2) the rotation of
the entire animal about the roll axis, ω_{rev}, which occurs at
a relatively smaller angular speed. The ratio of ω toω
_{rev} is given by Eqn
13. For the model adult individual,ω
/ω_{rev}=11.5 and for the juvenile,ω
/ω_{rev}=12.2.

Using ω=1 rotation s^{–1} (=6.3 rad
s^{–1}) for our sample calculation, ω_{rev}=(6.3
rad s^{–1})/11.5=0.55 rad s^{–1} and the total
rotational kinetic energy *K*_{rot} of an adult is:
(14)
Note that the kinetic energy involved in the obvious spinning motion around
the symmetry axes of each body part is substantially larger than the motion
around the roll axis.

The work done by a constant external torque Γ acting over an angleθ
in order to remove an energy *K*_{rot} is
W=Γθ. For θ=1, rotation=2π rad, in our case,Γ
=(78.1 J)/2π=12.4 N m. Furthermore, the shear force produced by this
torque acting over a lever arm equal to one half the width (*w*) of the
distal portion of the snout equals the torque, i.e.Γ
=*F*_{s}(0.5*w*), and therefore
*F*_{s}=2Γ/*w*=(2)(12.1 N m)/(0.18 m)=138 N. This
analysis does not account for any reduction in the ultimate strength of the
prey's tissue due to perforation by the alligator's dentition, which would
serve to significantly lower the shear force required for dismemberment.

For comparison, results for the juvenile *A. mississippiensis*
executing a death roll with ω=2 rotations s^{–1} show thatω
/ω_{rev}=12.2, which yields, along with relevant data
from Table 1,
*K*_{rot}=4.1×10^{–4} J. The corresponding
torques and shear force are 6.5×10^{–5} N m and 0.015 N,
respectively. Thus, an adult having a mass 1800 times that of a juvenile can
produce 200 000 times the energy and torque, and approximately 1000 times the
shear force.

### Force scaling relationships of alligators

The foregoing analysis permits the development of a scaling relation for
large adult individuals with lengths in the vicinity of 3 m. For 51
individuals ranging in length from 0.23 m to 3.75 m and in mass from 0.0318 kg
to 296.7 kg (Fig. 7), mass
*M* varies with length *L* according to the equation:
(15)
where *C*=3.6±0.4 and p=3.24±0.03, obtained by
least-squares regression. The 95% confidence interval for the value of p is
3.18–3.30. This interval does not overlap the predicted value of p=3.00
for isometric scaling. The increase in mass of alligators is therefore
positively allometric with respect to length. An implication of this
relationship is that shear forces will be predicted also to increase with
positive allometry.

If we assume that the masses and lengths of the head, body and tail are
distributed proportionally as in the individual in
Table 1, the various moments of
inertia can be determined in terms of the total length. These results, in
turn, can be used in Eqn 14 to
write *K*_{rot} in terms of the length of the alligator and its
angular rate of rotation as:
(16)
where *K*_{rot} is in joules (J). The shear force
*F*_{s} (in N) corresponding to this energy is:
(17)
where growth of the skull and snout is assumed to be isometric
(Dodson, 1975). Using
Eqn 15,
Eqn 17 can be rewritten so that
the shear force is given in terms of mass:
(18)
The shear force is extremely sensitive to changes in size of the alligator
(Fig. 8). For example, for the
same ω (rad s^{–1}), an adult whose length is just 10%
larger than another whose length is 3 m produces a shear force 50% greater.
Helfman and Clark (Helfman and Clark,
1986) provide values for ω of 0.6–1.1 rotations
s^{–1} for large (>3 m) crocodiles. Using ω=1 rotation
s^{–1}=6.3 rad s^{–1}, the record alligator of 5.8
m (Wood, 1976) would have a
*F*_{s} of 2326 N! Thus, shear forces generated by the spinning
maneuver are predicted to increase disproportionally with alligator size,
allowing dismemberment of large prey.

Along with crocodilians, spin feeding is used by other vertebrates with elongate bodies (Gans, 1974; Helfman and Clark, 1986; Measey and Herrel, 2006). Among these other species, spinning by eels occurs at higher rotation rates than similarly sized alligators and the mechanics of spinning may be different. A spinning force of 1.35 N was measured on rotationally feeding caecilians (Measey and Herrel, 2006). Although this spinning force was greater than the shear force calculated for alligators of approximately the same body length, these forces are not equivalent. The caecilians were handheld and were presumably pushing off the solid substrate during the maneuver, whereas alligators can generate their own internal torques to spin in water.

Rolling has largely been ignored as a maneuver for animals. Analyses of
maneuverability and agility have been confined to examination of pitching and
yawing motions (Frey and Salisbury,
2000; Webb, 2002;
Fish, 2002;
Rivera et al., 2006). While
pitch and yaw are typically associated with directional changes during
locomotion, roll is used for more varied behaviors. Spinner dolphins
(*Stenella longirostris*) perform aerial leaps and rotate around their
longitudinal axis up to seven times. This behavior was believed to function in
the removal of remoras from the body surface
(Fish et al., 2006), but may
function in acoustic communication. Birds will roll to use a component of lift
generated by the wings to produce a centripetal force to effect turning in
flight (Norberg, 1990).
Similarly, turning in water is facilitated by rolling in marine mammals and
penguins (Hui, 1995; Fish and Battle,
1995; Fish, 2002;
Fish et al., 2003;
Cheneval et al., 2007). Kasapi
et al. (Kasapi et al., 1993)
considered roll to be an important kinematic parameter in escape maneuvers by
knifefish (*Xenomystus nigri*). Female dugongs (*Dugon dugon*)
and right whales (*Eubalaena australis*) will roll onto their backs at
the water surface to prevent mating with unwanted suitors
(Payne, 1995;
Marsh, 2002). Grooming by sea
otters (*Enhydra lutris*) utilizes rolling to wash the fur
(Kenyon, 1969). Rolling
maneuvers are also involved in feeding behaviors. Fin whales (*Balaenoptera
physalus*) and other rorquals make lateral lunges involving a 90° roll
(Goldbogen et al., 2006). Gray
whales (*Eschrichtius robustus*) consume benthic invertebrates
(Pivorunas, 1979) by laterally
orienting the body as they plow thorough the soft sediment. The varied nature
of these behaviors provide a fruitful avenue for future studies of maneuvering
performance.

**List of symbols and abbreviations**

*a*_{B},*a*_{H}- semi-major axis of model ellipsoidal body and head;
length=2
*a* *b*_{B},*b*_{H}- semi-minor axes of model ellipsoidal body and head;
width=2
*b* - C
- proportionality constant in power law relation between
*L*and*M* - d
- perpendicular distance of snout tip to symmetry axis of body
*F*_{s}- shear force at snout
*i*_{B},*i*_{H},*i*_{T}- smallest value of the principal moments of inertia for the body, head and tail, respectively
*I*_{B},*I*_{H},*I*_{T}- largest value of the principal moments of inertia for the body, head and tail, respectively
*K*_{rot}- rotational kinetic energy
*l*_{T}- length of model right circular cone tail
- L
- total length of alligator
*L*_{B},*L*_{H},*L*_{T}- angular momenta of body, head and tail, respectively
*m*_{B},*m*_{H},*m*_{T}- masses of the body, head and tail, respectively
- M
- total mass of alligator
- p
- exponent in power law relation between
*L*and*M* - r
- radius of model right circular cone tail
- RR′
- roll axis
- w
- width
- x̂,ŷ
- unit vectors for each body part in Cartesian coordinates
- θ
- angle between symmetry axis of head and roll axis
- ϕ
- angle between symmetry axis of tail and roll axis
- Γ
- torque
- ω
- angular rate of rotation around symmetry axis of body part
- ωrev
- angular rate of rotation of alligator around roll axis

## ACKNOWLEDGEMENTS

We would like to express our appreciation to the Lauder Laboratory, Harvard University for providing morphometrics data and to the two anonymous viewers for their helpful comments on the manuscript. All experiments with the alligators were in compliance with the West Chester University Institution Animal Care and Use Committee.

- © The Company of Biologists Limited 2007