1. Intake
Resultant force 90°
Effective
force
Applied force
Yaw
Figure 5-49. Gyroscopic precession.
Applied force
Resultant force
Yaw
Effective
force
Figure 5-50. Raising tail produces gyroscopic precession.
gyroscope and thus has similar properties. Any time a force
is applied to deflect the propeller out of its plane of rotation,
the resulting force is 90° ahead of and in the direction of
rotation and in the direction of application, causing a pitching
moment, a yawing moment, or a combination of the two
depending upon the point at which the force was applied.
This element of torque effect has always been associated with
and considered more prominent in tailwheel-type aircraft
and most often occurs when the tail is being raised during
the takeoff roll. [Figure 5-50] This change in pitch attitude
has the same effect as applying a force to the top of the
propeller’s plane of rotation. The resultant force acting 90°
ahead causes a yawing moment to the left around the vertical
axis. The magnitude of this moment depends on several
variables, one of which is the abruptness with which the tail
is raised (amount of force applied). However, precession,
or gyroscopic action, occurs when a force is applied to any
point on the rim of the propeller’s plane of rotation; the
resultant force will still be 90° from the point of application
in the direction of rotation. Depending on where the force is
applied, the airplane is caused to yaw left or right, to pitch
up or down, or a combination of pitching and yawing.
It can be said that, as a result of gyroscopic action, any yawing
around the vertical axis results in a pitching moment, and any
pitching around the lateral axis results in a yawing moment.
To correct for the effect of gyroscopic action, it is necessary
for the pilot to properly use elevator and rudder to prevent
undesired pitching and yawing.
Asymmetric Loading (P-Factor)
When an aircraft is flying with a high AOA, the “bite” of
the downward moving blade is greater than the “bite” of the
upward moving blade. This moves the center of thrust to the
right of the prop disc area, causing a yawing moment toward
the left around the vertical axis. Proving this explanation is
complex because it would be necessary to work wind vector
problems on each blade while considering both the AOA of
the aircraft and the AOA of each blade.
This asymmetric loading is caused by the resultant velocity,
which is generated by the combination of the velocity of the
propeller blade in its plane of rotation and the velocity of the
air passing horizontally through the propeller disc. With the
aircraft being flown at positive AOAs, the right (viewed from
the rear) or downswinging blade, is passing through an area of
resultant velocity, which is greater than that affecting the left
or upswinging blade. Since the propeller blade is an airfoil,
increased velocity means increased lift. The downswinging
blade has more lift and tends to pull (yaw) the aircraft’s nose
to the left.
When the aircraft is flying at a high AOA, the downward
moving blade has a higher resultant velocity, creating more
lift than the upward moving blade. [Figure 5-51] This might
be easier to visualize if the propeller shaft was mounted
perpendicular to the ground (like a helicopter). If there
were no air movement at all, except that generated by the
propeller itself, identical sections of each blade would have
the same airspeed. With air moving horizontally across this
vertically mounted propeller, the blade proceeding forward
into the flow of air has a higher airspeed than the blade
retreating with the airflow. Thus, the blade proceeding into
the horizontal airflow is creating more lift, or thrust, moving
the center of thrust toward that blade. Visualize rotating the
vertically mounted propeller shaft to shallower angles relative
to the moving air (as on an aircraft). This unbalanced thrust
then becomes proportionately smaller and continues getting
smaller until it reaches the value of zero when the propeller
shaft is exactly horizontal in relation to the moving air.
The effects of each of these four elements of torque vary
in value with changes in flight situations. In one phase of
flight, one of these elements may be more prominent than
another. In another phase of flight, another element may be
more prominent. The relationship of these values to each
other varies with different aircraft depending on the airframe,
engine, and propeller combinations, as well as other design
features. To maintain positive control of the aircraft in all
flight conditions, the pilot must apply the flight controls as
necessary to compensate for these varying values.
Low angle of attack High angle of attack
Load on
upward moving
propeller blade
Load on
downward moving
propeller blade
Load on
downward moving
propeller blade
Load on
upward moving
propeller blade
Figure 5-51. Asymmetrical loading of propeller (P-factor).
Load Factors
In aerodynamics, the maximum load factor (at given bank
angle) is a proportion between lift and weight and has a
trigonometric relationship. The load factor is measured in
Gs (acceleration of gravity), a unit of force equal to the force
exerted by gravity on a body at rest and indicates the force to
which a body is subjected when it is accelerated. Any force
applied to an aircraft to deflect its flight from a straight line
produces a stress on its structure. The amount of this force
is the load factor. While a course in aerodynamics is not a
prerequisite for obtaining a pilot’s license, the competent
pilot should have a solid understanding of the forces that act
on the aircraft, the advantageous use of these forces, and the
operating limitations of the aircraft being flown.
For example, a load factor of 3 means the total load on an
aircraft’s structure is three times its weight. Since load factors
are expressed in terms of Gs, a load factor of 3 may be spoken
of as 3 Gs, or a load factor of 4 as 4 Gs.
If an aircraft is pulled up from a dive, subjecting the pilot to
3 Gs, he or she would be pressed down into the seat with a
force equal to three times his or her weight. Since modern
aircraft operate at significantly higher speeds than older
aircraft, increasing the potential for large load factors, this
effect has become a primary consideration in the design of
the structure of all aircraft.
With the structural design of aircraft planned to withstand
only a certain amount of overload, a knowledge of load
factors has become essential for all pilots. Load factors are
important for two reasons:
1. It is possible for a pilot to impose a dangerous overload
on the aircraft structures.
2. An increased load factor increases the stalling speed and
makes stalls possible at seemingly safe flight speeds.
Load Factors in Aircraft Design
The answer to the question “How strong should an aircraft
be?” is determined largely by the use to which the aircraft is
subjected. This is a difficult problem because the maximum
possible loads are much too high for use in efficient design.
It is true that any pilot can make a very hard landing or an
extremely sharp pull up from a dive, which would result in
abnormal loads. However, such extremely abnormal loads
must be dismissed somewhat if aircraft are built that take off
quickly, land slowly, and carry worthwhile payloads.
The problem of load factors in aircraft design becomes how
to determine the highest load factors that can be expected in
normal operation under various operational situations. These
load factors are called “limit load factors.” For reasons of
safety, it is required that the aircraft be designed to withstand
these load factors without any structural damage. Although
the Code of Federal Regulations (CFR) requires the aircraft
structure be capable of supporting one and one-half times
these limit load factors without failure, it is accepted that
parts of the aircraft may bend or twist under these loads and
that some structural damage may occur.
This 1.5 load limit factor is called the “factor of safety” and
provides, to some extent, for loads higher than those expected
under normal and reasonable operation. This strength reserve
is not something that pilots should willfully abuse; rather, it is
there for protection when encountering unexpected conditions.
The above considerations apply to all loading conditions,
whether they be due to gusts, maneuvers, or landings. The
gust load factor requirements now in effect are substantially
the same as those that have been in existence for years.
Hundreds of thousands of operational hours have proven
them adequate for safety. Since the pilot has little control over
gust load factors (except to reduce the aircraft’s speed when
rough air is encountered), the gust loading requirements are
substantially the same for most general aviation type aircraft
regardless of their operational use. Generally, the gust load
factors control the design of aircraft which are intended for
strictly nonacrobatic usage.
An entirely different situation exists in aircraft design with
maneuvering load factors. It is necessary to discuss this matter
separately with respect to: (1) aircraft designed in accordance
with the category system (i.e., normal, utility, acrobatic); and
(2) older designs built according to requirements that did not
provide for operational categories.
Aircraft designed under the category system are readily
identified by a placard in the flight deck, which states the
operational category (or categories) in which the aircraft
Load factor = 2 Gs
Centrifugal
force = 1.73 Gs
Gravity = 1G
60° 50° 40° 30° 20° 10°
Figure 5-52. Two forces cause load factor during turns.
7
6
5
4
3
2
1
00° 10° 20° 30° 40° 50° 60° 70° 80° 90°
Load factor (G units)
Bank angle
Figure 5-53. Angle of bank changes load factor in level flight.
is certificated. The maximum safe load factors (limit load
factors) specified for aircraft in the various categories are:
CATEGORY LIMIT LOAD FACTOR
Normal1 3.8 to –1.52
Utility (mild acrobatics,
including spins)
4.4 to –1.76
Acrobatic 6.0 to –3.00
1 For aircraft with gross weight of more than 4,000 pounds,
the limit load factor is reduced. To the limit loads given
above, a safety factor of 50 percent is added.
There is an upward graduation in load factor with the
increasing severity of maneuvers. The category system
provides for maximum utility of an aircraft. If normal
operation alone is intended, the required load factor (and
consequently the weight of the aircraft) is less than if the
aircraft is to be employed in training or acrobatic maneuvers
as they result in higher maneuvering loads.
Aircraft that do not have the category placard are designs that
were constructed under earlier engineering requirements in
which no operational restrictions were specifically given to
the pilots. For aircraft of this type (up to weights of about
4,000 pounds), the required strength is comparable to present-
day utility category aircraft, and the same types of operation
are permissible. For aircraft of this type over 4,000 pounds,
the load factors decrease with weight. These aircraft should
be regarded as being comparable to the normal category
aircraft designed under the category system, and they should
be operated accordingly.
Load Factors in Steep Turns
At a constant altitude, during a coordinated turn in any
aircraft, the load factor is the result of two forces: centrifugal
force and weight. [Figure 5-52] For any given bank angle,
the ROT varies with the airspeed—the higher the speed, the
slower the ROT. This compensates for added centrifugal
force, allowing the load factor to remain the same.
Figure 5-53 reveals an important fact about turns—the load
factor increases at a terrific rate after a bank has reached
45° or 50°. The load factor for any aircraft in a coordinated
level turn at 60° bank is 2 Gs. The load factor in an 80° bank
is 5.76 Gs. The wing must produce lift equal to these load
factors if altitude is to be maintained.
It should be noted how rapidly the line denoting load factor
rises as it approaches the 90° bank line, which it never quite
reaches because a 90° banked, constant altitude turn is not
mathematically possible. An aircraft may be banked to 90°
in a coordinated turn if not trying to hold altitude. An aircraft
that can be held in a 90° banked slipping turn is capable of
straight knife-edged flight. At slightly more than 80°, the
load factor exceeds the limit of 6 Gs, the limit load factor of
an acrobatic aircraft.
For a coordinated, constant altitude turn, the approximate
maximum bank for the average general aviation aircraft is 60°.
This bank and its resultant necessary power setting reach the
limit of this type of aircraft. An additional 10° bank increases
the load factor by approximately 1 G, bringing it close to the
yield point established for these aircraft. [Figure 5-54]
Load Factors and Stalling Speeds
Any aircraft, within the limits of its structure, may be stalled
at any airspeed. When a sufficiently high AOA is imposed,
the smooth flow of air over an airfoil breaks up and separates,
producing an abrupt change of flight characteristics and a
sudden loss of lift, which results in a stall.
A study of this effect has revealed that an aircraft’s stalling
speed increases in proportion to the square root of the
3
2
5
1
4
0 0 1 2 3 4 5 6 7 8 20 40 60 80 100 120 140 160 180 200 220 240 260
Ratio of acceleration Vs to unaccelerated Vs
Unaccelerated stall speed
“G” Load Accelerated stall speed
40
50
60
70
80
100
120
150
40
50
60
70
80
100
120
150
Figure 5-54. Load factor changes stall speed.
load factor. This means that an aircraft with a normal
unaccelerated stalling speed of 50 knots can be stalled at 100
knots by inducing a load factor of 4 Gs. If it were possible
for this aircraft to withstand a load factor of nine, it could
be stalled at a speed of 150 knots. A pilot should be aware
of the following:
• The danger of inadvertently stalling the aircraft by
increasing the load factor, as in a steep turn or spiral;
• When intentionally stalling an aircraft above its
design maneuvering speed, a tremendous load factor
is imposed.
Figures 5-53 and 5-54 show that banking an aircraft greater
than 72° in a steep turn produces a load factor of 3, and the
stalling speed is increased significantly. If this turn is made
in an aircraft with a normal unaccelerated stalling speed of
45 knots, the airspeed must be kept greater than 75 knots to
prevent inducing a stall. A similar effect is experienced in a
quick pull up or any maneuver producing load factors above
1 G. This sudden, unexpected loss of control, particularly in
a steep turn or abrupt application of the back elevator control
near the ground, has caused many accidents.
Since the load factor is squared as the stalling speed doubles,
tremendous loads may be imposed on structures by stalling
an aircraft at relatively high airspeeds.
The following information primarily applies to fixed-wing
airplanes. The maximum speed at which an airplane may
be stalled safely is now determined for all new designs.
This speed is called the “design maneuvering speed” (V A),
which is the speed below which you can move a single
flight control, one time, to its full deflection, for one axis
of airplane rotation only (pitch, roll or yaw), in smooth
air, without risk of damage to the airplane. V A must be
entered in the FAA-approved Airplane Flight Manual/
Pilot’s Operating Handbook (AFM/POH) of all recently
designed airplanes. For older general aviation airplanes,
this speed is approximately 1.7 times the normal stalling
speed. Thus, an older airplane that normally stalls at 60
knots must never be stalled at above 102 knots (60 knots ×
1.7 = 102 knots). An airplane with a normal stalling speed
of 60 knots stalled at 102 knots undergoes a load factor
equal to the square of the increase in speed, or 2.89 Gs (1.7
× 1.7 = 2.89 Gs). (The above figures are approximations to
be considered as a guide, and are not the exact answers to
any set of problems. The design maneuvering speed should
be determined from the particular airplane’s operating
limitations provided by the manufacturer.) Operating at or
below design maneuvering speed does not provide structural
protection against multiple full control inputs in one axis or
full control inputs in more than one axis at the same time.
Since the leverage in the control system varies with different
aircraft (some types employ “balanced” control surfaces while
others do not), the pressure exerted by the pilot on the controls
cannot be accepted as an index of the load factors produced
in different aircraft. In most cases, load factors can be judged
by the experienced pilot from the feel of seat pressure. Load
factors can also be measured by an instrument called an
“accelerometer,” but this instrument is not common in general
aviation training aircraft. The development of the ability to
judge load factors from the feel of their effect on the body is
important. A knowledge of these principles is essential to the
development of the ability to estimate load factors.
A thorough knowledge of load factors induced by varying
degrees of bank and the VA aids in the prevention of two of
the most serious types of accidents:
1. Stalls from steep turns or excessive maneuvering near
the ground
2. Structural failures during acrobatics or other violent
maneuvers resulting from loss of control
Load Factors and Flight Maneuvers
Critical load factors apply to all flight maneuvers except
unaccelerated straight flight where a load factor of 1 G is
always present. Certain maneuvers considered in this section
are known to involve relatively high load factors. Full
application of pitch, roll, or yaw controls should be confined
to speeds below the maneuvering speed. Avoid rapid and
large alternating control inputs, especially in combination
with large changes in pitch, roll, or yaw (e.g., large sideslip
angles) as they may result in structural failures at any speed,
including below VA.
Turns
Increased load factors are a characteristic of all banked
turns. As noted in the section on load factors in steep turns,
load factors become significant to both flight performance
and load on wing structure as the bank increases beyond
approximately 45°.
The yield factor of the average light plane is reached
at a bank of approximately 70° to 75°, and the stalling
speed is increased by approximately one-half at a bank of
approximately 63°.
Stalls
The normal stall entered from straight-and-level flight, or an
unaccelerated straight climb, does not produce added load
factors beyond the 1 G of straight-and-level flight. As the
stall occurs, however, this load factor may be reduced toward
zero, the factor at which nothing seems to have weight. The
pilot experiences a sensation of “floating free in space.” If
recovery is effected by snapping the elevator control forward,
negative load factors (or those that impose a down load on
the wings and raise the pilot from the seat) may be produced.
During the pull up following stall recovery, significant
load factors are sometimes induced. These may be further
increased inadvertently during excessive diving (and
consequently high airspeed) and abrupt pull ups to level
flight. One usually leads to the other, thus increasing the load
factor. Abrupt pull ups at high diving speeds may impose
critical loads on aircraft structures and may produce recurrent
or secondary stalls by increasing the AOA to that of stalling.
As a generalization, a recovery from a stall made by diving
only to cruising or design maneuvering airspeed, with a
gradual pull up as soon as the airspeed is safely above stalling,
can be effected with a load factor not to exceed 2 or 2.5 Gs. A
higher load factor should never be necessary unless recovery
has been effected with the aircraft’s nose near or beyond the
vertical attitude or at extremely low altitudes to avoid diving
into the ground.
Spins
A stabilized spin is not different from a stall in any element
other than rotation and the same load factor considerations
apply to spin recovery as apply to stall recovery. Since spin
recoveries are usually effected with the nose much lower than is
common in stall recoveries, higher airspeeds and consequently
higher load factors are to be expected. The load factor in a
proper spin recovery usually is found to be about 2.5 Gs.
The load factor during a spin varies with the spin characteristics
of each aircraft, but is usually found to be slightly above the
1 G of level flight. There are two reasons for this:
1. Airspeed in a spin is very low, usually within 2 knots
of the unaccelerated stalling speeds.
2. An aircraft pivots, rather than turns, while it is in a spin.
High Speed Stalls
The average light plane is not built to withstand the repeated
application of load factors common to high speed stalls.
The load factor necessary for these maneuvers produces a
stress on the wings and tail structure, which does not leave
a reasonable margin of safety in most light aircraft.
The only way this stall can be induced at an airspeed above
normal stalling involves the imposition of an added load
factor, which may be accomplished by a severe pull on the
elevator control. A speed of 1.7 times stalling speed (about
102 knots in a light aircraft with a stalling speed of 60 knots)
produces a load factor of 3 Gs. Only a very narrow margin
for error can be allowed for acrobatics in light aircraft. To
illustrate how rapidly the load factor increases with airspeed,
a high-speed stall at 112 knots in the same aircraft would
produce a load factor of 4 Gs.
Chandelles and Lazy Eights
A chandelle is a maximum performance climbing turn
beginning from approximately straight-and-level flight,
and ending at the completion of a precise 180° turn in a
wings-level, nose-high attitude at the minimum controllable
airspeed. In this flight maneuver, the aircraft is in a steep
climbing turn and almost stalls to gain altitude while changing
direction. A lazy eight derives its name from the manner in
which the extended longitudinal axis of the aircraft is made
to trace a flight pattern in the form of a figure “8” lying on
its side. It would be difficult to make a definite statement
concerning load factors in these maneuvers as both involve
smooth, shallow dives and pull-ups. The load factors incurred
depend directly on the speed of the dives and the abruptness
of the pull-ups during these maneuvers.
Generally, the better the maneuver is performed, the less
extreme the load factor induced. A chandelle or lazy eight
in which the pull-up produces a load factor greater than 2 Gs
will not result in as great a gain in altitude; in low-powered
aircraft, it may result in a net loss of altitude.
The smoothest pull-up possible, with a moderate load factor,
delivers the greatest gain in altitude in a chandelle and results
in a better overall performance in both chandelles and lazy
eights. The recommended entry speed for these maneuvers
is generally near the manufacturer’s design maneuvering
speed, which allows maximum development of load factors
without exceeding the load limits.
Rough Air
All standard certificated aircraft are designed to withstand
loads imposed by gusts of considerable intensity. Gust load
factors increase with increasing airspeed, and the strength used
for design purposes usually corresponds to the highest level
flight speed. In extremely rough air, as in thunderstorms or
frontal conditions, it is wise to reduce the speed to the design
maneuvering speed. Regardless of the speed held, there may
be gusts that can produce loads that exceed the load limits.
Each specific aircraft is designed with a specific G loading
that can be imposed on the aircraft without causing structural
damage. There are two types of load factors factored into
aircraft design: limit load and ultimate load. The limit load
is a force applied to an aircraft that causes a bending of the
aircraft structure that does not return to the original shape.
The ultimate load is the load factor applied to the aircraft
beyond the limit load and at which point the aircraft material
experiences structural failure (breakage). Load factors lower
than the limit load can be sustained without compromising
the integrity of the aircraft structure.
Speeds up to, but not exceeding, the maneuvering speed
allow an aircraft to stall prior to experiencing an increase in
load factor that would exceed the limit load of the aircraft.
Most AFM/POH now include turbulent air penetration
information, which help today’s pilots safely fly aircraft
capable of a wide range of speeds and altitudes. It is important
for the pilot to remember that the maximum “never-exceed”
placard dive speeds are determined for smooth air only. High
speed dives or acrobatics involving speed above the known
maneuvering speed should never be practiced in rough or
turbulent air.
Vg Diagram
The flight operating strength of an aircraft is presented
on a graph whose vertical scale is based on load factor.
[Figure 5-55] The diagram is called a Vg diagram—velocity
versus G loads or load factor. Each aircraft has its own Vg
diagram that is valid at a certain weight and altitude.
The lines of maximum lift capability (curved lines) are the
first items of importance on the Vg diagram. The aircraft in
Figure 5-53 is capable of developing no more than +1 G at
64 mph, the wing level stall speed of the aircraft. Since the
maximum load factor varies with the square of the airspeed,
the maximum positive lift capability of this aircraft is 2 G at
92 mph, 3 G at 112 mph, 4.4 G at 137 mph, and so forth. Any
load factor above this line is unavailable aerodynamically
(i.e., the aircraft cannot fly above the line of maximum lift
capability because it stalls). The same situation exists for
negative lift flight with the exception that the speed necessary
to produce a given negative load factor is higher than that to
produce the same positive load factor.
If the aircraft is flown at a positive load factor greater than the
positive limit load factor of 4.4, structural damage is possible.
When the aircraft is operated in this region, objectionable
permanent deformation of the primary structure may take place
and a high rate of fatigue damage is incurred. Operation above
the limit load factor must be avoided in normal operation.
There are two other points of importance on the Vg diagram.
One point is the intersection of the positive limit load factor
and the line of maximum positive lift capability. The airspeed
at this point is the minimum airspeed at which the limit load
can be developed aerodynamically. Any airspeed greater than
this provides a positive lift capability sufficient to damage
the aircraft. Conversely, any airspeed less than this does not
provide positive lift capability sufficient to cause damage
from excessive flight loads. The usual term given to this speed
is “maneuvering speed,” since consideration of subsonic
aerodynamics would predict minimum usable turn radius or
maneuverability to occur at this condition. The maneuver
speed is a valuable reference point, since an aircraft operating
below this point cannot produce a damaging positive flight
load. Any combination of maneuver and gust cannot create
damage due to excess airload when the aircraft is below the
maneuver speed.
0
–1
2
–2
1
–3
6
5
4
7
3
20 40 60 80 100 120 140 160 180 200 220 240
Load factor
Indicated airspeed (mph)
Structural damage
Caution range
Structural failure
Normal operating rangeAccelerated stall
Structural Damage
Normal stall speed
Maneuvering speed
Level flight at 1 G
Never exceed speed
Figure 5-55. Typical Vg diagram.
The other point of importance on the Vg diagram is the
intersection of the negative limit load factor and line of
maximum negative lift capability. Any airspeed greater than
this provides a negative lift capability sufficient to damage
the aircraft; any airspeed less than this does not provide
negative lift capability sufficient to damage the aircraft from
excessive flight loads.
The limit airspeed (or redline speed) is a design reference point
for the aircraft—this aircraft is limited to 225 mph. If flight
is attempted beyond the limit airspeed, structural damage or
structural failure may result from a variety of phenomena.
The aircraft in flight is limited to a regime of airspeeds
and Gs that do not exceed the limit (or redline) speed, do
not exceed the limit load factor, and cannot exceed the
maximum lift capability. The aircraft must be operated
within this “envelope” to prevent structural damage and
ensure the anticipated service lift of the aircraft is obtained.
The pilot must appreciate the Vg diagram as describing the
allowable combination of airspeeds and load factors for
safe operation. Any maneuver, gust, or gust plus maneuver
outside the structural envelope can cause structural damage
and effectively shorten the service life of the aircraft.
Rate of Turn
The rate of turn (ROT) is the number of degrees (expressed in
degrees per second) of heading change that an aircraft makes.
The ROT can be determined by taking the constant of 1,091,
multiplying it by the tangent of any bank angle and dividing
that product by a given airspeed in knots as illustrated in
Figure 5-55. If the airspeed is increased and the ROT desired
is to be constant, the angle of bank must be increased,
otherwise, the ROT decreases. Likewise, if the airspeed is
held constant, an aircraft’s ROT increases if the bank angle
is increased. The formula in Figures 5-56 through 5-58
depicts the relationship between bank angle and airspeed as
they affect the ROT.
NOTE: All airspeed discussed in this section is true airspeed
(TAS).
Airspeed significantly effects an aircraft’s ROT. If airspeed is
increased, the ROT is reduced if using the same angle of bank
used at the lower speed. Therefore, if airspeed is increased
as illustrated in Figure 5-57, it can be inferred that the angle
of bank must be increased in order to achieve the same ROT
achieved in Figure 5-58.
1,091 x tangent of the bank angle
airspeed (in knots)ROT =
1,091 x tangent of 30°
120 knotsROT =
1,091 x 0.5773 (tangent of 30°)
120 knotsROT =
ROT = 5.25 degrees per second
Example The rate of turn for an aircraft in a
coordinated turn of 30 ° and traveling at
120 knots would have a ROT as follows.
Figure 5-56. Rate of turn for a given airspeed (knots, TAS) and
bank angle.
1,091 x tangent of 30°
240 knotsROT =
ROT = 2.62 degrees per second
An increase in speed causes a decrease in the
ROT when using the same bank angle.
Example Suppose we were to increase the speed to 240
knots, what is the ROT? Using the same
formula from above we see that:
Figure 5-57. Rate of turn when increasing speed.
1,091 x tangent of X
240 knots ROT (5.25) =
240 x 5.25 = 1,091 x tangent of X
240 x 5.25 = tangent of X 1,091
1.1549 = tangent of X
49° = X
Example Suppose we wanted to know what bank angle
would give us a rate of turn of 5.25 ° per second
at 240 knots. A slight rearrangement of the formula
would indicate it will take a 49 ° angle of bank to
achieve the same ROT used at the lower airspeed
of 120 knots.
Figure 5-58. To achieve the same rate of turn of an aircraft traveling
at 120 knots, an increase of bank angle is required.
120 knots
11.26 x tangent of bank angle
R =
1202
11.26 x tangent of 30° R =
V2
R = 11.26 x 0.5773
14,400
R = 2,215 feet
The radius of a turn required by an aircraft traveling at 120 knots
and using a bank angle of 30° is 2,215 feet
Figure 5-59. Radius at 120 knots with bank angle of 30°.
What does this mean on a practicable side? If a given
airspeed and bank angle produces a specific ROT, additional
conclusions can be made. Knowing the ROT is a given number
of degrees of change per second, the number of seconds it
takes to travel 360° (a circle) can be determined by simple
division. For example, if moving at 120 knots with a 30° bank
angle, the ROT is 5.25° per second and it takes 68.6 seconds
(360° divided by 5.25 = 68.6 seconds) to make a complete
circle. Likewise, if flying at 240 knots TAS and using a 30°
angle of bank, the ROT is only about 2.63° per second and it
takes about 137 seconds to complete a 360° circle. Looking at
the formula, any increase in airspeed is directly proportional
to the time the aircraft takes to travel an arc.
So why is this important to understand? Once the ROT is
understood, a pilot can determine the distance required to
make that particular turn, which is explained in radius of turn.
Radius of Turn
The radius of turn is directly linked to the ROT, which
explained earlier is a function of both bank angle and
airspeed. If the bank angle is held constant and the airspeed
is increased, the radius of the turn changes (increases). A
higher airspeed causes the aircraft to travel through a longer
arc due to a greater speed. An aircraft traveling at 120 knots
is able to turn a 360° circle in a tighter radius than an aircraft
traveling at 240 knots. In order to compensate for the increase
in airspeed, the bank angle would need to be increased.
The radius of turn (R) can be computed using a simple
formula. The radius of turn is equal to the velocity squared
(V2) divided by 11.26 times the tangent of the bank angle.
R = V2
11.26 × tangent of bank angle
Using the examples provided in Figures 5-56 through 5-58, the
turn radius for each of the two speeds can be computed.
Note that if the speed is doubled, the radius is quadrupled.
[Figures 5-59 and 5-60]
Another way to determine the radius of turn is speed using
feet per second (fps), π (3.1415), and the ROT. In one of the
previous examples, it was determined that an aircraft with
a ROT of 5.25 degrees per second required 68.6 seconds to
make a complete circle. An aircraft’s speed (in knots) can
240 knots
11.26 x tangent of bank angle
R =
2402
11.26 x tangent of 30°R =
V2
R = 11.26 x 0.57735
57,600
R = 8,861 feet
(four times the radius at 120 knots)
The radius of a turn required by an aircraft traveling at 240 knots
using the same bank angle in Figure 4-51 is 8,861 feet. Speed
is a major factor in a turn.
Figure 5-60. Radius at 240 knots.
Figure 5-61. Another formula that can be used for radius.
r = speed (fps) x 360
ROT
Pi (π)
2
202.8 x 68.6
π
2
r =
13,912
π
2
r =
4,428
2r = = 2,214 feet
be converted to fps by multiplying it by a constant of 1.69.
Therefore, an aircraft traveling at 120 knots (TAS) travels
at 202.8 fps. Knowing the speed in fps (202.8) multiplied by
the time an aircraft takes to complete a circle (68.6 seconds)
can determine the size of the circle; 202.8 times 68.6 equals
13,912 feet. Dividing by π yields a diameter of 4,428 feet,
which when divided by 2 equals a radius of 2,214 feet
[Figure 5-61], a foot within that determined through use of
the formula in Figure 5-59.
In Figure 5-62, the pilot enters a canyon and decides to turn
180° to exit. The pilot uses a 30° bank angle in his turn.
Weight and Balance
The aircraft’s weight and balance data is important
information for a pilot that must be frequently reevaluated.
Although the aircraft was weighed during the certification
process, this information is not valid indefinitely. Equipment
changes or modifications affect the weight and balance data.
Too often pilots reduce the aircraft weight and balance into
a rule of thumb, such as: “If I have three passengers, I can
load only 100 gallons of fuel; four passengers, 70 gallons.”
Weight and balance computations should be part of every
preflight briefing. Never assume three passengers are always
of equal weight. Instead, do a full computation of all items
to be loaded on the aircraft, including baggage, as well as
the pilot and passenger. It is recommended that all bags be
weighed to make a precise computation of how the aircraft
CG is positioned.
The importance of the CG was stressed in the discussion
of stability, controllability, and performance. Unequal load
distribution causes accidents. A competent pilot understands
and respects the effects of CG on an aircraft.
Weight and balance are critical components in the utilization
of an aircraft to its fullest potential. The pilot must know
how much fuel can be loaded onto the aircraft without
violating CG limits, as well as weight limits to conduct
long or short flights with or without a full complement of
allowable passengers. For example, an aircraft has four seats
and can carry 60 gallons of fuel. How many passengers can
the aircraft safely carry? Can all those seats be occupied at
all times with the varying fuel loads? Four people who each
weigh 150 pounds leads to a different weight and balance
computation than four people who each weigh 200 pounds.
The second scenario loads an additional 200 pounds onto the
aircraft and is equal to about 30 gallons of fuel.
The additional weight may or may not place the CG outside
of the CG envelope, but the maximum gross weight could
be exceeded. The excess weight can overstress the aircraft
and degrade the performance.
Aircraft are certificated for weight and balance for two
principal reasons:
1. The effect of the weight on the aircraft’s primary
structure and its performance characteristics
2. The effect of the location of this weight on flight
characteristics, particularly in stall and spin recovery
and stability
Aircraft, such as balloons and weight-shift control, do not
require weight and balance computations because the load
is suspended below the lifting mechanism. The CG range
in these types of aircraft is such that it is difficult to exceed
loading limits. For example, the rear seat position and fuel
of a weight-shift control aircraft are as close as possible to
the hang point with the aircraft in a suspended attitude. Thus,
load variations have little effect on the CG. This also holds
true for the balloon basket or gondola. While it is difficult
to exceed CG limits in these aircraft, pilots should never
overload an aircraft because overloading causes structural
damage and failures. Weight and balance computations are
not required, but pilots should calculate weight and remain
within the manufacturer’s established limit.
5,000 feet
20 20
I0 I0
I0 I0
20 20
TESTSTBY PWR
2,215 feet
4,430 feet
5 , 0 0 0 f e e t
20 20
I0 I0
I0 I0
20 20
TESTSTBY PWR
3,014 feet
6,028 feet
V2
11.26 x tangent of the bank angle 30°
R = 1202
11.26 x 0.5773
14,400
6.50096
R = = = 2,215 feet
V2
11.26 x tangent of the bank angle 30°
R = 1402
11.26 x 0.5773
19,600
6.50096
R = = = 3,014 feet
120 knots
140 knots
Figure 5-62. Two aircraft have flown into a canyon by error. The canyon is 5,000 feet across and has sheer cliffs on both sides. The pilot
in the top image is flying at 120 knots. After realizing the error, the pilot banks hard and uses a 30° bank angle to reverse course. This
aircraft requires about 4,000 feet to turn 180°, and makes it out of the canyon safely. The pilot in the bottom image is flying at 140 knots
and also uses a 30° angle of bank in an attempt to reverse course. The aircraft, although flying just 20 knots faster than the aircraft in the
top image, requires over 6,000 feet to reverse course to safety. Unfortunately, the canyon is only 5,000 feet across and the aircraft will hit
the canyon wall. The point is that airspeed is the most influential factor in determining how much distance is required to turn. Many pilots
have made the error of increasing the steepness of their bank angle when a simple reduction of speed would have been more appropriate.