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Copyright © 1996-2001 jsd
19 The Laws of Motion
There is no gravity.
The earth sucks.
— Physicist's bumper sticker
This chapter pulls together some basic physics ideas
that are used in several places in the book.
We will pay special attention to rotary motion, since
it is less familiar to most people than ordinary straight-line
motion. Gyroscopes, in particular, behave very differently from
ordinary non-spinning objects. It is amazing how strong the gyroscopic
effects can be.
19.1 Straight-Line MotionFirst, let's review the physical laws that govern
straight-line motion. Although the main ideas go back to Galileo,
we speak of Newton's laws, because he generalized the ideas and
codified the laws.
The first law of motion states: ``A body
at rest tends to remain at rest, while a body in motion tends
to remain in motion in a straight line unless it is subjected
to an outside force''. Although that may not sound like a
very deep idea, it is one of the most revolutionary statements
in the history of science. Before Galileo's time, people omitted
frictional forces from their calculations. They considered friction
``natural'' and ubiquitous, not requiring explanation;
if an object continued in steady motion, the force required to
overcome friction was the only thing that required explanation.
Galileo and Newton changed the viewpoint. Absence of friction
is now considered the ``natural'' state, and frictional
forces must be explained and accounted for just like any others.
The second law of motion says that if there
is any change in the velocity of an object, the force required
(F) is proportional to the mass (m) of the object,
and proportional to the acceleration (a). (Acceleration
is defined to be the rate-of-change of velocity.) In symbols,
The following restatement of the second law is often
useful: since momentum is defined
to be mass times velocity, and since the mass is not supposed
to be changing, we conclude that the force is equal to the rate-of-change
of the momentum. To put it the other way, change in momentum
is force times time.
The third law of motion says that if a force
is applied to an object, an equal and opposite force must be applied
somewhere else. This, too, can be restated in terms of momentum:
if we impart a certain momentum to an object, we must impart an
equal and opposite amount of momentum to something else.
This means that the total momentum of the world cannot change. This
principle — conservation of momentum
— is one of the
most fundamental principles of physics, on a par with the conservation
of energy discussed in chapter 1.
* Force is Not MotionAs simple as these laws are, they are widely misunderstood.
For example, there is a widespread misconception that an airplane
in a steady climb requires increased upward force and a steady descent
requires reduced upward force.
Remember, lift is a force, and any unbalanced force would cause an
acceleration, not steady flight.
In unaccelerated flight (including steady climbs
and steady descents), the upward forces (mainly lift) must balance
the downward forces (mainly gravity). If the airplane had an
unbalanced upward force, it would not climb at a steady rate —
it would accelerate upwards with an ever-increasing vertical
Of course, during the transition from level flight
to a steady climb an unbalanced vertical force must be applied
momentarily, but the force is rather small. A climb rate of 500
fpm corresponds to a vertical velocity component of only 5 knots,
so there is not much momentum in the vertical direction. The
kinetic energy involved in the vertical motion is negligible.
In any case, once a steady climb is established, all the forces are in
19.2 Sitting in a Rotating FrameSuppose Moe is riding on a turntable; that is, a large, flat, smooth,
horizontal rotating disk. Moe has painted an X, Y grid on the
turntable, so he can easily measure positions, velocities, and
accelerations relative to the rotating coordinate system. His friend
Joe is nearby, observing Moe's adventures and measuring things
relative to a nonrotating coordinate system.
Moe quickly notices that Newton's first law does not apply in a
rotating reference frame. A hockey puck initially at rest on the
floor (anywhere except right at the center of rotation) will be
In Joe's nonrotating frame, Newton's laws do apply. In a
nonrotating frame, there is no such thing as centrifugal acceleration.
Moe holds the puck in place with a rubber band
(one end attached to the puck, the other end attached to the floor).
Joe sees the same rubber band and the same force. The elongation of
the rubber band indicates the magnitude of the force.
Moe says the puck is not moving relative to his reference frame. The
rubber band compensates for the centrifugal force.
Joe says that the puck's momentum is constantly
changing due to the rotation. The rubber band provides the necessary
19.3 Moving in a Rotating FrameWe now consider what happens to an object that is moving
relative to a rotating reference frame.
Suppose Moe has another hockey puck, which he attaches by means of a
rubber band to a tiny tractor. He drives the tractor in some
arbitrary way. We watch as the puck passes various marks (A, B,
etc.) on the floor.
Moe sees the puck move from mark A to mark B. The marks obviously
are not moving relative to his reference frame.
Joe agrees that the puck moves from mark A to mark B, but he must
account for the fact that the marks themselves are moving.
So let's see what happens when Joe analyzes the compound motion,
including both the motion of the marks and the motion of the puck
relative to the marks. He actually sees four contributions:
We can say a few words about each of these contributions from Moe's
point of view:
If the puck is accelerating relative to Moe's rotating frame,
Joe agrees and counts that as a contribution to the acceleration.
Both observers agree on how much force this requires.
- From Joe's point of view, mark A is not only moving; its
velocity is changing. Changing this component of the puck's velocity
requires a force. (From Moe's point of view, this is the force needed
to oppose centrifugal acceleration, as discussed previously.)
- The velocity of mark B is different from the velocity of mark
A. As the puck is towed along the path from point A to point B,
the rubber band must provide a force in order to change the velocity
so the puck can ``keep up with the Joneses''.
- The velocity of the puck relative to the marks is also a
velocity, and it must also rotate as the system rotates. This change
in velocity also requires a force.
Contribution #3 is numerically equal to contribution #4. The total
effect is just twice what you would get from either contribution
separately. Together these two contributions are called the
This ``F=ma'' contribution is completely unsurprising. It
is independent of position, independent of velocity, and independent
of the frame's rotation rate.
- The centrifugal contribution depends on position, but is
independent of the velocity that Moe measures relative to his rotating
reference frame. It is also independent of any acceleration created
by Moe's tractor. It is proportional to the square of the frame's
- This contribution is independent of position. It is
proportional to the velocity that Moe measures, and is always
perpendicular to that velocity. It is also proportional to the first
power of the frame's rotation rate.
- This contribution is also independent of position. It is also
proportional to the velocity relative to the rotating frame, and is
perpendicular to that velocity, and is proportional to the first power
of the frame's rotation rate.
The Coriolis effect can be described as an acceleration
(proportional to the object's velocity), and equivalently it can be
described as a force (proportional to the object's momentum).
Let's consider a reference frame attached to an eastward-rotating
rotating planet, such as the earth. Near the north pole, the Coriolis
acceleration is always toward your right, if you are facing forward
along the direction of motion. Northward motion produces a
Coriolis acceleration to the east; a very real westward force is
necessary to oppose it if you want to follow a straight line painted
on the earth. Eastward motion produces a Coriolis acceleration to the
south; a very real northward force is necessary to oppose it.
The Coriolis argument only applies to motion in the plane of rotation.
Momentum parallel to the axis is unaffected. In all cases the Coriolis
acceleration lies in the plane of rotation and perpendicular to the motion.
Near the equator, we have to be careful, because the plane of rotation
is not horizontal. In this region, eastward motion produces a Coriolis
acceleration in the upward direction, while westward motion produces a
Coriolis acceleration in the downward direction. In this region, north/south
motions are parallel to the axis of rotation and produce no Coriolis
To reiterate: The Coriolis effect and the centrifugal field are two
separate contributions to the story. The Coriolis effect applies
only to objects that are moving relative to the rotating
reference frame. The centrifugal field affects all objects in the
rotating frame, whether they are moving or not.
* Magnitude of the EffectSuppose you are in an airplane, flying straight ahead at 120 knots
along the shortest path between two points on the earth's surface.
Because of the rotation of the earth, the airplane will be subject to
a Coriolis acceleration of about 0.001G. This is too small to be
Now suppose you and a friend are standing 60 feet apart, playing catch
in the back of a cargo plane while it is performing a standard-rate
turn (three degrees per second). If your friend throws you the ball
at 60 mph, it will be subject to a horizontal Coriolis acceleration of
more than a quarter G. That means the ball will be deflected
sideways about 2
feet before it gets to you — which is enough to
be quite noticeable. In normal flying, though, we don't often throw
things far enough to produce large Coriolis effects.
The wind, moving relative to the rotating earth, is subject to a
Coriolis acceleration that is small but steady; the cumulative effect
is tremendously important, as discussed in section 20.1.
19.4 Centrifuges with and without Gravity
19.4.1 The Centrifugal Field is RealAn airplane in a turn, especially a steep turn, behaves like a
centrifuge. There are profound analogies between centrifugal and
The gravitational field at any given point is an acceleration. It
acts on objects, producing a force in proportion to the object's mass.
The centrifugal field at any given point is also an
acceleration. It, too, acts on objects, producing a force in
proportion to the object's mass.
Strictly speaking, neither gravity nor centrifugity is a ``force''
field. Each is
really an acceleration field. Of course there is a force
involved, but it is always a force per unit mass, which is properly
called an acceleration.
Einstein's principle of equivalence states that at any given point,
the gravitational field is indistinguishable from an
acceleration of the reference frame.4 In a freely-falling reference
frame, such as a freely-orbiting space station, everything is
My laboratory is not a free-falling inertial frame.
It is being shoved skyward as the earth pushes on its foundations.
If you measure things relative to the laboratory walls,
you will observe gravitational accelerations.
Similarly, the cabin of a centrifuge is clearly
not an inertial frame. If you measure things
relative to the cabin, you will observe
From a modern-physics point of view, both gravity and centrifugity
emerge as consequences of working in an accelerated frame. There
is nothing wrong with doing so, provided the work is done carefully.
Accounting for centrifugal effects is not much trickier than
accounting for gravitational effects. When people think this
can't be done, it is just because they don't know how to do it.
For a ground-bound observer analyzing the flight of an airplane,
it may be convenient to use a reference frame where gravity exists
and centrifugity does not. However, the pilot and passengers
usually find it convenient to use a frame that includes both gravity
The centrifugal field is not crude or informal or magical. (The
problem with magic is that it can explain false things just as
easily as true things.) Like the gravitational field, it is a
precise way of accounting for what happens when you work in a
non-freely-falling reference frame.
To get a better understanding of the balance of forces in a turning and/or
slipping airplane, consider the centrifuge shown in
figure 19.1. For the moment we will neglect the effects of
gravity; imagine this centrifuge is operating in the weightless
environment of a space station. We are riding inside the centrifuge
cabin, which is shown in red. We have a supply of green tennis
balls. At point A (the southernmost point of our path) we drop a
tennis ball, whereupon it flies off as a free particle. Our
centrifuge continues to follow its circular path.
Case 1a: Consider the point of view of a bystander
(not riding in the centrifuge). The dropped tennis ball moves
in a straight line, according to the first law of motion. Contrary
to a common misconception, the bystander does not see the
ball fly radially away from the center of the centrifuge. It just
continues with the purely eastward velocity it had at point A,
Case 1b: Consider our point of view as we ride in
the centrifuge. At point A, the tennis ball has no velocity
relative to us. For the first instant, it moves along with us,
but then gradually it starts moving away. We do see the
ball accelerate away in the purely radial direction. The tennis
ball — like everything else in or near the centrifuge — seems
to be subjected to a centrifugal acceleration field.
Einstein's principle of equivalence guarantees that our
viewpoint and the bystander's viewpoint
are equally valid. The bystander says that the centrifuge cabin
and its occupants accelerate away from the freely moving tennis
ball, while we say that the tennis ball accelerates away from
us under the influence of the centrifugal field.
There is one pitfall that must be avoided: you can't freely mix the
two viewpoints. It would be a complete fallacy for the bystander to
say ``gee, the folks in the cabin told me the tennis ball accelerated
outward; therefore it must move to the south starting from point
A''. In fact, the free-flying ball does not accelerate
relative to the bystander. It will not wind up even one millimeter
south of point A. It will indeed wind up south of our
centrifuge cabin, but only because we have peeled off to the north.
Case 2a: Consider from the bystander's point
of view what happens to a ball that has not been released,
but is just sitting on a seat in the centrifuge. The bystander
sees the ball subjected to an unbalanced force, causing it to
move in a non-straight path relative to the earth.
Case 2b: Consider the seated ball from the centrifuge-riders'
point of view. The force on the ball exerted by the seat is just
enough to cancel the force due to centrifugal acceleration, so
the forces are in balance and the ball does not move.
When analyzing unsteady motion, or when trying to
calculate the motion of the centrifuge itself, it is often simpler
to analyze everything from the bystander's point of view, in which
the centrifugal field will not appear. On the other hand, in
a steady turn, is often easy and natural to use the centrifuge-riders'
point of view; in which all objects will be subject to centrifugal
19.4.3 Centrifuge and GravityNow that we understand the basic idea, let's see what happens when our
centrifuge operates in the normal gravitational field of the
earth. This is shown in figure 19.2. When the tennis ball
departs the centrifuge, it once again travels in a purely easterly
direction, but this time it also accelerates downward under the
influence of gravity.
: An Object Departing a Centrifuge, with Gravity
Once again, from inside the cabin we observe that the tennis ball
initially accelerates away in the direction exactly away from the
pivot of the centrifuge. This is no coincidence; it is because the
only difference between our motion and the free-particle motion
comes from the force in the cable that attaches us to the pivot.
(The foregoing applies only to the initial acceleration of the
dropped ball. As soon as it picks up an appreciable velocity relative
to us, we need to account for Coriolis acceleration as well as
Remember, the equivalence principle says that at
each point in space, a gravitational field is indistinguishable
from an accelerated reference frame. Therefore we need not know
or care whether the tennis ball moves away from us because we
are being accelerated, or because there is a gravitating planet
in the vicinity, or both.
19.5 Centrifugal Effects in a Turning AirplaneLet's examine the forces felt by the pilot in a turning airplane.
We start with a coordinated turn, as shown in figure 19.3.
In figures such as this, whenever I am analysing things using the
pilot's point of view, the figure will include a rectanglular
``frame'' with a little stick figure (the observer) standing in the
corner. It is important to carefully specify what frame is being
used, because even simple questions like ``which way is down'' have
answers that depend on which observer you ask. In particular, I
define N-down (Newtonian down) to mean the direction straight
down toward the center of the earth. In contrast, I define
E-down (effective down, or Einsteinian down) to be the direction
in which a free particle departs if you drop it. In a turning
airplane, the two directions are not the same.
Using your inner ear, the seat of your pants, and/or
the inclinometer ball, you can tell which way is E-down. Using
the natural horizon and/or the artificial horizon, you can tell
which way is N-down.
In figure 19.3, assume the airplane's mass is one ton.
Real gravity exerts a force of one ton, straight down toward the
center of the earth. The airplane is an a 45° bank, so there is
one ton of centrifugal force, sideways, parallel to the earth's
horizon. All in all, the wings are producing 1.41 tons of lift,
angled as shown in the figure.
Figure 19.4 shows the same situation, and analyzes the
forces on the inclinometer ball. Real gravity exerts a downward force
on the ball, and centrifugity exerts a sideways force. The tubular
race that contains the ball exerts a force perpendicular to the wall
of the race (whereas the ball is free to roll in the direction along
the race). The race-force balances the other forces when the ball is
in the middle, confirming that this is a coordinated turn.
As a contrast, consider figure 19.5, which
shows what happens if you make a boat turn.
(Boat turns are discussed in section 8.10.)
Because the airplane is turning, it and everything
in it will be subjected to a centrifugal acceleration (according
to the viewpoint of the centrifuge riders).
Figure 19.6 shows how the inclinometer ball responds to a
boat turn. Gravity still exerts a force on the ball, straight
down. Centrifugity exerts a force sideways toward the outside of the
turn. The ball is subject to a force of constraint, perpendicular to
the walls of the race. (It is free to roll in the other direction.)
The only place in the race where this constraint is in a direction to
balance the other forces is shown in the figure. The ball has been
``centrifuged'' toward the outside of the turn. This is a
quantitative indication that the E-down direction is not perpendicular
to the wings, and some force other than wing-lift is acting on the
Finally, figure 19.7 shows the forces on the airplane in an
ordinary nonturning slip. The right rudder pedal is
and the port wing has been lowered just enough that the horizontal
component of lift cancels the horizontal force due to the crossflow
over the fuselage. The airplane is not turning. The bystander and
the riders agree that there is no centrifugal field. Figure 19.8 shows the forces on the inclinometer ball in the same
19.6 Angles and Axes
19.6.1 Axes: Yaw, Pitch, and RollThe standard definitions of the yaw, pitch, and roll
axes are shown in figure 19.9.
Many older books refer to the pitch axis as the lateral
axis and the roll axis as the longitudinal axis,
which are sensible enough, but then they refer to pitch-axis stability
as longitudinal stability and roll-axis stability as
lateral stability — which seems completely reversed and is
confusing at best. Reference 22 calls the yaw axis the
normal axis, since it is normal (i.e. perpendicular) to the
other axes — but that isn't very helpful since all the axes
are normal to all the others. Other references call the yaw axis the
vertical axis, but that is very confusing since if the bank
attitude or pitch attitude is not level, the yaw axis will not be
Obviously the new terminology is an improvement. The situation is
summarized in the following table.
19.6.2 Attitude: Heading, Pitch, BankThe term attitude describes the orientation of the airplane's
axes relative to the earth. Attitude is specified in terms of three
angles: the heading, the pitch attitude, and the bank attitude.
To place the airplane in a specified attitude, start
with the airplane in level flight (i.e. X axis and Y axis
horizontal) and with the airplane pointed north. Then
As discussed below (section 19.6.4),
it is important to perform these rotations in the order specified.
- Rotate the airplane around the yaw axis by the
specified heading angle. A positive
angle specifies a clockwise rotation as seen from above, so that
a heading of 090 degrees corresponds to pointing east and a heading
of 180 degrees corresponds to pointing south.
- Rotate the airplane around the pitch axis by
the specified pitch attitude
angle. A positive angle specifies a nose-up attitude.
- Rotate the airplane around the roll axis by the
specified bank attitude angle. A positive angle corresponds to
clockwise as seen from the rear.
We have just seen how, given a set of angles, we
can put the airplane into a specified attitude. We now consider
the reverse question: given an airplane in some attitude, how
do we determine the angles that describe that attitude?
Answer: just figure out what it would take to return
the airplane to level northbound flight. The rotations must be
undone in the reverse of the standard order:
- First, rotate the aircraft around the roll axis
until the wings are level. This determines the bank attitude.
- Second, rotate the aircraft around the pitch
axis until the X axis is level. This determines the pitch attitude.
Note that this rotation is not performed around the original
pitch axis, but rather around the new pitch axis, which is horizontal
as a consequence of the previous step.
- Finally, rotate the aircraft around the (new)
yaw axis until the nose is pointing north. This determines the
19.6.3 Angle TerminologyThe following table summarizes the various nouns
and verbs that apply to angles and motions around the three axes.
||the pitch attitude
||the bank attitude
There are a few more fine points of X-axis terminology:
- Saying that the airplane is ``banking'' or ``in a bank''
refers to a definite bank attitude.
- In contrast, saying that the airplane is ``rolling'' or ``in a
roll'' refers to a definite rate of rotation, i.e. a changing bank
- Pitch angle usually means the same thing as pitch
- Bank angle usually means the same thing as bank
- The term roll angle is a rarely-used synonym for bank angle.
* Other AnglesTo define the angle of attack of the fuselage, take the
direction of flight (or its reciprocal, the relative wind) and project
it onto the XZ plane. The angle of attack is the angle between this
projection and the X axis or some other convenient reference.
To define the slip angle, take the direction of flight (or the
relative wind) and project it onto the XY plane. The slip angle is
the angle between this projection and the X axis.
Some aerodynamics texts use the term sideslip angle, which is
synonymous with slip angle. This may be confusing because
some pilot-training books try to draw a distinction between a
forward slip and a side slip, even though the difference
is imaginary, as is discussed in conjunction with figure 11.1.
19.6.4 Yaw Does Not Commute with PitchIt is a fundamental fact of geometry that the result
of a sequence of rotations depends on the order in which
the rotations are performed.
Note that for a sequence of ordinary non-rotational
movements, the ordering does not matter. That is, suppose I have
two small objects that start out at the same place. I move one
object move two feet north, and then three feet west. I move
the other object the same distances in the other order: three
feet west and then two feet north. Assuming there are no obstructions,
both objects will arrive at the same destination. The ordering
of the movements does not matter.
However, angles don't play by the same rules as distances.
For instance, it is possible to change the yaw angle (i.e., the
heading) by 37 degrees (or any other amount) without ever yawing
the plane. That is:
If the aircraft (and its occupants) can tolerate
heavy G loads, such maneuvers are perfectly
fine ways to make tight turns at high airspeed.
- You can pull up into aerobatic vertical flight,
use the ailerons to rotate 37 degrees around the now-vertical
roll axis, and then push back to level flight.
- Another way to do the same thing is to roll into
a ninety-degree bank, pull on the yoke to rotate around the now-vertical
pitch axis by 37 degrees, and then roll back to wings-level
In non-aerobatic flight, a less-extreme statement
applies: a rotation in a purely horizontal plane is not
a pure yaw when the aircraft is not in a level attitude. For
instance, suppose you are in level flight, steadily turning to
the left. This is, of course, a turn around an axis that runs
vertically through the center of the earth. Further suppose that
you have a nose-up pitch attitude, while still maintaining a level
flight path. This means that the airplane's yaw axis (Z axis)
is not vertical. You could, in principle, perform the required
heading change by pitching down to level pitch attitude, performing
a pure yaw, and then pitching back up, but since rotations are
not commutative this is not equivalent to maintaining your pitch
attitude and performing a pure yaw. Performing the required change
of heading without pitching down requires mostly pure leftward
yaw, but involves some rightward rotation around the airplane's
roll axis as well.
The analysis in the previous paragraph is 100% accurate, but
completely irrelevant when you are piloting the airplane.
In particular, the rotation around the roll axis
does not raise the angle of attack on one wingtip or lower it on the
other. Arguing about whether the heading change is a pure yaw or a
yaw plus roll is almost like arguing about whether a glass of water is
half full or half empty — the physics is the same. In this case the
physics is simple: the inside (left) wing follows a horizontal
circular path, while the outside (right) wing follows a slightly
longer horizontal path around a larger circle.
It is easy to see why that is so: The turn requires a rotation around
a vertical axis. Such a rotation moves the wingtips (and
everything else) in purely horizontal directions. As long as the
airplane's center-of-mass motion is also horizontal, the rotation can
only change the speeds, not the angles, of the airflow.
Now, things get more interesting when the direction
of flight is not horizontal. Therefore let us consider a new
example in which you are climbing while turning. That means your
flight path is inclined above the horizontal. As before, you
are turning to the left at a steady rate.
In any halfway-reasonable situation, the yaw axis will be nearly
to the direction of flight. Having it not exactly perpendicular is
just a distraction from the present topic, so I hereby define a
new ``yaw-like'' axis exactly perpendicular the direction of flight.
It is also perpendicular to the pitch axis (which remains parallel to
the wingtip-to-wingtip direction). The third axis is a new
``roll-like'' axis perpendicular to the other two. If we assume zero
slip angle, it will be aligned with the direction of flight.
As the airplane flies from point to point along its curving path, its
heading must change. This is a rotatation around a purely vertical
axis. In climbing flight, the yaw-like axis is not vertical, so the
turn is not pure yaw. The turn moves the inside wingtip horizontally
backwards, relative to where it would be if there were no heading
change. In contrast, a pure yaw-like rotation would have moved the
wing back and down. Therefore we need not just leftward
yaw-like rotation but also some rightward roll-like rotation to keep
the wingtip moving along the actual flight path.
This roll-like motion means that (other things being equal) the inside
wingtip would fly at a lower angle of attack during a climbing turn.
Less lift would be produced. You need to deflect the ailerons to the
outside to compensate.
Note that I said less lift ``would be'' produced, not ``is'' produced.
Because you have compensated with the ailerons, both wings are in fact
producing the same amount of lift, as they should. Remember that this
is a steady turn, so no force is required to maintain the steady
roll rate. (Remember, according to Newton's laws, an unbalanced force
would create an acceleration around the roll axis, which is not
what is happening here.) There are widespread misconceptions about
this. Because of the roll-like motion, the air will arrive at the two
wings from two different directions.
You will have to deflect the ailerons, not in order to create a
magnitude of lift, but rather to keep from creating a different
magnitude of lift.
Although the magnitude of the lift remains the same, the direction of
the lift will be twisted, as discussed in
section 8.8.4; see in particular
figure 8.7. You will need to deflect the rudder to
overcome the resulting yawing moment. This will be in the usual
direction: right rudder in proportion to right aileron deflection, and
left rudder in proportion to left aileron deflection.
In a climbing turn, the differential relative wind combines with the
differential wingtip velocity to create a large overbanking tendency.
In an ordinary descending turn, the relative wind effect tends to
oppose the velocity effect. In a spin, the differential relative wind
is a key ingredient, as discussed in section 18.6.1, including
figure 18.6. Also, section 9.7
analyzes climbing and descending turns in slightly different words and
gives a numerical example.
19.6.5 Yaw Does Not Commute with BankAs stated above, a rotation in a purely horizontal
plane is not a pure yaw when the aircraft is not in a level attitude.
In the previous section we considered the consequences of a non-level
pitch attitude, but the same logic applies to a non-level bank
attitude. The latter case is in some sense more significant,
since although not all turns involve a non-level pitch attitude,
they almost always involve a bank.
You could perform the required rotation by rolling
to a level attitude, performing a pure yaw, and then rolling back
to the banked attitude. This is not equivalent to performing
a pure yaw while maintaining constant bank. For modest bank angles,
the constant-bank maneuver is mostly pure yaw, but involves
some rotation around the pitch axis as well. Because of this
rotation around the pitch axis, the relative wind hits the wing
and the tail at slightly different angles. You will need to pull
back on the yoke slightly to compensate. This pull is in addition
to whatever pull you might use for controlling airspeed during
the turn. You can see that the two phenomena are definitely distinct,
by the following argument: suppose that you maintain constant
angle of attack during the turn, so that the required load factor
is produced by increased airspeed not increased angle of attack.
You would still need to pull back a little bit, to overcome the
19.7 Torque and MomentJust as Newton's first law says that to start an object moving you
have to apply a force, there is a corresponding law that says to start
an object turning you need to apply a torque.
You may have heard of the word ``torque''
in conjunction with left-turning tendency on takeoff, and you
may have heard of the word ``moment'' in conjunction with
weight & balance problems. In a deep sense, the two words
are two names for the same thing. In particular,
A familiar example: fuel and cargo cause a pitching
moment, depending on how far forward or aft they are loaded.
By the same token, they will cause a rolling moment if they are
loaded asymmetrically left or right.
- A rolling moment is a torque around the roll
- A pitching moment is a torque around the pitch
- A yawing moment is a torque around the yaw axis.
Another familiar example: gyroscopic effects can
cause torque around the yaw axis. By the same token, they can
cause torque around the pitch axis as well.
I will use the terms ``torque'' and ``moment'' interchangeably.
Torque is not the same as force. Of the two, force
is the more familiar concept. If you attach a string to an object
and pull, the object is subjected to a force in the direction
of the string. Force is measured in pounds or newtons.
To apply a torque,
you need a force and a lever-arm. The amount of torque is given by
the following formula:
where the distance is a vector representing the separation between the
pivot-point6 and the point where the force is
applied. In this formula, we are using the vector
cross product,7 indicating
that we only count the component of the distance perpendicular to the
force. This quantity (the component of the distance perpendicular to
the force) is called the lever arm, which makes it convenient
to say that the torque is the force times the lever arm.
torque = force × distance
Torque is measured not in pounds but in footpounds (that is, feet times
pounds); the corresponding metric unit is newtonmeters.
Figure 19.10 shows a situation where all the forces and
torques are in balance. On the right side of the
bar, a group of three springs is exerting a force of 30 pounds. On
the left side of the bar, there is a group of two springs (exerting a
force of 20 pounds) and a single spring (exerting a force of 10
pounds). Since the total leftward force equals the total rightward
force, the forces are in balance.
To show that the torques are in balance requires
a separate check. Let's choose the point marked ``x''
as our pivot point. The rightward force produces no torque, because
it is attached right at the pivot point — it has a zero-length
lever arm. The group of two springs produces a counterclockwise
torque, and the single spring produces a clockwise torque of the
same magnitude, because even though it has half as much force
it has twice the lever arm. The torques cancel. The system is
Figure 19.11 shows a different situation.
The forces are in balance (20 pounds to the right, 20 pounds
total to the left) but the torques are not in balance. One of
the left-pulling springs has twice the lever arm, producing a
net clockwise torque. If you tried to set up a system like this,
the bar would immediately start turning clockwise. The system
is out of equilibrium.
Torque is a vector; that is, it has direction as well as magnitude. A
torque around a given axis is represented by a torque vector
along that axis. In particular, the clockwise torque in figure 19.11 is represented by a vector perpendicular to the
paper, directed away from the reader.
19.8 Angular MomentumThe notion of angular momentum is the key to really
understanding rotating objects.
Angular momentum is related to ordinary straight-line
momentum in the same way that torque is related to ordinary straight-line
force. Here is a summary of the correspondences:
Torque (equals force times lever arm)
Angular momentum (equals ordinary momentum times lever arm)
The ordinary momentum of a system
won't change unless a force is applied.
The angular momentum of a system
won't change unless a torque is applied.
The change in straight-line momentum is the force times the time.
The change in angular momentum is the torque times the time.
When I give lectures, I illustrate conservation of angular
momentum using a demo you can
easily set up for yourself. As
illustrated in figure 19.12, tie some kite string to a small
bean-bag and swing it in a circle. When you pull on the free end of
the string (reducing the radius of the circle) the bean-bag speeds up.
When you let out the string (increasing the radius of the circle) the
bean-bag slows down.9
In typical textbooks, conservation of angular momentum is exemplified
by spinning ice skaters, but I find it easier to travel with a
bean-bag (rather than an ice skater) in my luggage.
In the demonstration, there are some minor torques due to friction
than will eventually slow down the bean-bag whether or not you shorten
or lengthen the string, but if you perform the experiment quickly
enough the torques can be neglected, and the angular momentum of the
system is more or less constant. Therefore, if you decrease the lever
arm by a factor of N, the straight-line momentum must increase by a
factor of N (since their product cannot change).10
Since the tangential velocity increases by a factor of N, and
the radius decreases by a factor of N, the rate of turn (degrees
per second) increases by a factor of N squared.
The energy of the system also increases by a factor of N
squared. You can feel that you added energy to the system when
you pull on the string, pulling against tension.
So far we have analyzed the situation from the point of view of a
bystander in a non-rotating reference frame. You can reach the same
conclusion by analyzing the situation in the rotating reference frame,
as would apply to an ant riding on the bean-bag. The ant would say
that as the string is pulled in, the bean-bag accelerates sideways
because of the Coriolis effect, as discussed in
Conservation of angular momentum applies to airplanes as well
as bean-bags. For instance, consider an airplane in a flat spin,
as discussed in section 18.6.4. In order to recover
from the spin, you need to push the nose down. This means whatever
mass is in the nose and tail will move closer to the axis of rotation.
The angular momentum of the airplane doesn't change (in the short
run), so the rotation will speed up (in the short run). More
rotation may seem like the opposite of what you wanted, but remember
you are trying to get rid of angular momentum, not just angular
rate. You should persevere and force the nose down. Then the
aerodynamic forces (or, rather, torques) will carry angular momentum
out of the system and the rotation will decrease.
Angular momentum is a vector; that is, it has direction as well as
magnitude. The angular momentum vector points
more-or-less11 along the axis of rotation.
In particular, the angular momentum of an aircraft propeller (rotating
clockwise as viewed from behind) is directed forward along the
axis of rotation.
For any normal object (such as a book) if you apply
a force in a given direction, it will respond with motion in that
direction. People are so accustomed to this behavior that they
lose sight of the fact that force and motion are not exactly the
same thing, and they don't always go together.
In particular, for a gyroscope, if you apply a torque around one axis,
it will respond with motion around a different axis. When I give my
``See How It Flies'' lectures, I carry around a bicycle
handles, as shown in figure 19.13. The indicated direction of spin
corresponds to a normal American engine and propeller, if the nose of
the airplane is toward the left side of the diagram.
To demonstrate the remarkable behavior of a gyroscope,
I stand behind the ``propeller'' (on the right side of
the diagram) and support its weight by lifting the rear handle
only. The force of gravity acts on the center of the system,
so there is a pure nose-down / tail-up pitching moment. If this
were a normal, non-spinning object, it would respond by pitching
in the obvious way, but the gyroscope actually responds with a
pure yawing motion. I have to turn around and around to my left
to stay behind the wheel.
It is really quite amazing that the wheel does not pitch down. Even
though I am applying a pitching force, the wheel doesn't pitch down;
it just yaws around and around.
This phenomenon, where a gyro responds to a torque about one axis with
a motion about another axis, is called gyroscopic precession.
Precession is often quite noticeable at the point where a taildragger
raises the tail, early in the takeoff roll. If the airplane were an
ordinary non-spinning object, you could raise the tail using the
flippers alone. But note that airflow over the flippers does not
actually dictate the motion of the airplane; it just produces a
force and a torque around the pitch axis. For a gyroscope, a
torque around the pitch axis produces a motion around the yaw axis.
If you try to raise the tail of a real airplane using flippers alone,
it will yaw to the left because of precession.
To get a gyroscope to actually move around the pitch axis, you
need to apply a torque around the yaw axis — using the rudder.
Of course, an airplane has some plain old mass in
addition to its gyroscopic properties. In order to lift this
ordinary mass you need to use the flippers. Therefore, the tail-raising
maneuver requires both flippers and rudder — flippers
to change the pitch of the ordinary mass, and rudder to change
the pitch of the gyroscope.
19.9.2 The Torque is 90 Degrees Ahead of the MotionNow that you've got the idea that a torque around the pitch axis
causes a motion around the yaw axis, you may be wondering how to
figure out the direction. Will it be yaw to the right, or yaw
to the left? There is a simple rule: the axis of the applied torque
will be 90 degrees ahead of the axis of the motion.12 An application of this
rule is depicted in figure 19.15.
: The Applied Torque is 90 Degrees
Ahead of The Motion
If you apply forward pressure on the yoke, it applies a torque around
the pitch axis in the direction that your fingers curl in
Now rotate your hand 90 degrees in the direction that the propeller is
spinning. Your fingers now specify the direction of motion,
as shown in figure 19.15(b). The nose will move to the left.
Conclusion: forward pressure on the yoke makes the
nose move to the left.
As another application of this rule, you can convince yourself that
right rudder deflection will cause a nose-down / tail-up pitching
The applied torque is 90 degrees ahead of the motion.
Superficially, this gyroscope law is reminiscent of the Coriolis effect
(force in one direction, motion in a perpendicular direction) but they
do not represent the same physics. The Coriolis law only applies to
objects that are moving relative to a rotating observer. In contrast,
the gyroscope law applies to a stationary observer; no parts of
the gyroscope are moving relative to each other.
19.9.3 Angular Momentum is the Key IdeaPrecession, and all the other gyro effects discussed
in this section, are direct consequences of the angular momentum
laws discussed in the previous section.
Figure 19.16 is the same as figure 19.14,
but with the torque and angular momentum
vectors drawn in. The torque around the pitch axis is represented
by a vector along the pitch axis — to my left if I am standing
behind the wheel. The angular momentum is represented by a vector
along the axis of rotation, directed away from me since the wheel
is spinning clockwise as seen from behind. The change in angular
momentum is (according to the laws of motion) equal to the torque
times the time.
The figure shows what happens during one unit of
time (one second). The new angular momentum is the vector sum
of the old angular momentum, plus the torque times the time.
Since we are considering one second, the change vector
is equal in magnitude to the torque vector.
During subsequent seconds, the torque vector will
be a new direction because the whole system has rotated. The
successive changes will pull the angular momentum vector around
in a horizontal circle.
Gyroscopic effects only occur when the orientation of the gyro axis is
changing. You can take a gyro and transport it north/south,
east/west, or up/down, without causing any precession, as long as the
gyro axis remains pointed in the same direction. You can even roll an
airplane without seeing gyroscopic effects, since the roll leaves the
gyro axis pointing in the same direction.
If you have a lightweight airframe and a heavy, rapidly
spinning propeller, watch out: the flippers will cause yawing
motion and the rudder will cause pitching motion.
If you want to make a gyro change orientation quickly,
it will take more torque than doing it slowly.
19.9.4 Inertial PlatformWe now consider what happens when a gyro is not subjected to any
Suppose we support a gyroscope on gimbals. The gimbals
support its weight but do not transmit any torques to it, even if the
airplane to which the gimbals are mounted is turning. We call this a
free gyro since it is free to not turn when the airplane turns.
Even though the gyro is small, it has a huge amount of angular
momentum, because it is spinning so rapidly. Any small torque applied
to the gyro (because of inevitable imperfections in the gimbals) will,
over time, change the angular momentum — but over reasonably short
times the change is negligible compared to the total.
In such a situation,
the gyro will tend to maintain fixed orientation in space. We say
that the gyro is an inertial platform with respect to
rotations.14 Other books say the gyro exhibits rigidity in
space but that expression seems a bit odd to me.
19.10 Gyroscopic InstrumentsWe now discuss the principles of operation of the
three main gyroscopic instruments: artificial horizon gyro (attitude
indicator), the directional gyro (heading indicator), and the
rate of turn gyro (turn needle or turn coordinator).
19.10.1 Heading IndicatorThe directional gyro is a free gyro. It establishes an
The gyro spins in some vertical plane; that is, its
angular momentum vector points in some arbitrary horizontal direction.
A system of gears measures the angle that the angular momentum
vector makes in the XY plane15
and displays it to the pilot. The trick is to measure the angle
and support the gyro while minimizing the accidental torques on
it. Imperfections in the mechanism cause the gyro to precess;
therefore, every so often the heading indication must be corrected,
typically by reference to a magnetic compass.
19.10.2 Artificial HorizonThe artificial horizon (also known as the attitude
indicator) is another free gyro. This gyro's plane of rotation is
horizontal; that is, its angular momentum vector is vertical. A
mechanical linkage measures the angle that this vector makes in the
YZ (bank) and XZ (pitch) planes, and displays it to the pilot.
It is instructive to compare the horizon gyro (which
tells you which way is ``down'') with the inclinometer
ball or a plumb-bob on a string (which has a different notion
of which way is ``down''). The distinction is that the
plumb-bob tells you which way is E-down, while the gyro is designed
to tell you which way is N-down (toward the center of the earth).
Whenever the airplane is being accelerated (e.g. during the takeoff
roll or during a turn), the two directions are quite different.
As seen in figure 19.17, during a turn the E-down vector
gets centrifuged to the outside of the turn; the N-down vector
always points to the center of the earth.
To a first approximation, the horizon gyro works
just by remembering which way is N-down. However, no gyro
can remember anything forever, so the instrument contains an ``erecting
mechanism'' that makes continual small adjustments. You would
like it to align the gyro axis with N-down — but the mechanism
doesn't know which way is N-down! It knows which way is E-down
(the same way the plumb-bob does), but according to Einstein's
principle of equivalence, it
cannot possibly know what components
of E-down are due to gravity and what components are due to acceleration.
The erecting mechanism does, in fact, continually nudge the gyro
axis toward E-down, but the result is a good approximation to
N-down, for the following reason: if you average the E-down
vectors over an entire turn, they average out to N-down. As you
can see in the figure:
If you average the discrepancies over an entire turn, they cancel.
- sometimes E-down points a little to the north of N-down,
- sometimes E-down points a little to the west of N-down,
- sometimes E-down points a little to the south of N-down,
- sometimes E-down points a little to the east of N-down.
* Artificial Horizon ErrorsOf course, if you only make half a turn, the
discrepancies don't average to zero, and the attitude indicator
will be slightly inaccurate for a while. Analogous errors occur
during takeoff, because the gyro's estimate of ``down''
gets dragged backwards by the acceleration, so the artificial
horizon will be a little bit below the true forward horizon for
a while thereafter. The averaging time for a typical instrument
is about five minutes. When the instrument gets old, its memory
gets shorter because the gyro isn't spinning as fast, so the errors
19.10.3 Rate-of-Turn GyroThere are two slightly different types of rate-of-turn gyro: (a) the
rate-of-turn needle, and (b) the turn coordinator.
In both cases, the gyro is not free; it is a rate gyro. That
is, its axis of rotation is more-or-less firmly attached to the
airplane. It does not have gimbals. It is forced to change
orientation when the airplane yaws, i.e. rotates around its Z
axis.16 The instrument measures how much torque is
required to re-orient the gyro.
Sometimes the rate-of-turn needle is built with its spin axis
running parallel to the Y axis, in which case the airplane's yawing
motion requires a torque around the X axis. Other models have the
spin axis in the X direction, in which case yaw requires a torque
around the Y axis. In principle, you could use any pair of axes
that lie in the XY plane.17
The required torque is proportional to (a) the rate of change of
orientation, and (b) the angular momentum of the gyro. Therefore an
accurate rate-of-turn gyro must spin at exactly the right speed, not
too fast or too slow. (This is in contrast to the directional gyro
and the artificial horizon gyro, which just have to spin ``fast
Many rate gyros incorporate a sneaky trick. They spin around the Y
axis, with the top of the gyro spinning toward the rear. That is, the
angular momentum vector points in the +Y direction. They also use a
spring that is weak enough to allow the gyro to precess a little
around the X axis. In a turn to the left, precession will tilt the
gyro a little to the right. That means that during a turn, the gyro's
tilt compensates for the airplane's bank, leaving the gyro somewhat
more aligned with the earth's vertical axis. The
goal, apparently, is to create an instrument that more nearly
indicates heading change (relative to the earth's vertical axis)
rather than simply rotation around the airplane's Z axis. Since the
relationship between bank angle and rate of turn depends on airspeed,
load factor, et cetera, this trick can't possibly achieve the goal
except under special conditions.
The turn coordinator is very similar to the rate-of-turn
needle. It displays a miniature airplane instead of a needle. The key
operational difference is that it is slightly sensitive to rate of
roll as well as rate of heading change. To create such an
instrument, all you have to do is take a rate-of-turn instrument, tilt
the mechanism nose-up by 20 or 30 degrees, and change the display.
The advantage of a turn coordinator is that it helps
you anticipate what actions you need to take. That is, if the
airplane has its wings level but is rolling to the right,
it will probably be turning to the right pretty soon, so you might
want to apply some aileron deflection. The disadvantage has to
do with turbulence. Choppy air oftentimes causes the airplane
to roll continually left and right. The roll rate can be significant,
even if the bank angle never gets very large. The chop has relatively
little effect on the yaw axis. In such conditions a plain old
rate-of-turn needle gives a more stable indication than a turn
It is rather unfortunate that the display on a turn coordinator is a
miniature airplane that banks left and right. This leads some people
to assume, incorrectly, that the instrument indicates bank angle,
which it most definitely does not.
- This is traditionally expressed in terms of an ``action''
and an ``equal and opposite reaction'', but the meaning of
those words has drifted over the centuries. Momentum is the modern
- Troublemakers sometimes point out that lift actually
is slightly reduced in a steady descent, since part of the
weight is being supported by drag. To this I retort: (a) this is an
obscure technicality, based on details of the definitions of the four
forces (as given in section 4.1); (b) the magnitude of
the reduction is negligible in ordinary flying, (c) the lift is
reduced for climbs as well as descents — so this technicality
certainly does not explain the motion, and (d) when we consider the
total upward force, there is no reduction.
- It is easy to find hand-waving
explanations of the Coriolis effect that overlook one or the other of
the two contributions, and are therefore off by a factor of two.
- If you
consider multiple widely-separated points, you can distinguish gravity
versus centrifugity versus straight-line acceleration by checking for
nonuniformities in the fields. However, an airplane is so small
compared to the planet, and so small compared to its turning radius,
that these nonuniformities do not provide a very practical way of
telling one field from another.
- It might be relevant if you are designing an airplane or
a flight simulator.
- In ordinary cases (specifically, when you know
the forces are in balance and you are trying to figure out whether the
torques are in balance) it doesn't matter what point in the airplane
you choose as the pivot-point, provided you measure all lever arms
from the same point.
- Contrast this with
the dot product used in section 4.4.
- Sometimes you see these written as hyphenated words
(foot-pounds or newton-meters)
in which case the hyphen should not be mistaken for a minus sign.
A foot-pound is a foot times a pound, not a foot minus a pound.
- It is best to feed the
string through a small smooth tube, rather than just your bare hand.
You might use a poultry baster, or the axial hole in a spool of
bean-bag acquires the necessary straight-line
momentum, and energy, via the string. It cannot acquire angular
momentum from the string, since that would require a lever arm
perpendicular to the force. Since the string can only exert a force
parallel to itself, the lever arm is zero, so the torque is
- For an object rotating around an axis of
symmetry, the angular momentum is exactly aligned with the axis; for
an oddball axis this might not be true.
- The word
``motion'' here refers to moving or re-orienting the airplane as a
whole, not to the rotation of the propeller.
- If you use your right hand, as
shown, the torque vector is pointed in the direction of your
thumb. Engineers would call this a negative pitching moment, since the
torque vector points opposite to the standard (left to right)
direction of the pitch axis.
- An even fancier inertial platform would keep a
position (not just orientation) independent of straight-line
- See figure 19.9 for the
definition of the X, Y, and Z directions.
- The instrument is not directly sensitive to any change
in the direction the airplane is going, just to changes in the
direction it is pointing.
- The X, Y,
and Z directions are defined in figure 19.9.
[Comments or questions]
Copyright © 1996-2001 jsd