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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 Motion

Let’s start by reviewing the physical laws that govern straight-line motion.

19.1.1  First Law

The first law of motion states: “A body at rest will remain at rest, and a body in motion will remain in motion, namely uniform straight-line motion, 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 dramatically 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 first law applies in the absence of outside forces. Any situation involving outside forces is covered by the second law, as we now discuss.

This is often called Newton’s first law of motion, which is quite unfair to Galileo. Newton was the first to set this law at the top of a numbered list, but he did not originate the law itself.

19.1.2  Second Law

The second law of motion says that if there is any change in the velocity of an object, the force (Fu) is proportional to the mass (m) of the object, and proportional to the acceleration vector (a). In symbols,

Fu = m a              (19.1)

The acceleration vector is defined to be the rate-of-change of velocity. See below for more about accelerations. Here Fu is the force exerted upon the object by its surroundings, not vice versa.

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.

19.1.3  Third Law

The third law of motion expresses the idea that momentum can neither be created nor destroyed. It can flow from one region to an adjoining region, but the total momentum does not change in the process. This is called conservation of momentum. As a corollary, it implies that the total momentum of the world cannot change. An application of this principle appears in section 19.2. Conservation of momentum is one of the most fundamental principles of physics, on a par with the conservation of energy discussed in chapter 1.

In simple situations, the third law implies that if object A exerts a force on object B, then object B exerts an equal and opposite1 force on object A.2 (In complicated situations, keeping track of equal-and-opposite forces may be impractical or impossible, in which case you must rely on the vastly more fundamental notion of conservation of momentum.)

Note the contrast:

The third law implies that if we add the force exerted by object A on object B plus the force exerted by object B on object A, the two forces add to zero. These are two forces acting on two different objects. They always balance.   Equilibrium means that if we add up all the forces exerted on object A by its surroundings, it all adds up to zero. These forces all act on the same object. They balance in equilibrium and not otherwise.

There is also a law of conservation of angular momentum. This is so closely related to conservation of ordinary linear momentum that some people incorporate it into the third law of motion. Other people leave it as a separate, unnumbered law of motion. We will discuss this starting in section 19.3.

19.1.4  Two Notions of Acceleration

The quantity a = F/m that appears in equation 19.1 was carefully named the acceleration vector. Care was required, because there is another, conflicting notion of acceleration:

Alas, everyone uses both of these conflicting notions, usually calling both of them “the” acceleration. It is sometimes a struggle to figure out which meaning is intended. One thing is clear, though: the quantity a = F/m that appears in the second law of motion is a vector, namely the rate-of-change of velocity.

Do not confuse velocity with speed. Velocity is a vector, with magnitude and direction. Speed is the magnitude of the velocity vector. Speed is not a vector.

Suppose you are in a steady turn, and your copilot asks whether you are accelerating. It’s ambiguous. You are not speeding up, so no, there is no scalar acceleration. However, the direction of the velocity vector is changing, so yes, there is a very significant vector acceleration, directed sideways toward the inside of the turn.

If you wish, you can think of the scalar acceleration as one component of the vector acceleration, namely the projection in the forward direction.

Try to avoid using ambiguous terms such as “the” acceleration. Suggestion: often it helps to say “speeding up” rather than talking about scalar acceleration.

19.1.5  Force is Not Motion

As 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.3 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 speed.

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 of ordinary (non-aerobatic) vertical motion is negligible.

In any case, once a steady climb is established, all the forces are in balance.

19.2  Momentum in the Air

We know from the third law of motion that if object A exerts a force on object B, then object B exerts an equal and opposite force on object A, as discussed in section 19.1.

There are many such force-pairs in a typical flight situation, as shown in figure 19.1 and figure 19.2.

Figure 19.1: Force and Momentum in Straight Flight
Figure 19.2: Force and Momentum in Curved Flight

In each of these numbered force-pairs, the “a” part always balances the “b” part exactly, in accordance with the third law of motion, whether or not the system is in equilibrium. In fact, figure 19.2 shows a non-equilibrium situation: the weight (1b) exceeds the lift (2a), so there is an unbalanced downward force, and the airplane is following a downward-curving flight path.

Note: In reality, these forces are all nearly aligned, all acting along nearly the same vertical line. (In the figure, they are artificialy spread out horizontally to improve readability.) Also, for simplicity, we are neglecting the effect of gravity on the air mass itself.

The arrows representing forces are color-coded according to which item they act upon: Blue arrows act upon the wing; brown arrows act upon the ground; green arrows act upon the light-green air parcel, et cetera.

For simplicity, we choose to analyze this from the viewpoint of an unaccelerated bystander. This means there will be no centrifugal field associated with the curved flight path in figure 19.2.

Let us now return to the scenario of unaccelerated flight, as shown by figure 19.1. In this scenario, the airplane weighs less than the airplane in figure 19.2, while all the other forces remain the same. The weight (1b) now equals the lift (2a), as it should for unaccelerated flight.

Since force is just momentum per unit time, the same process can be described by a big “closed circuit” of momentum flow. The earth transfers downward momentum to the airplane (by gravity). The airplane transfers downward momentum to the air (by pressure near the wings). The momentum is then transferred from air parcel to air parcel to air parcel. Finally the momentum is transferred back to the earth (by pressure at the surface), completing the cycle. In steady flight, there is no net accumulation of momentum anywhere.

You need to look at figure 3.29 to really understand the momentum budget. Looking only at figure 3.2 doesn’t suffice, because that figure isn’t large enough to show everything that is going on. You might be tempted to make the following erroneous argument:

To solve this paradox, remember that figure 3.2 shows only the flow associated with the bound vortex that runs along the wing, and does not show the flow associated with the trailing vortices. Therefore it is only valid relatively close to the wing and relatively far from the wingtips.

Look at that figure and choose a point somewhere about half a chord ahead of the wing. You will see that the air has some upward momentum at that point. All points above and below that point within the frame of the figure also have upward momentum. However, it turns out that if you go up or down from that point more than a wingspan or so, you will find that all the air has downward momentum. This downward flow is associated with the trailing vortices. Near the wing the bound vortex dominates, but if you go higher or lower the trailing vortices dominate. If you are wondering how this is possible, consider the following contrast. It helps to think carefully about what we mean by “vortex”:

A vortex line or vortex core is just a line, with zero thickness. The core of the trailing vortex extends behind the wing only.   The flow pattern of the vortex extends throughout all space. The speed of flow falls off only gradually as a function of distance from the vortex core. The trailing vortex flow pattern affects the air ahead of the wing.

If you know the location and strength of the core, you can determine the entire flow pattern ... but the core and the flow are two different concepts.

If you add up all the momentum in an entire column of air, for any vertical column ahead of the wing, you will find that the total vertical momentum is zero. The total momentum associated with the trailing vortices exactly cancels the total momentum associated with the bound vortex.

If you consider points directly ahead of the wing (not above or below), a slightly different sort of cancellation occurs. The flow associated with the trailing vortices is never enough to actually reverse the flow associated with the bound vortex; there is always some upwash directly ahead of the wing, no matter how far ahead. However, the contribution associated with the trailing vortices greatly reduces the magnitude, so the upwash pretty soon becomes negligible. Therefore, to a reasonable approximation, we can speak of “undisturbed” air ahead of the airplane.

Behind the wing there is no cancellation of any kind; the downwash of the wing is only reinforced by the downward flow associated with the trailing vortices. There is plenty of downward momentum in any air column behind the wing.

This gives us a simple picture of the airplane’s interaction with the air: There is downward momentum in any air column that passes through the vortex loop (such as the loop shown in figure 3.29). There is no such momentum in any air column that is ahead of the wing, outboard of the trailing vortices, or aft of the starting vortex.

So now we can understand the momentum balance:

  1. As the airplane flies along minute by minute, it imparts more and more downward momentum to the air, by enlarging the region of downward-moving air behind it.
  2. The air imparts downward momentum to the earth.
  3. The gravitational interaction between earth and airplane completes the circuit.

19.3  Sitting in a Rotating Frame

If we measure motion relative to a rotating observer, we cannot directly apply the laws of motion in the simple form that Newton published in the late 1600s. In this section and the next, we will use what we know about non-rotating reference frames to deduce the correct laws for rotating frames.

Suppose Moe is riding on a turntable; that is, a large, flat, smooth, horizontal rotating disk, as shown in figure 19.3. 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.

Figure 19.3: Rotating and Non-Rotating Coordinate Systems

We will assume that friction between the puck and the turntable is negligible.

The two observers analyze the same situation in different ways:

Moe immediately observes that the first law of motion, it its simplest form, does not apply in rotating reference frames.   In Joe’s nonrotating frame, the simple laws do apply.

Relative to the turntable, an unconstrained hockey puck initially at rest (anywhere except right at the center) does not remain at rest; it accelerates outwards. This is called centrifugal acceleration.   In a nonrotating frame, there is no such thing as centrifugal acceleration. The puck moves in a straight line, maintaining its initial velocity, as shown in figure 19.4.

To oppose the centrifugal acceleration, Moe holds the puck in place with a rubber band, which runs horizontally from the puck to an attachment point on the turntable. By measuring how much the rubber band stretches, Moe can determine the magnitude of the force.   Joe can observe the same rubber band. Moe and Joe agree about the magnitude and direction 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 force.

There are additional contributions to the acceleration if the rate of rotation and/or direction of rotation are unsteady. For simplicity, we will consider only cases where the rotation is steady enough that these terms can be ignored.

The centrifugal acceleration varies from place to place, so we call it a field. Section 19.5.1 discusses the close analogy between the centrifugal field and the familiar gravitational field.

The centrifugal field exists in a rotating reference frame
and not otherwise.

It must be emphasized that what matters is the motion of the reference frame, not the motion of the airplane. You are free to choose whatever reference frame you like, but others are free to choose differently. Pilots usually find it convenient to choose a reference frame comoving with the aircraft, in which case there will be a centrifugal field during turns. Meanwhile, however, an engineer standing on the ground might find it convenient to analyze the exact same maneuver using a non-rotating reference frame, in which case there will be no centrifugal field.

The centrifugal field comes from
the rotation of the reference frame
not the rotation of any particular object(s).

19.4  Moving in a Rotating Frame

We 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 turntable.

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.

So far, we have identified four or five contributions (which we will soon collapse to three):

Description   Scaling Properties

1. Suppose the puck is accelerating relative to Moe’s rotating frame (not just moving, but accelerating). Joe sees this and counts it as one contribution to the acceleration.   This “F=ma” contribution is completely unsurprising. Both observers agree on how much force is required for this part of the acceleration. It is independent of position, independent of velocity, and independent of the frame’s rotation rate.

2. 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. This “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 rotation rate.

3. 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”.   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.

4. 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.   Just like contribution #3, this contribution is independent of position, proportional to the velocity relative to the rotating frame, perpendicular to that velocity, and proportional to the first power of the frame’s rotation rate.

5. We continue to assume that the frame’s rotation rate is not changing, and its plane of rotation is not changing. Otherwise there would be additional contributions to the equations of motion in the rotating frame.  

Contribution #3 is numerically equal to contribution #4. The total effect is just twice what you would get from either contribution separately. We lump these two contributions together and call them the Coriolis effect.4

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 in the other direction (parallel to the axis of rotation) 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 perpendicular to the plane of rotation and produce no Coriolis effects.

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.

The Corilois effect exists when there is motion
in the plane of rotation of the reference frame
and not otherwise.

*   Magnitude of the Effect

Suppose 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 noticeable.

Now suppose you and a friend are standing 60 feet apart, playing catch in the back of a cargo airplane 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 2½ 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.

19.5  Centrifuges with and without Gravity

19.5.1  The Centrifugal Field is as Real as Gravity

An airplane in a turn, especially a steep turn, behaves like a centrifuge. There are profound analogies between centrifugal and gravitational fields:

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. There is often 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.5 Relative to a freely-falling reference frame, such as a freely-orbiting space station, everything is weightless.

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 centrifugal accelerations.

From a modern-physics point of view, both local gravity and local 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. To paraphrase Harry Emerson Fosdick:

Person saying it can’t be done
is liable to be interrupted by persons doing 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 and centrifugity.

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.

19.5.2  Centrifuge

To get a better understanding of the balance of forces in a turning and/or slipping airplane, consider the centrifuge shown in figure 19.4. 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.

Figure 19.4: An Object Departing a Centrifuge

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, moving tangentially.

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 “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 accelerations.

19.5.3  Centrifuge and Gravity

Now 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 19.5. 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.

Figure 19.5: 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.

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 initially moves away from us because we are being accelerated, or because there is a gravitating planet in the vicinity, or both.

(The previous two paragraphs apply 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 gravitational and centrifugal acceleration.)

19.6  Centrifugal Effects in a Turning Airplane

Let’s examine the forces felt by the pilot in a turning airplane. We start with a coordinated turn, as shown in figure 19.6.

Figure 19.6: Airplane in a Coordinated Turn
Figure 19.7: Airplane in a Nonturning Slip
Figure 19.8: Airplane in a Boat Turn

In figures such as this, whenever I am analysing things using the pilot’s point of view, the figure will include a rectangular “frame” with a little stick figure (the observer) standing in it. It is important to carefully specify what frame is being used, because even simple questions like “which way is down” can have answers that depend on which observer you ask. In particular, I define L-down (Lab-frame down) to mean the downward direction as observed some nearby terrestrial laboratory frame. In contrast, I define A-down (aircraft down) to be the downward direction in the reference frame attached to the aircraft. When the aircraft is turning, 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 A-down. Using the natural horizon and/or the artificial horizon, you can tell which way is L-down.

In figure 19.6, 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.

The lower part of figure 19.6 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.

Next, we consider the forces on the airplane in an ordinary nonturning slip, as shown in figure 19.7. The right rudder pedal is depressed, 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. Everybody agrees there is no centrifugal field.

As a third example, we consider what happens if you make a boat turn, as shown in figure 19.8. (For more about boat turns in general, see section 8.11.) 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).

The lower part of figure 19.8 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 A-down direction is not perpendicular to the wings, and some force other than wing-lift is acting on the plane.

19.7  Angles and Rotations

19.7.1  Axes and Planes of Rotation: Yaw, Pitch, and Roll

Any rotation can be described by specifying the plane of rotation and the amount of rotation in that plane. (Note that in this chapter, the word “airplane” is always spelled out, using eight letters. In contrast, the word “plane” will be reserved to refer to the thin, flat abstraction you learned about in geometry class.)

Figure 19.9: Rotations: Yaw, Pitch and Roll

Three particularly simple planes of rotation are yaw, pitch, and roll, as shown in figure 19.9. If you want a really precise definition of these three planes, proceed as follows: First: The airplane has a left-right mirror symmetry, and it is natural to choose the plane of symmetry as the plane of pitch-wise rotations. Secondly: Within the symmetry plane, we somewhat-arbitrarily choose a reference vector, attached to the airplane, that corresponds to zero pitch angle. It is conventional to choose this so that level cruising flight corresponds to zero pitch. The exact choice is unimportant. The roll-wise plane is perpendicular to this vector. Thirdly: The yaw-wise plane is perpendicular to the other two planes.

Any plane of rotation – not just the three planes shown in figure 19.9 – can be quantified in terms of bivectors, as discussed in section 19.8.

Older books often speak in terms of the axis of rotation, as defined in figure 19.10. In the end, it comes to the same thing: for example, yaw-wise rotation is synonymous with a rotation about the Z axis.

We prefer to speak in terms of the plane of rotation. This is more modern, more sophisticated, and more in accord with the way things look when you’re in the cockpit: For example, in normal flight, when the airplane yaws, it is easy to picture the nose moving left or right in a horizontal plane. This is easier than thinking about the Z axis.

Figure 19.10: Axes: X, Y and Z

Beware that older books give peculiar names to some of the axes. They refer to the Y axis as the lateral axis and the X axis as the longitudinal axis, which are sensible enough, but then they refer to Y-axis stability as longitudinal stability and X-axis stability as lateral stability — which seems completely reversed and causes needless confusion. Reference 17 calls the Z axis the normal axis, since it is normal (i.e. perpendicular) to the other axes — but that isn’t very helpful since every one of the axes is normal to each of the others. Other references call the Z axis the vertical axis, but that is very confusing since if the bank attitude or pitch attitude is not level, the Z axis will not be vertical. The situation is summarized in the following table.

This BookOlder Terminology
yaw bivector     vertical axis
XY plane      Z axis
yaw-wise stability     directional stability
pitch bivector     lateral axis
ZX plane      Y axis
pitch-wise stability     longitudinal stability
roll bivector     longitudinal axis
YZ plane      X axis
roll-wise stability     lateral stability

19.7.2  Attitude: Heading, Pitch, Bank

The term attitude describes the orientation of the airplane relative to the earth. Attitude is specified in terms of three angles: heading, pitch, and bank. (These are sometimes called the Euler angles.)

Oddly enough, it turns out that heading, pitch attitude, and bank angle are not always equivalent to rotations around the yaw, pitch, and roll axes, although they are intimately related. The relationship can be established by following a simple recipe, as we now discuss.

To construct a specified attitude, imagine that the airplane starts in level flight attitude with the X axis pointed due north; then:

For reasons discussed in section 19.7.4, it is important to perform these rotations in the order specified: yaw, then pitch, then roll.

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 attitude. The rotations must be undone in the reverse of the standard order:

If you do not follow the recipes given above, all bets are off, as you can see from the following example: Suppose you start out in level flight, and then roll 90 degrees. This is sometimes called the knife-edge attitude. Then:

See section 19.7.4 and section 19.7.5 for more about this.

19.7.3  Angle Terminology

The following table summarizes the various nouns and verbs that apply to angles and motions in the three principal directions:

XY planeZX planeYZ plane
Motionit yawsit pitchesit rolls
Anglethe headingthe pitch attitudethe bank attitude

Here are a few more fine points of angle-related terminology:

*   Other Angles

To 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. It can be most easily perceived with the help of a slip string, as discussed in section 11.3.

Some aerodynamics texts use the term sideslip angle, which is synonymous with slip angle. Don’t forget the somewhat-subtle distinction, as discussed in section 11.5.1:

The sideslip angle is exactly the same thing as the slip angle.   The sideslip maneuver is different from the forward slip maneuver.

19.7.4  Yaw Does Not Commute with Pitch

It 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 on a flat surface. 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, there are ways of changing the yaw angle (i.e., the heading) by 37 degrees (or any other amount) without ever yawing the airplane. That is, starting from straight and level flight:

If the aircraft (and its occupants) can tolerate heavy G loads, such maneuvers are perfectly fine ways to make tight turns at high airspeed.

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 in a purely horizontal plane. Further suppose that you have a nose-up pitch attitude, while still maintaining a level flight path, as could happen during slow flight. This means that the plane of yaw-wise rotations is is not exactly horizontal. 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 roll-wise rotation also.

The analysis in the previous paragraph is 100% accurate, but completely irrelevant when you are piloting the airplane.6 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 in a horizontal plane. 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 direction of flight will very nearly lie in the plane of yaw-wise rotations. Having it not exactly in the plane is just a distraction from the present topic, so I hereby define a new plane of “yaw-like” rotations which is defined by the direction of flight and the good old Y axis (the wingtip-to-wingtip direction). The pitch-wise rotations remain the same, and we define a new plane of “roll-like” rotations perpendicular to the other two. We assume zero slip angle for simplicity.

As the airplane flies from point to point along its curving path, its heading must change. This is a rotatation in a purely horizontal plane. In climbing flight, the yaw-like direction is not exactly horizontal, 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. That’s because I’m assuming you have compensated with the ailerons, so that both wings are 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 the laws of motion, an unbalanced force would create an acceleration in the roll-wise direction, 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 deflect the ailerons, not in order to create a wing-versus-wing difference in the magnitude of lift, but rather to avoid creating such a difference.

The best you can do is to keep the magnitude of the lift the same. The direction of the lift will be twisted, as discussed in section 8.9.5; 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.

19.7.5  Yaw Does Not Commute with Bank

As 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 in the pitch-wise direction as well. Because of this pitch-wise rotation, 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 noncommutativity.

19.8  Torque and Moment

Just as the first law of motion says that to start an object moving byou 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. When pilots talk about moment, they usually mean a particular type of moment that is equal to a torque. In other contexts, there exist other types of moments that are not equal to torque; examples include moment of inertia and dipole moment. We don’t need to discuss such things in detail, but you should be aware that they exist. In the present context, you can more-or-less assume that moment means torque. 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.

Another familiar example: gyroscopic effects are known for causing yaw-wise torques. By the same token, they can cause pitch-wise torques as well.

Torque is not the same as force. Of the two, force is the more familiar concept. In introductory physics courses, they focus attention on zero-sized pointlike particles, in which case there is only one place where the force can be applied, and you don’t need to worry about torque.

To apply a torque, you need a force and a lever-arm. The amount of torque is defined by the following formula:

torque = arm ∧ force              (19.2)

where the arm (also called lever arm) is a vector representing the separation between the pivot-point7

and the point where the force is applied. In this formula, we are multiplying vectors using the geometric wedge product, denoted “∧”.8 The wedge product of two vectors is called a bivector, and is represented by an area, namely the area of the parallelogram spanned by the two vectors, as shown in figure 19.11. All five bivectors in the figure are equivalent, as you can confirm by counting squares.

Figure 19.11: Torque: Equivalent Bivectors

A vector (such as force) has geometric extent in one dimension. The drawing of a vector has a certain length. This is in contrast to scalars, which have no geometric extent. They are zero-dimensional, and are drawn as points with no size.   A bivector (such as torque) has geometric extent in two dimensions. The drawing of a bivector has a certain area. In particular, the torque in figure 19.13 is represented by an area in the plane of the paper.

A vector points in a definite direction. It is drawn with an arrowhead on one end.   A bivector has a definite direction of circulation. It is drawn with arrowheads on its edges.

When constructing a bivector from two vectors, such as A ∧ F, you determine the direction of circulation by going in the A direction then going in the F direction, not vice versa. In particular, F ∧ A = − A ∧ F, which tells us the two bivectors are equal-and-opposite.

When the force and the lever-arm are perpendicular, the magnitude of the torque is equal to the magnitude of the force times the length of the lever-arm, which makes things simple. If the two vectors are not perpendicular, pick one of them. Then keep the component of that vector perpendicular to the other vector, throwing away the non-perpendicular component. What remains is two perpendicular vectors, and you can just multiply their magnitudes.

Torque is measured not in pounds but in footpounds (that is, feet times pounds); the corresponding metric unit is newtonmeters. 9

Figure 19.12 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.

Figure 19.12: Forces and Torques 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 in equilibrium.

Figure 19.13: Forces in Balance but Torques NOT in Balance

Figure 19.13 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.

19.9  Angular Momentum

The 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:

Straight-line concept
  Angular concept

Force   Torque (equals force times lever arm)

Momentum   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.

Force equals momentum per unit time.   Torque equals angular momentum per unit 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 19.14, 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.10

Figure 19.14: Conservation of Angular Momentum

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).11

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 section 19.4.

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 bivector, like torque (section 19.8). It lies more-or-less12 in the plane of rotation.

19.10  Gyroscopes

19.10.1  Precession

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 in one direction it will respond with motion in a different direction. When I give my “See How It Flies” lectures, I carry around a bicycle wheel with handles, as shown in figure 19.15. 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.

Figure 19.15: Bicycle Wheel with Handles

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 pitch-wise torque, the wheel doesn’t pitch down; it just yaws around and around.

Figure 19.16: Gyroscopic Precession

This phenomenon, where a gyro responds to a torque in one direction with a motion in another direction, is called gyroscopic precession.

For a gyroscope, a torque in the pitch-wise direction produces a motion in the yaw-wise direction. If you try to raise the tail of a real airplane using flippers alone, it will yaw to the left because of precession.

This effect is particularly noticeable early in the takeoff roll in a taildragger, when you raise the tail to keep the airplane on the ground while you build up speed. If the airplane were an ordinary non-spinning object, you could raise the tail just by pushing on the yoke. However, note that airflow over the flippers does not actually dictate the motion of the airplane; it just produces a torque in the pitch-wise direction. When you combine this torque to the angular momentum of the engine, the result is pronounced precession to the left. You need to apply right rudder to compensate.

Another place where this is noticeable is during power-on stall demonstrations. You need a downward pitch-wise torque to make the non-rotating parts of the airplane pitch down. However, this same pitch-wise torque, when added to the angular momentum of the engine, causes yaw-wise precession to the left. You need right rudder to compensate.

To get a gyroscope to actually move in the pitch-wise direction, you need to apply a torque in the yaw-wise direction — using the rudder.

Of course, an airplane has some ordinary non-rotating 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.

19.10.2  Precession: Which Way and How Much

Let’s try to understand what causes precession, so we can predict which way the airplane will precess, and how much. Consider what happens when a torque is applied for a certain small time interval (one second or so). This will contribute some angular momentum to the system. Remember: torque is angular momentum per unit time. Then we just add this contribution to the initial angular momentum, and the result is the final angular momentum.

Angular momentum is a bivector. Figure 19.17 shows the bivectors involved in the precession, and figure 19.18 is an exploded view showing how to add bivectors. We add bivectors edge-to-edge, in analogy to the way we add ordinary vectors tip-to-tail. In this example, edge b adds tip-to-tail to edge x to form the top edge of the sum. Similarly, edge z adds tip-to-tail to edge d to form the bottom edge of the sum. Edge c cancels13 edge w since they are equal and opposite. Edges a and y survive unchanged to become the vertical edges of the sum.

Figure 19.17: Angular Momentum Explains Precession
Figure 19.18: Addition of Bivectors – Exploded View

We see that the new angular momentum differs from the old angular momentum by a yaw to the left. That’s the correct answer.

During subsequent time intervals, the torque will be a new direction because the whole system has rotated. The successive changes will cause the system (wheel, axle, and everything attached to it) to keep turning in the horizontal plane, yawing to the left.

Beware: This gyroscope law might seem roughly similar to 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, and a wheel precesses even though no part of the wheel is moving relative to other parts.

Gyroscopic effects only occur when the there is a change in the orientation of the gyro’s plane of rotation. You can take a gyro and transport it north/south, east/west, or up/down, without causing any precession, as long as the gyro’s plane of rotation remains parallel to the original plane of rotation. You can even roll an airplane without seeing gyroscopic effects due to engine rotation, since the roll leaves the engine’s plane of rotation undisturbed.

You can figure it out by adding the bivectors. Right rudder deflection will cause a pitch-wise precession in the nose-down / tail-up direction. Pushing on the yoke causes a yaw-wise precession to the left.

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.

19.10.3  Inertial Platform

We now consider what happens when a gyro is not subjected to any large torques.

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.

19.11  Gyroscopic Instruments

We now discuss the principles of operation of the three main gyroscopic instruments: artificial horizon (attitude indicator), directional gyro (heading indicator), and rate of turn gyro (turn needle or turn coordinator).

19.11.1  Heading Indicator

The directional gyro is a free gyro. It establishes an inertial platform.

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.

19.11.2  Artificial Horizon

The 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 A-down, while the gyro is designed to tell you which way is L-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 19.19, during a turn the A-down vector gets centrifuged to the outside of the turn; the L-down vector always points to the center of the earth.

Figure 19.19: A-Down versus L-down During a Turn

As you can see in figure 19.19,

To a first approximation, the horizon gyro works just by remembering which way is L-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 L-down — but the mechanism doesn’t know which way is L-down! It knows which way is A-down (the same way the plumb-bob does), but according to Einstein’s principle of equivalence, it cannot possibly know what components of A-down are due to gravity and what components are due to acceleration. The erecting mechanism does, in fact, continually nudge the gyro axis toward A-down, but the result is a good approximation to L-down, for the following reason: if you average the A-down vectors over an entire turn, they average out to L-down.

If you average the discrepancies over an entire turn, they cancel. This is why a gyro is vastly more valuable than a plumb-bob: The gyro can perform long-term averaging, whereas a plumb-bob can’t.

Technical note: Even in the ordinary terrestrial lab frame, you need to account for a small amount of centrifugal field, because the earth itself is rotating. That means that the conventional notion of “down” does not point toward the center of the earth. At temperate latitudes, it’s off by about a tenth of a degree. This is not immediately noticeable, for several reasons:

On the other hand, the earth’s rotation is certainly noticeable if you look closely enough: (a) You can easily see that the earth rotates relative to the stars. (b) Centrifugal forces cause the earth to be ellipsoidal rather than spherical, and you can notice this if you have a good-enough map: A degree of latitude is about 1% longer at the pole than at the equator. (c) Consider the directional gyro in an aircraft. A simple gyroscope will drift relative to the earth – or, rather, the earth will drift relative to the gyro – even if the aircraft is sitting on the taxiway, with the brakes set. This is most obvious at the north pole, where the earth rotates 15 per hour, while the simple gyro does not.

A real directional gyro instrument is not so simple. For one thing, it is only sensitive to rotation in the aicraft’s XY plane, which is normally horizontal. Therefore, at the equator it doesn’t respond to the earth’s rotation at all. More generally, the effect is proportional to the sine of the latitude. Secondly, the instrument has a so-called latitude nut that can be adjusted to cancel the rotation of the earth, for some chosen latitude. However, if you fly somewhere else, the cancellation will be imperfect.

*   Artificial Horizon Errors

Let’s look again at figure 19.19. 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.

Sometimes you find an old, worn-out instrument in which the gyro isn’t spinning as fast as it should. As a result, its memory gets shorter, and the systematic errors become larger.

19.11.3  Rate-of-Turn Gyro

There 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 plane 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.16 The instrument measures how much torque is required to re-orient the gyro.

Sometimes the rate-of-turn needle is built to spin in the pitch-wise (ZX) plane, in which case the airplane’s yawing motion requires a torque in the roll-wise (YZ) direction. Other models spin in the roll-wise (YZ) plane, in which case yaw requires a torque in the pitch-wise (ZX) direction. In principle, the spin and the torque could be in any pair of planes perpendicular each other and perpendicular to the yaw-wise (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 enough”.)

Many rate gyros incorporate a sneaky trick. They spin around the pitch-wise (ZX) plane, with the top of the gyro spinning toward the front. They also use a spring that is weak enough to allow the gyro to precess a little in the roll-wise (YZ) direction. 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 in the airplane’s yaw-wise (XY) plane, which is not exactly horizontal during the turn. Since the relationship between bank angle and rate of turn depends on airspeed, load factor, et cetera, this trick cannot possibly do the job perfectly.

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 heading. In such conditions a plain old rate-of-turn needle gives a more stable indication than a turn coordinator does.

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, as you can demonstrate by performing a boat turn (section 8.11).

19.12  Some Fine Points about Energy and Conservation

The main discussion of energy is in chapter 1 and chapter 7. This section clarifies a few fine points, for the benefit of experts.

19.12.1  Vernacular «Energy»

Beware that the physics notion of energy differs from the vernacular notion of «energy».

Physics Energy   Vernacular «Energy»

The physics energy is simple, well-defined, and well-behaved.   When the Department of Energy talks about «energy», they are referring to some vernacular notion of «available» energy or «useful» energy. This is super-important, but very hard to define.

In this book, when we use the word energy (except within scare quotes) we are talking about the physics energy.   Questions of «usefulness» belong more to economics or philosophy than to physics, and the answers tend to be highly subjective and context-dependent.

The physics energy cannot be created or destroyed. There are no sources. There are no sinks. The laws of nature guarantee it.   The DoE is always looking for new sources of «energy». Obviously they’re not talking about the physics energy.

It must be emphasized that the vernacular notion of «energy» is not wrong; it’s just different from the physics energy. You should not try to unlearn the vernacular notion, but you must learn the physics notion also. This is confusing, but there’s no alternative. It has nothing to do with physics; it’s just how the language works. Words have multiple meanings. For example, a lap in the swimming pool is quantitatively, qualitatively, and conceptually different from a lap on the race track. Context matters. In physics context, you have to use the physics definition.

19.12.2  Vernacular «Conservation»

Beware that the physics notion of conservation differs from the vernacular notion of «conservation».

Physics Conservation   Vernacular «Conservation»

Physics conservation (such as conservation of energy or conservation of electric charge) means that something can flow from place to place, but can never be created or destroyed.   Vernacular «conservation» (such as «conservation» of endangered wildlife) means to protect something from waste or loss.

Combining this with section 19.12.1, we see that when the DoE asks you to «Conserve Energy» they are using neither the physics notion of energy nor the physics notion of conservation. If you’re not careful, you can suffer from two profound misconceptions in a single two-word phrase.

19.12.3  Energy versus Entropy

Loosely speaking, we sometimes say that energy is “dissipated” during an irreversible process. That’s not entirely wrong, but if you’re not careful it can be misleading. The fact is, in all cases, dissipation and irreversibility can be understood in terms of entropy and not otherwise.

Energy and entropy are two different things. Energy is governed by the first law of thermodynamics (i.e. conservation of energy), while entropy is governed by the second law of thermodynamics. Energy is not defined in terms of entropy, nor vice versa.

In particular, you should avoid any notion of «degraded» energy. In a steam engine, there are three things to keep track of: energy, water, and entropy. Dissipation does not «degrade» the energy any more than it «degrades» the water. Dissipation leaves you with the same energy, the same water, and extra newly-created entropy.

Dissipation and irreversibility can be understood
in terms of entropy and not otherwise.

The details of what entropy is and how it behaves are beyond the scope of this book. Qualitative, intuitive notions of irreversibilty are adequate for present purposes.

The expression “equal and opposite” refers to vectors that are equal in magnitude and opposite in direction.
In the olden days, this was 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 term.
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. Beware.
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.
The pivot-point is also known as the datum. Force is measured in pounds or newtons; torque is measured in inch·pounds or newton·meters.    
In some cases you already know the forces are in balance and you are just trying to figure out whether the torques are in balance. In such a case, 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.
Some other books try to calculate the torque using a “cross product” but the wedge product is much nicer. The wedge product is in some sense complimentary to the dot product used in section 4.5.
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 thread.
The 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 zero.
For an object rotating around an axis of symmetry, the angular momentum lies exactly in the plane of rotation; for odd off-axis rotations this might not be true.
As we have just discussed, it is OK join c to w. Alternatively, you could join a to y and get the same answer. In contrast, it would not make sense to join c to y, for the following reasons: Edge y is not opposite to c, so this pair would not drop out of the sum. Also, this would make us unable to add z to d tip-to-tail; they woud be tail-to-tail. Similarly, it would make us unable to add x to b tip-to-tail; they would be tip-to-tip.
An even fancier inertial platform would keep a position (not just orientation) independent of straight-line accelerations.
See figure 19.10 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 19.10.

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