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Copyright © 1996-2001 jsd
4 Lift, Thrust, Weight, and Drag
It is better to be on the ground wishing you were
flying, rather than up in the air wishing you were on the ground.
— Aviation proverb.
4.1 DefinitionsThe main purpose of this chapter is to clarify the concepts of
lift, drag, thrust, and weight. Pilot books call
these the four forces.
It is not necessary for pilots to have a super-precise
understanding of the four forces. The concept of energy
(discussed in chapter 1) is considerably more important.
In the cockpit (especially in critical situations like final
approach) I think about the energy budget a lot, and think about
forces hardly at all. Still, there are a few situations that
can be usefully discussed in terms of forces, so we might as well
learn the terminology.
The relative wind acting on the airplane produces a certain amount of
force which is called (unsurprisingly) the total aerodynamic
force. This force can be resolved into components, called lift and
These are the official definitions.
- Lift is the component of
aerodynamic force perpendicular to the relative wind.
- Drag is the component of
aerodynamic force parallel to the relative wind.
- Weight is the force directed
downward from the center of mass of the airplane towards the center of
the earth. It is proportional to the mass of the airplane times the
strength of the gravitational field.
- Thrust is the force produced by
the engine. It is directed forward along the axis of the engine
(which is usually more or less parallel to the long axis of the
Figure 4.1 shows the orientation
of the four forces when the airplane in ``slow flight''
— descending with a nose-high attitude, with the engine producing
some power. Similarly, figure 4.2 shows the four forces
when airplane in a high-speed descent. The angle of attack is
much lower, which is consistent with the higher airspeed. Finally,
figure 4.3 shows the four forces when the airplane
is in a climb. The angle of attack, the lift, and the drag have
the same magnitude as in figure 4.2.
Note that the four forces are defined with respect
to three different coordinate systems: lift and drag are defined
relative to the wind, gravity is defined relative to the earth,
and thrust is defined relative to the orientation of the airplane.
In level flight these coordinate systems usually don't differ
too much, but in figure 4.1 you can see the
This situation seems a little complicated, and it
is: for instance, thrust, lift and drag all have vertical components
that combine to oppose the weight; similarly the thrust and lift
both have forward horizontal components.
In ordinary cruising flight, the situation is simpler.
When all three coordinate systems coincide, lift must balance
weight and thrust must balance drag.
4.2 Balance of ForcesFigure 4.4 proves that it isn't safe to assume
that lift always matches weight, or thrust exactly matches drag.
A bomb falling straight down has no
lift and no thrust; when it reaches terminal velocity its weight
is supported purely by drag. Another interesting case is a moon
lander (figure 4.5)
hovering on its rocket plume — it has no lift and no drag; its
weight is supported by its thrust.
You may think lift, thrust weight, and drag are defined
in a crazy way, but the definitions aren't going to change anytime
soon. They have too much history behind them, and they actually
have advantages when analyzing complex situations.
The good news is that these subtleties are really
quite unimportant. First of all, the angles in figure 4.1
are greatly exaggerated. In ordinary transportation (as opposed
to aerobatics), even in climbs and descents, deck angles are generally
quite small, so thrust is always nearly horizontal. Also,
the relative wind differs from horizontal by only a few degrees,
so drag is always nearly horizontal, and lift is nearly
vertical except in turns.
The most important situation where you have to worry
about the forces on the airplane is during a turn. In a steeply-banked
turn, the lift vector is inclined quite a bit to the left or right
of vertical. In order to support the weight of the airplane and
pull the airplane around the turn, the lift must be significantly
greater than the weight. This leads us to the notion of load
factor, which is discussed in section 6.2.3.
The bottom line is that thrust is usually nearly equal (and opposite)
to drag, and lift is usually nearly equal (and opposite) to weight
times load factor.
If we don't like the technical definitions of lift,
drag, thrust and weight, we are free to use other terms. In particular,
we can make the following sweeping statement: in unaccelerated
flight, the upward forces balance the downward forces, and the
forward forces balance the rearward forces. This statement is
true whether or not we calculate separately the contributions
of lift, drag, thrust and weight.
Before going on, let me mention a couple of petty
paradoxes. (1) In a low-speed, high-power climb, lift is less
than weight — because thrust is supporting part of the weight.
It sounds crazy to say that lift is less than weight during climb,
but it is technically true. (2) In a low-power, high-speed descent,
lift is once again less than weight — because drag is supporting
part of the weight.
These paradoxes are pure technicalities, consequences
of the peculiar definitions of the four forces. They have no
impact on pilot technique.
There is some additional discussion of the balance
of forces in section 19.1.
4.3 Types of DragWe have seen that the total force on the airplane
can be divided into lift and drag. We now explore various ways
of subdividing and classifying the drag.
When a force acts on a surface, it is often useful
to distinguish processes that act parallel to the surface (friction
along the surface) versus forces that act perpendicular to the
surface (pressure against the surface).
Figure 4.6 illustrates the idea of
pressure drag. If the tea table is moving
from right to left,
you can oppose its motion by putting your hand against the front
vertical surface and pushing horizontally.
Figure 4.7 illustrates the idea of
friction drag. Another
way to oppose the motion of the tea-table is to put your hand in the
middle of the horizontal surface and use friction to create a force
along the surface. This might not work too well if your hand is
wet and slippery.
Figure 4.8 shows a situation where air flowing along a
surface will create lots of friction drag. There is a large area
where fast-moving air is next to the non-moving surface. In contrast,
there will be very little pressure drag because there is very little
frontal area for anything to push against.
Friction drag is proportional to viscosity (roughly, the
``stickiness'' of the fluid). Fortunately, air has a rather low
viscosity, so in most situations friction drag is small compared to
pressure drag. In contrast, pressure drag does not depend very
strongly on viscosity. Instead, it depends on the mass density of the
Friction drag and pressure drag both create a force in proportion to
the area involved, and to the square of the airspeed. Part of the
pressure drag that a wing produces depends on the amount of lift it is
producing. This part of the drag is called induced
drag. The rest of the drag —
everything except induced drag — is called parasite
The part of the parasite drag that is not due to
friction is called form drag.
That is because it is extremely sensitive to the detailed form
and shape of airplane, as we now discuss.
A streamlined object such as the one shown in figure 4.9 can have ten times less form drag than a non-streamlined
object of comparable frontal area (e.g. the flat plate in figure 4.10). The peak pressure in front of the two shapes will
be the same, but (1) the streamlined shape causes the air to
accelerate, so the region of highest pressure is smaller, and more
importantly, (2) the streamlined shape cultivates high pressure
behind the object that pushes it forward, canceling most of the
pressure drag, as shown in figure 4.9. This is called
Any object moving through the air will have a high-pressure
region in front, but a properly streamlined object will have a high-pressure region in back as well.
The flow pattern1
near a non-streamlined object is not symmetric fore-and-aft because
the stream lines separate from the object as
they go around
the sharp corners of the plate. Separation is discussed at more
length in chapter 18.
Streamlining is never perfect; there is always at
least some net pressure drag. Induced drag also contributes to
the pressure drag whenever lift is being produced (even for perfectly
streamlined objects in the absence of separation).
Except for very small objects and/or very low speeds,
pressure drag is larger than friction drag (even for well-streamlined
objects). The pressure drag of a non-streamlined object is much
larger still. This is why on high-performance aircraft, people
go to so much trouble to ensure that even the smallest things
(e.g. fuel-cap handles) are perfectly aligned with the airflow.
An important exception involves the air that has
to flow through the engine compartment to cool the engine. A
lot of air has to flow through narrow channels. The resulting
friction drag — called cooling drag
— amounts to 30% of the total drag of some airplanes.
Unlike pressure drag, friction drag cannot possibly
be canceled, even partially. Once energy is lost to friction,
it is gone forever.
The various categories of drag are summarized in
figure 4.11. The way to reduce induced drag (while
maintaining the same amount of lift) is to have a longer wingspan
and/or to fly faster. The way to minimize friction drag is to
minimize the total wetted area (i.e. the total area that has high-speed
air flowing along it). The way to reduce form drag is to minimize
separation, by making everything streamlined.
4.4 Coefficients, Forces, and PowerThe word ``drag'', by itself, usually refers to a force (the force of
drag). Similarly, the word ``lift'', by itself, usually refers to a
force. But there are other ways of looking at things.
* CoefficientsIt is often convenient to write the drag force as a dimensionless
number (the coefficient of drag) times a bunch of factors that
characterize the situation:
where r (the Greek letter ``rho'') is the density of the air,
V is your true airspeed, and the relevant area is typically taken
to be the wing area (excluding the surface area of the fuselage,
drag force =
× coefficient of drag × area
Similarly, there is a coefficient of lift:
We used these equations back in section 2.12
to explain why the airspeed indicator is a good source of information
about angle of attack.
lift force =
× coefficient of lift × area
One nice thing about these equations is that the
coefficient of lift and the coefficient of drag depend on the
angle of attack and not much else. If you could (by magic) hold
the angle of attack constant, the coefficient of lift and the
coefficient of drag would be remarkably independent of airspeed,
density, temperature, or whatever.
The coefficient of lift is a ratio2 that basically measures how effectively
the wing turns the available dynamic pressure into useful average
suction over the wing. A typical airfoil can achieve a coefficient
of lift around 1.5 without flaps; even with flaps it is hard to
achieve a coefficient of lift bigger than 2.5 or so. For data
on real airfoils, see figure 3.14 and/or reference 5.
Figure 4.12 shows how the various coefficients depend
on angle of attack. The left side of the figure corresponds to the
highest airspeeds (lowest angles of attack). Note that the
coefficient-of-lift curve has been scaled down by a factor of ten to
make it fit on the same graph as the other curves. Airplanes are
really good at making lots of lift with little drag.
In the range corresponding to normal flight (say 10 degrees angle
of attack or less) we can make the following approximations, which I
will call the basic lift/drag model:
At higher angles of attack (approaching or exceeding the critical
angle of attack) these approximations break down. The coefficient of
parasite drag will rapidly become quite large, and the induced drag
will probably be quite large also. There will be no simple
proportionality relationships. The details aren't of much interest to
most pilots, for the following reason: Typically you recover from a
stall as soon as you notice it, so you don't spend much time in the
stalled regime. If you do happen to be interested in stalled flight
and spins, see chapter 18.
- the coefficient of lift is proportional to the
angle of attack,
- the coefficient of induced drag is proportional
to the square of the angle of attack, and
- the coefficient of parasite drag is essentially
In flight, we are not free to make any amount of lift we want. The
lift is nearly always equal to the weight times the load factor. This
leads us to rearrange the lift equation as follows:
where the airspeed and load factor are more-or-less the only variables
on the right-hand side of the equation. Because of the factor of
airspeed squared, the airplane must fly at a very high coefficient of
lift in order to support its weight at low airspeeds.
coefficient of lift =
(weight × load factor)
Figure 4.13 plots the same four curves
against airspeed. Now the left side of the plot corresponds to
the lowest airspeeds (highest angles of attack).
* ForcesFigure 4.14 shows the corresponding forces.
We see that whereas the coefficient of parasite drag was
more or less constant, the force of parasite drag increases
with airspeed. If somebody says ``the drag is a ... function
of airspeed'' you have to ask whether ``drag'' refers
to the drag coefficient, the drag force, or (as discussed below)
the drag power.
We can also see in the figure that the lift force curve is perfectly
constant, which is reassuring, since the figure was constructed using
the principle that the lift force must equal the weight of the
airplane; this is how I converted angle of attack to airspeed.
The lowest point in the total drag force curve corresponds to
VL/D, and gives the best lift-to-drag ratio. Using the standard
lift/drag model and a little calculus, it can be shown that this
occurs right at the point where the induced drag force curve crosses
the parasite drag force curve.
* PowersFigure 4.15 shows the amount of dissipation due to
drag, for the various types of drag. Dissipation is a form of power,
i.e. energy per unit time.
Dissipation is related to force by the simple rule:
In this equation, we are multiplying two vectors using the dot
product (·),3 which means that
only the velocity component in the direction of the force counts.
power = force ·
In the case of drag, we have specifically:
The lowest point in the curve for total drag power corresponds to
VY, and gives the best rate of climb. Using the standard
lift/drag model and a little calculus, it can be shown that at this
speed, the occurs right at the point where the induced drag power is
is 3/4ths of the total, and the parasite drag power is 1/4th of the
total. Actually, in the airplane represented in these figures, VY
is so close to the stalling speed that the standard lift/drag model is
starting to break down, and the 3:1 ratio is not exactly accurate.
dissipation = force of drag ·
In the case of lift, the lift force is (by its definition)
perpendicular to the relative wind, so there is no such thing as
dissipation due to lift. (Of course the physical process that
produces lift also produces induced drag, but the part of the force
properly called lift isn't the part that contributes to the power
4.5 Induced vs. Parasite DragThere are several useful conclusions we can draw from
these curves. For starters, we see that the curve of total power
required to overcome dissipation has a familiar shape; it is just
an upside-down version of the power curve that appears in
section 1.2.5 and elsewhere throughout this book.
We can also see why the distinction between induced
drag and parasite drag is significant to pilots:
In the high-speed regime (which includes normal cruise), the power
required increases rapidly with increasing airspeed. Eventually it
grows almost like the cube of the airspeed. The reason is easy
to see: parasite drag is the dominant contribution to the coefficient
of drag, and it more-or-less independent of airspeed.4 We pick up two factors of V from
equation 4.2 and one from equation 4.4.
Knowing this cube law is useful for figuring out the shape of your
airplane's power curve (section 7.6.2), and for
figuring out how big an engine you need as a function of speed
(section 7.6.4) and altitude
- In the mushing regime, most of the drag is induced
drag. As you go slower and slower, induced drag increases
dramatically and parasite drag becomes almost negligible.
- At high airspeeds, parasite drag is dominant
and induced drag becomes almost negligible.
- Figure 4.10 is not as precise as the other airflow diagrams in
the book. My flow software is not capable of properly modeling
the wake of the flat plate, so I had to take some liberties.
is a dimensionless number, not measured in pounds or seconds or
anything, just a pure number.
- Contrast this with the
cross product used in section 19.7.
drag decreases as the airspeed increases, but this is a relatively
minor contribution in this regime.
[Comments or questions]
Copyright © 1996-2001 jsd