Aircraft Weight and Balance

By Robert Waldvogel

1. Introduction to Physics

Physics is the study of the entire natural, or physical, world and can be subdivided into two categories:

a). Classical physics.

b). Modern physics.

It employs the scientific method of analysis, which itself is the application of a logical process of reasoning. This process entails five integral steps:

1. Observation.

2. Hypothesis.

3. Experimentation.

4. Theory or law.

5. Prediction.

"Prediction" implies that, once an accurate model of nature has been established, that its theory results in a law of physics.

From the physics perspective, the entire world is describable in terms of four fundamental parameters:

1. Length.

2. Mass.

3. Time.

4. Change.

Physics is not necessarily the obscure subject you study at a university in order to complete your degree requirements, but instead something with which you interact countless times per day because you live in the physical world, although you have probably never thought about it like this before.

Every time you sit down somewhere, for instance, you stay there, instead of floating up to the ceiling, because of gravity. Gravity is part of physics. If you sling an item across a table or surface, it will not slide indefinitely, but will ultimately be arrested and stop because of the presence of friction on that surface. This is also physics. Why is it more dangerous to drive on ice than on pavement? Ice is obviously more slippery than pavement and hence provides less "grip" with the surface. According to physics, every surface or substance has a frictional value and the frictional value of ice is far lower than that of pavement or concrete. The fact that your car equally does not float above this pavement when you drive it indicates that both gravity and friction are present.

Listening to someone speak sounds mundane, but the very fact that there is "sound" at all indicates that sound waves are being carried from the speaker to the listener. This, again, is physics.

Physics, in fact, is all around you. You interact with its concepts and principles every second of everyday and cannot escape this fact, yet still exist in and negotiate the physical world. You, too, are, in essence, "physics," since you are part of this world.

Weight and balance, therefore, inescapably entails physics.

2. Language of Physics

Like many disciplines, physics has its own language. If you ask someone to say "aircraft" in Spanish, for example, he would probably respond, "El avion." If you equally ask someone to translate the same word into German, he would likely answer, "Das Flugzeug." If, on the other hand, you ask someone how he would say "work" in physics, he would respond, "W=fd." This is not an odd language, but a mathematical formula, because mathematics is the language of physics.

Work = Force x Displacement

W = Fd

According to the formula, every time you pick up something from, say, a desk, and place it somewhere else, you have exerted a force on the object and produce the physics principle of "work." Work is the scalar product of force multiplied by distance (displacement) whose unit of measurement is the "joule."

If, however, you exert tremendous effort to lift a 150-pound object above your head and sweet profusely to do so, but return it to the exact spot from which you had taken it, by the physics formula, you have performed no "work," since the object's displacement is zero (0), although this may defy ordinary logic.

Physics laws are precise. Any result must be mathematically proven. Opinion is valueless in this discipline.

3. The Physics Principle Behind Weight and Balance

Weight and balance also involves physics and its formulas. Consider a seesaw with a 100-pound box placed on its right side, and answer these two questions.

a). Which way will the seesaw go?

b). Why?

The answer to the first question is obvious. The seesaw will rotate to the right, toward the side on which the box has been placed. The answer to the second is not so obvious. If you say it is because of the "weight" on the right end of it, you would only be partially correct. "Weight" in physics is "mass" and there are two other elements exerting their effects on this movement.

A closer look at this diagram indicates that there are three elements involved, all of which, when combined, result in a physics law and an associated formula. The first of these, as already indicated, is the weight of the box on the right side of the seesaw, but here it cannot be considered "weight," but instead a "force," which causes the seesaw to rotate downward on the right side. The first element is therefore "force."

F = Force

The second element is the distance that force is applied from the seesaw's fulcrum or pivot point. We shall designate this distance by the small letter "r."

r = Distance

Finally, this force and distance combination rotates, one way or the other, at the fulcrum and that fulcrum creates an angle. The angle, generically designated "sine theta," constitutes the third interactive element.

Angle = sine theta

All these elements comprise what is known as the torque formula.

Torque = distance x force x sin theta

T = r F sine theta

Torque, derived from the Latin verb "torquere," means "to twist." Several additional definitions entail the following.

a). The vector is at 90 degrees to the plane of both the force and the displacement.

b). Torque results in a perpendicular motion along the axis of rotation to the line along which the force acts.

c). Torque is the component of force perpendicular to the line joining the center of a circle and the point where the force is applied multiplied by the distance from the center of a circle to the point where the force is applied.

d). Torque is the measurement of the amount of "twisting" one can obtain from a given force.

e). Contrasted with work, torque is a force multiplied by a perpendicular distance whereas work is a force multiplied by a parallel distance.

One of the three integral torque formula elements--namely, that of the angle--can be omitted. The angle, in this case, is 90 degrees, and the sine of 90 degrees equals one (1). Feed that into your calculator and verify its results.

sine theta = sine 90

sine 90 = 1

Since any number multiplied by one (1) always equals itself, its inclusion serves no purpose. For example:

100 (1) = 100

Therefore, its inclusion results in no numerical change or effect, and the torque formula can be simplified to the following:

T = rF1 = rF

T = rF

Although the formula can be reduced to two elements, torque is still the product of them, which is why the answer of "weight" alone in the initial question had been incorrect and incomplete. The elements are multiplied together.

The physics unit of "force" is neither pounds nor kilograms, but Newtons, abbreviated "N," and the distance this force is applied from the fulcrum is expressed in meters (m). Their product is the "Newton-meter" or "mN."

F = Newtons (N)

r = meters (m)

Newtons x meters = Newton-meters (mN)

The concept of torque is something you actually use many times per day, but have not been aware of it. Every time you exert a force to open a door, for example, you use the physics principle of torque and the vector product of the force you apply on the doorknob and how far that doorknob is located from the door hinge results in torque. Although you may never have been aware that its distance had been helping you open the door, it actually reduced the amount of effort you had exerted to do so, and this fact can be illustrated with a few simple examples. Suppose you applied 10 Newtons of force to a door whose handle were located 1.22 meters from the hinge (which is the door's point of axis). By substituting these figures into the formula, you could calculate the amount of torque you generated, as follows:

T = rF

T = 1.22(10) = 12.2 mN

T = 12.2 mN

Suppose the door handle were relocated to a position only halfway to the hinge. Would you generate the same amount of torque? Let us see.

1.22/2 =.61

T = rF

.61(10) = 6.1 mN

T = 6.1 mN

You have obviously generated only half the amount of torque.

6.1/12.2 = ½

Could you generate the same amount of torque with this now relocated door handle? If you have halved the distance of the lever arm, you would obviously have to double the force to produce the same amount of torque-which means that you would have to work twice as hard to generate the same results.

2(10) = 20

T = rF

.61(20) = 12.2 mN

T = 12.2 mN

If you relocated the door handle to the hinge line, how much torque would you generate?

T= rF

T = 0(10) = 0 mN

T = 0 mN

You would not, in fact, have generated any torque because torque is the product of the distance and the force, and since the distance here is zero, zero multiplied by any number equals zero. You could not have opened this door! Try it!

Torque does not depend upon vertical orientation...only on axis of rotation. An example of this horizontal, torque-generating arrangement is the previously considered seesaw in which the fulcrum is the "hinge" and the weights of two children are the "forces." Force, which is derived by multiplying the mass of an object with the acceleration of the object, is the mathematical expression of Sir Isaac Newton's Second Law of Motion which states that a force not only causes an acceleration of a body, but that that acceleration is directly proportional to the force and in the direction of the force. Since there are two sides to a seesaw, there can potentially be two generated torques...torque 1 or "T1" on the left side and torque 2 or "T2" on the right side. If the two torques, T1 and T2, are equal, the seesaw will be in balance, but only calculation can determine this, as follows:

Torque 1 = distance 1 x force 1

T1 = r1F1

Torque 2 = distance 2 x force 2

T2 = r2F2

If two 21-Newton children sat on either side of a seesaw 1.5 meters from its center, its balance could be mathematically determined by inserting the proper numbers into the formula, as indicated below.

T1 = r1F1

1.5(21) = 31.5 mN

T2 = r2F2

1.5(21) = 31.5 mN

31.5 = 31.5

T1 = T2

This seesaw is, indeed, in balance. Children quickly learn that, when one weighs more than the other, one will have to sit closer to the fulcrum in order to effectuate a condition of rotational equilibrium or the seesaw will provide only one direction of rotation...until the ground intercepts its travel. Which child here would have to sit closer to the fulcrum...the heavier or the lighter one?

If a child weighing 30 Newtons sat 1.83 meters from the fulcrum, how far would an 18-Newton child have to sit in order to balance the seesaw? You calculate the distance.

Consider, further, a seesaw on which one 95-Newton object has been placed on either of its ends. Would it be in balance?

Because physics is a precise science and results can only be proven mathematically, the balance of this seesaw cannot be determined or calculated, despite appearances to the contrary, since only the forces have been given and not their distances from the fulcrum. Opinions are, again, valueless in physics.

If you understand the physics principle of torque, then you can answer the following questions:

a). Why does a torque wrench have a long handle?

b). If you turned a bicycle over and wished to spin one of its wheels, where on the spoke would you place your fingers to do so with as little effort as possible...near the hub or near the rim?

c). Why are you not able to lift up the side of your car with your bare hands when you need to change a tire, but you are able to do so with the aid of a jack?

4. From Seesaws to Airplanes

How does torque relate to aviation? Let us now move from doors to airplanes. We shall take our door and make two fundamental changes to it.

a). We shall rotate the door from the vertical to the horizontal.

b). We shall relocate the hinges from the side to the center.

c). The way the door can be equated to the seesaw, so too can the seesaw be analogous to an aircraft.

In order to understand how the torque formula elements can be applied to the weight and balance of an aircraft, each can be progressively transposed from a representative seesaw to an aircraft itself. The elements, as you will recall, had included the following:

F = Force.

r = Distance.

Angle = Sine Theta.

The first of these, the distance, can be equated to an aircraft's lower-deck holds, its main deck passenger cabin zones, and its fuel tanks. The way the two sides of the seesaw had provided the distance from the fulcrum, all these locations provide the same purpose on an airplane.

While the children, or anything else, had provided the forces exerted on the seesaw's sides, payload comprises this element on an aircraft. Payload, which implies something an airline carries to "pay" the way, includes the passengers on the main deck, the baggage, cargo, and mail on the lower deck, and the fuel distributed in the various fuel tanks.

Finally, the angle at which an aircraft rotates, or the demarcation line between the forward, negative, and aft, positive, torque, is the center of gravity. Center of gravity is defined as that point at which an object, if suspended by a single cord, would balance. Although the angle, sine theta, had been omitted in the torque formula's calculation, it nevertheless remains in reality, since torque involves a vector, not a scalar, product.

Consider, now, an aircraft with a nose-heavy loading condition. How could this situation be remedied? The solution lay in the torque formula, with changing either its distance or its force parameters.

1. Add load in the aft hold.

2. Move the load further aft.

3. Transfer load, from forward to aft.

4. Remove load from the forward hold.

5. Move the forward load in the forward hold further aft in the same hold.

Although it is not possible to affect only one element of the torque formula without affecting the other, since they are intertwined by multiplication, these remedies had attempted to target one side of the torque elements, force or distance, more than the other. They have been repeated below, indicating which of the two they had predominantly targeted and why.

1. Add load in the aft hold - F (force), which did not previously exist.

2. Move the load further aft - r (distance), force was neither added nor removed, only placed at a different distance from the center of gravity.

3. Transfer load, from forward to aft. - r (distance), force was neither added nor removed, only placed at a different distance from the center of gravity.

4. Remove load from the forward hold - F (force), because payload was removed, not redistributed.

Therefore, the distance in the torque formula had been reduced.

5. Move the forward load in the forward hold further aft in the same hold - r (distance), force was neither added nor removed, only placed at a different disitance from the center of gravity.

The lower deck of an Airbus A-330-200 aircraft, for example, is divided into two distinct areas, the forward hold and the aft hold, and these comprise the physical space. However, they are further subdivided into compartments 1 and 2 in the forward hold and compartments 3, 4, and 5 in the aft hold for weight and balance purposes in order to assign a single distance, or "r," in the torque formula from them to the center of gravity, regardless of where their contents, such as cargo or baggage, are located in that compartment. The total weight of load in that compartment is also considered a single force, or, "F" in the torque formula.

The main, or passenger, deck of an A-330 serves the same purpose: it is subdivided into four zones for weight and balance purposes. All passengers in a particular zone are considered to be one distance, or "r," from the center of gravity in the torque formula, while their collective weight is considered one force, or "F," in the torque formula.

Resultantly, there are five torque calculations taken into consideration on the lower deck and four on the main deck on an A-330, each with its own force and distance parameters.

The torques ahead of the center of gravity are given negative (-) signs and therefore produce counterclockwise rotations, while the torques behind the center of gravity are given positive (+) signs and produce clockwise rotations. If the sum of the negative torques equals the sum of the positive torques, the airplane is in a state of rotational equilibrium...which can be considered "optimum trim." If not, the aircraft will either be nose-heavy, resulting in a forward trim, or tail-heavy, resulting in an aft trim. Because the net angular acceleration always acts in the direction of the larger torque, that torque will be the determining factor of the airplane's balance...the same way the distances and weights of the two children determined whether the seesaw remained in balance.

Although an equality between the forward, negative torques and the aft, positive ones results in the idealized, perfect balance on both seesaws and aircraft, the latter seldom take off with such a condition, departing instead with either a forward or aft trim. If the aircraft is not perfectly in balance, as hitherto striven for, how can it safely take off and travel to its destination?

The answer is in understanding its three axes of flight: longitudinal, lateral, and vertical.

Unlike the many experimenters who had preceded the Wright Brothers, it had been the Wright Brothers themselves who had succeeded in conquering controlled, powered, heavier-than-air flight because they had applied a systematic approach to solving its aerodynamic problems. They had focused on three principle areas:

1. Lift

2. Propulsion

3. Balance and control

It had been the latter which ultimately provided the inflight corrective force of either a nose- or tail-heavy aircraft with a less-than-optimum trim.

The horizontal stabilizers counterbalance a nose-heavy (negative torque) or tail-heavy (positive torque) condition in flight and the degree of its travel is determined by the aircraft manufacturer. These parameters are indicated by the respective aircraft's balance envelope. Provided that an aircraft's trim is within these limits, it can be safely dispatched for flight. If it is out-of-trim, the horizontal stabilizers cannot counteract this tendency and it cannot be dispatched for flight. The rule of thumb here is that, if an aircraft is out-of-trim on the ground, it will be out-of-trim in the air.

Many other calculations are made with the aid of the balance table. The front side of it, for instance, yields index corrections for:

1. Load in lower-deck compartments.

2. Passengers in main-deck cabins.

3. Fuel in both the main tanks and the trim tank.

Each of these index corrections is the result of the already-incorporated torque formula. All weight placed in any weight & balance station is considered one "F" and any station is considered one "r" in the torque formula: T=rF. Index corrections correspond to a range of weights, not a single one. The 6321-6682 kilograms for compartment 4 on an A-330-200, for example, yields an index correction of +18. The balance table incorporates the already calculated torque by multiplying the force and the distance and dividing the result by a constant to equal a reduction factor index unit.

5. Torque--and Its Role in Fuel Burn Reduction

Although an aircraft experiences rotations about all three axes in flight and these complicated calculations extend far beyond this discussion, the torque concept can further illustrate how a properly loaded airplane can result in a reduction in fuel burn and therefore lower trip-mile costs. The further aft the loaded center of gravity is, the further aft the "door hinge" is...which, in this case, is the CG and the distance is the line between it and the horizontal tail. As you know, torque is the product of the distance and the force and the shorter that distance, the smaller is the resultant torque which must be counteracted. The tail actually serves the purpose of "correcting" both the less-than-optimum trim and the unstable air which induces its rotation. Unlike an unbalanced seesaw, an aircraft cannot rely on the ground to intercept the angular rotation of its heavier end and therefore employs the horizontal tail surfaces to counteract this tendency...hence the term, "horizontal stabilizers". If your aircraft utilizes the earth for this purpose, you know your flight is "out-of-trim"...which, in physics, can be translated as "severe rotational disequilibrium."

Although this may sound complex, we are still dealing, in the simplest form, with "T = rF." While torque may aid the opening of a door on the ground, it works against an aircraft's economy in flight. When an airplane cruises in a greater state of "rotational equilibrium," it is less prone to respond to air current-induced torque and this, in turn, results in a reduction in fuel burn.

Can you believe that your infant-learned ability to open a door could lead to something so profound as aircraft weight and balance?

A graduate of Long Island University-C.W. Post Campus with a summa-cum-laude BA Degree in Comparative Languages and Journalism, I have subsequently earned the Continuing Community Education Teaching Certificate from the Nassau Association for Continuing Community Education (NACCE) at Molloy College, the Travel Career Development Certificate from the Institute of Certified Travel Agents (ICTA) at LIU, and the AAS Degree in Aerospace Technology at the State University of New York - College of Technology at Farmingdale. Having amassed almost three decades in the airline industry, I managed the New York-JFK and Washington-Dulles stations at Austrian Airlines, created the North American Station Training Program, served as an Aviation Advisor to Farmingdale State University of New York, and devised and taught the Airline Management Certificate Program at the Long Island Educational Opportunity Center. A freelance author, I have written some 70 books of the short story, novel, nonfiction, essay, poetry, article, log, curriculum, training manual, and textbook genre in English, German, and Spanish, having principally focused on aviation and travel, and I have been published in book, magazine, newsletter, and electronic Web site form. I am a writer for Cole Palen's Old Rhinebeck Aerodrome in New York. I have made some 350 lifetime trips by air, sea, rail, and road.

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