Vector addition is more complicated than scalar addition. Vector quantities are conveniently shown by arrows. The length of the arrow represents the magnitude of the quantity, and the orientation of the arrow represents the directional property of the quantity. For example, if we consider the top of this page as representing north and we want to show the velocity of an aircraft flying east at an airspeed of 300 kts., the velocity vector is as shown in Figure 1.3. If there is a 30‐kts. wind from the north, the wind vector is as shown in Figure 1.4.
To find the aircraft’s flight path, groundspeed, and drift angle, we add these two vectors as follows. Place the tail of the wind vector at the head of the arrow of the aircraft vector and draw a straight line from the tail of the aircraft vector to the head of the arrow of the wind vector. This resultant vector represents the path of the aircraft over the ground. The length of the resultant vector represents the groundspeed, and the angle between the aircraft vector and the resultant vector is the drift angle (Figure 1.5).
Figure 1.3 Vector of an eastbound aircraft.
Figure 1.4 Vector of a north wind.
Figure 1.5 Vector addition.
The groundspeed is the hypotenuse of the right triangle and is found by use of the Pythagorean theorem
The drift angle is the angle whose tangent is Vw/Va/c = 30/300 = 0.1, which is 5.7° to the right (south) of the aircraft heading.
Vector Resolution
It is often desirable to replace a given vector by two or more other vectors. This is called vector resolution. The resulting vectors are called component vectors of the original vector and, if added vectorially, they will produce the original vector. For example, if an aircraft is in a steady climb, at an airspeed of 200 kts., and the flight path makes a 30° angle with the horizontal, the groundspeed and rate of climb can be found by vector resolution. The flight path and velocity are shown by vector Va/c in Figure 1.6.
Figure 1.6 Vector of an aircraft in a climb.
Figure 1.7 Vectors of groundspeed and rate of climb.
In Figure 1.7, to resolve the vector Va/c into a component Vh parallel to the horizontal, which will represent the groundspeed, and a vertical component, Vv, which will represent the rate of climb, we simply draw a straight line vertically upward from the horizontal to the tip of the arrow Va/c. This vertical line represents the rate of climb and the horizontal line represents the groundspeed of the aircraft. If the airspeed Va/c is 200 kts. and the climb angle is 30°, mathematically the values are
MOMENTS
If a mechanic tightens a nut by applying a force to a wrench, a twisting action, called a moment, is created about the center of the bolt. This particular type of moment is called torque (pronounced “tork”). Moments, M, are measured by multiplying the amount of the applied force, F, by the moment arm, L:
The moment arm is the perpendicular distance from the line of action of the applied force to the center of rotation. Moments are measured as foot–pounds (ft‐lb) or as inch–pounds (in.‐lb). If a mechanic uses a 10 in.‐long wrench and applies 25 lb of force, the torque on the nut is 250 in.‐lb.
The aircraft moments that are of particular interest to pilots include pitching moments, yawing moments, and rolling moments. If you have ever completed a weight and balance computation for an aircraft, you have calculated a moment, where weight was the force and the arm was the inches from datum. Pitching moments, for example, occur when an aircraft’s elevator is moved. Air loads on the elevator, multiplied by the distance to the aircraft’s center of gravity (CG), create pitching moments, which cause the nose to pitch up or down. As you can see from Eq. 1.2, if a force remains the same but the arm is increased, the moment increases.
Several forces may act on an aircraft at the same time, and each will produce its own moment about the aircraft’s CG. Some of these moments may oppose others in direction. It is therefore necessary to classify each moment, not only by its magnitude, but also by its direction of rotation. One such classification could be by clockwise or counterclockwise rotation. In the case of pitching moments, a nose‐up or nose‐down classification seems appropriate.
Mathematically, it is desirable that moments be classified as positive (+) or negative (−). For example, if a clockwise moment is considered to be a + moment, then a counterclockwise moment must be considered to be a − moment. By definition, aircraft nose‐up pitching moments are considered to be + moments.
Figure 1.8 Balance Lever.
EQUILIBRIUM CONDITIONS
Webster defines equilibrium as “a state in which opposing forces or actions are balanced so that one is not stronger or greater than the other.” A body must meet two requirements to be in a state of equilibrium:
1 There must be no unbalanced forces acting on the body. This is written as the mathematical formula ∑F = 0, where ∑ (cap sigma) is the Greek symbol for “sum of.” Figure 1.2 illustrates the situation where this condition is satisfied (lift = weight, thrust = drag, etc.)
2 There