La Tecnología apoya el Aprendizaje de la Matemática

So far, we have established that lift is generated by a wing because, as it moves through the air at a certain angle of attack, the wing turns the air downwards and, itself, experiences a reaction force acting in an upwards direction. You have already had a glimpse, then, at how lift is accounted for by Newton’s Laws of Motion. You have also learnt that a wing of aerofoil cross section is more efficient than a flat plate in turning air downwards and, thus, in generating lift. We must now look a little more closely at the nature of this turning action on the air and examine how that action can be interpreted in the light of Newton’s Laws to explain the nature of lift.

The angle between the wing’s chord line and the

relative airflow is called the angle of attack.

First of all, you must not fall under the misconception that when airflow meets a wing the air is in any way deflected downwards by “bouncing” off the inclined undersurface of the wing. This is not what happens.

Experiments show that if a solid body, such as a wing, moves through a fluid, such as air, and if the body is so shaped or inclined that it deflects or turns the fluid from its relative path, this turning action occurs because the fluid tends to stay in contact with the body and, so, is influenced by the body’s shape or angle of inclination. (See

Figure 3.9).

So, when looking for an explanation of lift, we should note, primarily, that the wing is changing the direction of flow of an air mass. Now, even if the speed of the airflow were to remain unaltered, the fact that the wing is changing the direction of flow of the air means that the wing is causing a change in the velocity of the airflow, because velocity has both magnitude (speed) and direction. If either speed or direction changes, the velocity changes. Now a change in velocity over a given time is, by definition, a rate of change of velocity, in other words, an acceleration (See Page 16). So, in turning the air downwards, the wing is causing the air to change its velocity and, thus, to accelerate. The force which the wing applies to the air, in order to accelerate it, is given by the following formula derived from Newton’s 2nd Law of Motion:

Force = mass × acceleration

But not only does the wing exert a force on the air causing it to change its velocity downwards (that is, causing it to accelerate downwards) but this change in velocity of the air also generates a reaction force on the wing acting in an upwards direction.

This principle - every action has an equal and opposite reaction - is the principle expressed by Newton’s 3rd Law of Motion (See Page 14) and which helps explain the nature of lift.

Figure 3.9 The downward turning of an airflow (downwash) by a wing of aerofoil cross- section.

The reaction force experienced by the wing is the total reaction that we have already considered. The component of the total reaction acting at right angles to the relative airflow is the lift force. The higher the aircraft’s speed, the greater is the rate of change of velocity (acceleration) imparted to the air by the wing, the greater is the total reaction force experienced by the wing, and, so, the greater is the lift force.

(See Figure 3.8.)

As we mentioned earlier on, Newton’s Laws concern themselves with the principle of the conservation of momentum which is one of the fundamental principles of Physics. So let us take a very slightly different perspective on lift generation than the one we have just taken and consider the momentum implications in the production of lift. The air, because of its mass and its velocity relative to the moving wing, possesses momentum. Momentum is a concept which expresses the quantity of motion possessed by a body or substance (See Page 8). Momentum is related to mass and velocity as follows:

Momentum = mass × velocity

Therefore, in imparting a downwash to the air and, thus, causing a change in the velocity of the airflow, the wing is also bringing about a change in momentum of the air. Now, Newton’s 1st Law teaches us that any physical substance which is in motion will continue moving at the same velocity (that is, at the same speed and in the same straight line direction) unless acted upon by a resultant force. So, in order to turn the air downwards, thereby changing the velocity and, thus, momentum of the air flowing over it, the wing must exert a force on the air. If we measure this change of momentum of the air, over a given lapse of time, we observe a particular rate of change of momentum. Newton’s 2nd Law states that the magnitude of

the resultant force acting on a body is proportional to the rate of change of momentum of the body brought about by that force. At the same time, in order

to satisfy the principle of conservation of momentum, Newton’s 3rd Law tells us that any resultant force which acts on a body gives rise to an equal and opposite reaction force which acts on the object which was the cause of the first action. In the case we are considering, the reaction predicted by Newton’s 3rd Law is an explanation of the generation of lift by a wing.

Lift can be explained then, by considering the momentum implications in the following

way:

In accordance with Newton’s 2nd Law, the downwards acting force, F, exerted by a wing on the air flowing over it is equal to the rate of change of momentum of the airflow.

Now, momentum = mass × velocity and, considering a unit mass of air, m with the initial velocity of air, v1

being changed to a final velocity, v2 over a period of time, t,

the Force, F, exerted by the wing on the airflow may be expressed in a simplified

The component of the Total Reaction,

acting at right angles to the relative airflow, is called Lift.

manner by the formula:

F = m × ( v2 - v1 ) t

Of course, ( v2 - v1 ) expresses the rate of change of velocity, which is the same t

as acceleration, a.

Therefore, we arrive again at the formula F = m × a which is the definition of Force given by Newton’s 2nd Law. It follows, then, by Newton’s 3rd Law, which states that action and reaction are equal and opposite, and act on different bodies, that the wing experiences a reaction force, known as the total reaction, TR, which can be expressed by:

TR = m × a

Lift, itself, of course, as you have learned, is the vertical component of the total reaction which acts perpendicularly to the relative airflow.

The above explanation of lift, then, shows how the lift force acting on a wing is accounted for by Newton’s Laws of Motion. Scientists are able to confirm that the aerodynamic lift force acting on a wing in a wind tunnel, (and which can easily be measured directly by mechanical means), can be predicted accurately by Newton’s Laws and the principle of the conservation of momentum.