About 40 years ago, Brian Pullan was Head of the Department of Medical Biophysics in the University of Manchester. Late last year, he sent me a problem to think about – it’s in the next paragraph.
An ideal plane flies in a straight line with a constant altitude at a constant speed. Does it need to consume fuel?
There were also two supplementary questions – they are in the next two paragraphs.
Are the forces acting on the plane analogous to those maintaining a brick on top of a wall?
To what extent do we need to think about repulsion between electrons when we consider the forces on the brick?
I don’t believe there is a “right” answer to these questions. But, in this post, I explain my approach to them.
Let’s start with the main question. To answer it, we need to consider what we mean by an “ideal plane”. The answer to the question will depend on how we decide to define this idea. I decided that my ideal plane had the characteristics listed below.
- It flies through air of negligible viscosity.
- It flies much slower than the speed of sound, so that the air flow over its wings can be considered incompressible.
- There is no turbulence, so that the air flow can be considered laminar.
We can then use Bernoulli’s equation to answer the question.
According to Newton’s third law of motion, the plane will continue to fly in a straight line (therefore, at a constant altitude) provided there is no external force acting on it; it would then not need to use any fuel. But there is an external force acting on the plane – gravity.
Returning to Bernoulli’s equation, we know that
where Δp is the pressure difference between the upper and lower surfaces of the wing, ρ is the density of air, v is the speed of the plane, h is its height and g is the modulus of the gravitational field (see post 17.16). C is a constant. For incompressible flow at constant temperature, ρ will also be constant.
We are considering what happens when v is constant. Then Δp, and hence Δp/ρ, will be constant because the speed of the plane and the area of its wing do not change (see post 17.16). So we can then rewrite Bernoulli’s equation as
where C’ is a new constant. Since g is a constant, h is constant. This means that the plane continues to fly at a constant altitude – it has no need to consume fuel because it doesn’t need to do work (and, therefore, use energy) to overcome the gravitational force.
My answer doesn’t describe the motion of a real plane. It is a simple model that provides some useful insights – not an exact description of reality. My three characteristics of an idea plane assume that the viscosity of air is zero, that air is completely incompressible and that there is no turbulence. These three assumptions are, at best, approximations to reality. In addition, a plane may need to use fuel to overcome the effects of wind, and stay on its intended flight path even when it travels at constant speed at a constant height.
The stability of the flying plane is not the same as the stability of the brick on top of a wall. Understanding the stability of the brick is a problem in statics. An observer, in the frame of reference of the wall, observes no movement. According to Newton’s third law of motion the gravitational force exerted on the brick is opposed by an equal and opposite force exerted by the wall (the bricks and mortar below the top brick) on the top brick.
But the flying plane can’t be considered as a static problem. An observer in the frame of reference of the plane observes air moving over the wings. An observer in the frame of reference of the ground observes a moving plane. The flying plane is stable because it is moving (an example of dynamic stability) but the system (the air and the plane) is not in equilibrium because an observer will always detect that part of the system is moving. The plane is stable but not in equilibrium – it has dynamic stability.
We have used Newton’s third law of motion to answer the second part of the question. But to answer the third part we need to think about how the force opposing gravity is generated. The gravitational force acting on the brick compresses the wall beneath it. Some of the work done in compressing the wall is stored as energy. This energy can be used in recoil to generate a force that tends to oppose the force of gravity. All this is explained in post 20.3.
How are electrons involved in the elastic energy stored by the compressed wall? Compression pushed the atoms in the bricks and mortar closer together. But they are pushed apart by the repulsive van der Waals’ forces between atoms. These van der Waals’ forces arise because of temporary fluctuations in the positions of electrons in atoms. So, ultimately, the behaviour of electrons could be considered as responsible for the stability of the brick.
But we don’t have to think about electrons to understand the stability of the brick. On one level, we can explain this stability using Newton’s third law of motion. If we want to understand how the forces that oppose the gravitational force are generated, we need to think about van der Waals’ forces. And, if we want to understand van der Waals’ forces, we need to think about electrons. But, once we understand the idea of van der Waals’ forces, we don’t need to think about electrons to consider the forces on the brick. Similarly, if we accept Newton’s third law, we don’t need to think about how the force opposing gravity is generated.
I think that the previous paragraph exposes a fallacy in the way some scientists (especially some physicists) think about science. Some scientists appear to believe that some ideas are more fundamental than others and that, by implication, these fundamental ideas are more important. I accept that some people may find the search for the elementary particles, that make up protons and neutrons, interesting and that others are interested in a “theory of everything”. But we can understand the conduction bands in metals and semi-conductors and chemical reactions without worrying about anything more fundamental than the behaviour of electrons. And to understand metabolism, the chemical reactions that enable us to gain energy from food, we don’t (usually) need to think about electrons. The trick in finding scientific explanations is often to identify the simplest hierarchical level (electrons, atoms, molecules, biological cell etc.) needed. For example, genes are made up of molecules whose stability depends on the behaviour of electrons. But we don’t need to know anything about electrons to understand Mendel’s ideas about genetics.