It is hard to know how Newton came by his ideas because it was the fashion of the times to publish one’s ideas in as concise a form as possible and let other people try to follow and see the consequences. It almost looks like a game of “ this is what I think now prove me wrong”.
Newton’s laws of motion are nothing to do with gravity as such, they tell us how a body behaves when it is subjected to a force or a system of forces.
Newton tells us first that a force cannot exist by itself, there must be something resisting the force.[1] That something might be a force or a combination of forces but it might also be the inertia of a body that is accelerating under the action of the force. A force might be a push or a pull but it might also be a friction force. Clearly we shall find all sorts of forces and bodies undergoing all sorts of accelerations.
Newton then makes a bold statement that relates force and acceleration. He asserts that bodies have mass and that force = mass times acceleration. We still do not know any more about mass than Newton knew but his assertion has enabled us to create the whole science of dynamics and use it very successfully indeed.
Finally Newton tells us what happens when a body is moving in the absence of any net force on it. He says that it will move in a straight line at constant speed or, of course, be at rest. This raises the question of a reference frame for measuring speed. The reference frame is that of the distant stars that, in human terms, are fixed. Engineers work on a much, much smaller frame and they work relative to Earth. Usually they are interested in the corollary to this law. They note that if a flow of fluid is moving in a path other than in a straight line there will be a force or system of forces acting on it somewhere.
It may not have astonished Newton when he recognised that action at a distance without any apparent means was possible but the more recent extension of the idea to say that it acts at the astronomical distances between galaxies is unimaginable even if you can get used to the idea. Fortunately we have only a need for gravity on a terrestrial scale.
It is our daily experience that a vertical force is exerted on our bodies continually and without letting up for just an instant. That force can easily be measured on a set of bathroom scales. We do not need to question that it exists but when we try to think of a “mechanism” by which it might be seen to operate then it is very troublesome. It is fun to think about it but we do not need anything complicated. We just need Newton’s laws. Newton postulated that the force of attraction between two masses would depend directly on the product of the masses and inversely as the distance between them. He wrote this in mathematical shorthand as:-
the force where m1 and m2 are the masses, d is the distance between their centres of mass and G is a constant[2].
For most purposes this simple model of the gravitational attraction between two bodies can be used exactly in this form. Only in astrophysics do we require a more refined model.
Newton’s model of the relationship between the gravitational force exerted by the Earth on a body that is not in contact with the Earth seems to stand up if the mass of the Earth is regarded as being at a point at the centre of the Earth. According to the model the force on a body should decrease with distance from the centre. This has been checked by experiment and appears to be correct.
The earth is very large when compared with any device we might make in engineering. It is a large, ever-present body that gives every other body a weight that does not change much wherever it may be.
Clearly we shall need to be able to quantify the force due to the gravitational attraction of the earth. We use the statement that force = mass times acceleration. The gravitational force on any body can be measured. Its acceleration in free fall can be measured and it is usually taken to be 9.81 m/s/s. This gives a relationship between mass and weight and all we need is some system of units.
Before we look at units there is an important feature of a gravitational field to be noted. Work can be stored in it without loss. A mass can be raised and in doing so work is done equal to the weight times the increase in height. This work can be recovered in its entirety by lowering the mass without acceleration. The gravitational field seems to be perfectly elastic. (I cannot think of any other system for which this is true.)
How then shall we visualise the gravitational field of the Earth? The answer is that it depends what you plan to use your field model for. If you want to fly satellites in orbits round the Earth then it may well be necessary to let the field take into account the concentrations of mass that occur in various points in and on the Earth. (This data is readily available.) For the purpose of engineering we can manage with Newton’s model and visualise the field as a series of concentric spherical surfaces. The force exerted on a given mass by the Earth would be the same for any point on a given spherical surface. The value of g decreases with height and it is easy to show that where gh is the value of g at height h and is the value of g at the mean radius of the Earth. As the radius of the Earth is about 6500 km at a height h equal to 1 kilometre . For our purposes g is constant at 9.80665 m/s2 and for most purposes in this book this can be rounded to 9.81 m/s2.
The gravitational force is thought of as being towards the centre of the Earth and this completes our model of the field. This raises another important consequence.
Fluids exist like the rest of the material in a gravitational field and the thing that is special is that, because they can flow, gravity determines where they end up. A fluid will just go on flowing perhaps over thousands of years until it can get no lower because there are solid impermeable surfaces under it. Water might flow through a watercourse and so to the sea but it gets into the watercourse either over the surface of the land or by percolation through the ground. This could not happen to a solid. Water and lots of other liquids can flow in large quantities and in very small quantities because the ability to flow comes from its molecular structure. It flows under the influence of forces that are created by gravity and rigid but not necessarily impervious surfaces.
The atmosphere surrounds the solid surface of the Earth. Air is compressible but it too gets as low as it can but, being compressible, it has greater density at the surface simply because of the weight of air above it. It is about 10 km thick.[3]
[1] This statement has troubled me for 65 years. We have the idea of North and South poles to a magnet and that a North pole, say, cannot exist by itself. So the pole is not an entity. Newton gives us a concept of a force as an entity and then says that it cannot exist by itself. Is that contradictory? Somehow this must be rationalised.
[2] This simple equation has a well-defined character when d is large. The gravitational force that is very very small decreases very slowly with distance.
[3] Have a think about this. Weather systems are often 1500 km in diameter and they clearly have structure, eg, they rotate. How can a fluid system be so thin and yet be coherent?