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By Root 725 0

a framework for describing the motion of things under the influence of a force. The first law describes what happens to an object when no forces act: the object either just sits still or moves in a straight line at constant speed. We shall be looking for an equivalent statement for quantum particles later on, and it’s not giving the game away too much to say that quantum particles do not just sit still – they leap around all over the place even when no forces are present. In fact, the very notion of ‘force’ is absent in the quantum theory, and so Newton’s second law is bound for the wastepaper basket too. We do mean that, by the way – Newton’s laws are heading for the bin because they have been exposed as only approximately correct. They work well in many instances but fail totally when it comes to describing quantum phenomena. The laws of quantum theory replace Newton’s laws and furnish a more accurate description of the world. Newton’s physics emerges out of the quantum description, and it is important to realize that the situation is not ‘Newton for big things and quantum for small’: it is quantum all the way.

Although we aren’t really going to be very interested in Newton’s third law here, it does deserve a comment or two for the enthusiast. The third law says that forces come in pairs; if I stand up then my feet press into the Earth and the Earth responds by pushing back. This implies that for a ‘closed’ system the net force acting on it is zero, and this in turn means that the total momentum of the system is conserved. We shall use the concept of momentum throughout this book and, for a single particle, it is defined to be the product of the particle’s mass and its speed, which we write p = mv. Interestingly, momentum conservation does have some meaning in quantum theory, even though the idea of force does not.

For now though, it is Newton’s second law that interests us. F = ma says that if you apply a known force to something and measure its acceleration then the ratio of the force to the acceleration is its mass. This in turn assumes we know how to define force, but that is not so hard. A simple but not very accurate or practical way would be to measure force in terms of the pull exerted by some standard thing; an average tortoise, let us say, walking in a straight line with a harness attaching it to the object being pulled. We could term the average tortoise the ‘SI Tortoise’ and keep it in a sealed box in the International Bureau of Weights and Measures in Sèvres, France. Two harnessed tortoises would exert twice the force, three would exert three times the force and so on. We could then always talk about any push or pull in terms of the number of average tortoises required to generate it.

Given this system, which is ridiculous enough to be agreed on by any international committee of standards,1 we can simply pull an object with a tortoise and measure its acceleration, and this will allow us to deduce its mass using Newton’s second law. We can then repeat the process for a second object to deduce its mass and then we can put both masses into the law of gravity to determine the force between the masses due to gravity. To put a tortoise-equivalent number on the gravitational force between two masses, though, we would still need to calibrate the whole system to the strength of gravity itself, and this is where the symbol G comes in.

G is a very important number, called ‘Newton’s gravitational constant’, which encodes the strength of the gravitational force. If we doubled G, we would double the force, and this would make the apple accelerate at double the rate towards the ground. It therefore describes one of the fundamental properties of our Universe and we would live in a very different Universe if it took on a different value. It is currently thought that G takes the same value everywhere in the Universe, and that it has remained constant throughout all of time (it appears in Einstein’s theory of gravity too, where it is also a constant). There are other universal constants of Nature that we’ll meet in this