I have known what potential energy is for a very long time, but it is only of late I think I have truly began to appreciate its subtly in describing the world around us. Anyone that took A-level physics in the UK (and I presume other countries around the world) will have come up against the “classic” potential energy formulas; the main one of which I speak being PE = mgh used to describe the gravitational potential energy with m representing mass, g the gravitational constant (9.8 on Earth) and h representing the height of the object above the ground. It’s a neat little nugget of information and it helps to rationalise the changes in energy types when say an object falls from rest towards the earth – ignoring other forces etc, we model this as gravitational potential energy being converted to kinetic energy (½mv²) and all go and get on with our lives.
But of course, there is more to it than this. At school, as so often is the case we only scratched the surface and the reality is humbling. It is worth noting that a lot of the ideas and constructs I now speak of are derived from classical mechanics. What does this mean? For most this simply means the observable world around us. For the more advanced physicist it is worth including the caveat that we talk about a set of intertial reference frames in which such laws are true; but we do not speak of the set of all intertial reference frames.
Which of F = ma and E = mc² is the post famous law of Physics? I don’t know. But that I do know is that they are both very well known. Today we consider the first. When you learnt differentiation in mathematics and forces in science; somewhat annoyingly nobody put the two together; and said take the position of an object x. Well the velocity is represented by dx/dt (the variation of x with t) and the acceleration is represented by dv/dt or the second time derivative of the position – x double dot. I don’t know why not but I think it is crazy because this is what really starts to illustrate what these constructs are and how they work. But from this point onwards we need to stop considering position,velocity and acceleration as three separate idea; but rather rates of change of position and so on.
Now we are properly considering physics with rates of change – all we are left to do is consider what the force is. Well the force is infact the rate of change of the potential energy. This is true for all conservative forces which include gravity, magnetism, electricity and spring force. What we are really saying, in words when we consider F=ma for a system is:
The rate of change of the potential energy is equal to the mass multiplied by the second time derivative of the position of x.
Naturally since most systems are such that mass is a constant – all we are saying is the rate of change of potential energy is proportional to the second time derivative of the position by a constant factor – equal to the mass. Now that is much more elegant – but more to the point that really gets behind what is actually going on here. It actually tells us about the mathematics and it tells us about the physics, without patronising us with over simplified formulas. This is just one of many neat little examples how when we put back the detail into out formula we get a better understanding of the world.