Just a quickie today, since I’m wrestling with an enormous kraken of a post about fission and radiation and that’s probably not going to be ready until next week.

Gravitational potential energy has come up a couple of times on this blog, and I think it’s worth exploring because bitter experience has shown me that not everyone understands what it is. *I* didn’t understand what it was exactly until I started studying physics at university. This is probably down to the way gravitational potential energy is taught in secondary school science, which is to get the teacher to throw the equation

at you and then run screaming for the door. This equation might be a simple way of rendering gravitational potential energy as a number, but it tells you bugger all about what gravitational potential energy actually *is*, which is where the problem comes from.

Breaking it down, the equation says that gravitational potential energy is the mass of the object times the gravitational acceleration of the Earth times the distance between the object and the Earth’s surface (or the height). My problem with this is that this form of the equation is an approximation meant only to apply to small distances close to the Earth’s surface, and by making that approximation the people who came up with it neatly removed every single term that might have given a clue as to where exactly the energy is coming from. Mass and height are properties of the object, while g is mostly taught as a simple scalar number (9.81 ms^{-1}, or 10 ms^{-1} for the particularly hard of thinking). Why does the simple act of raising an object up high imbue it with energy? That’s not a question the syllabus or the equation want to deal with, and as a result we get a generation of kids who learn the equation by rote but who don’t understand what it *means*. This leads to situations like the one I had a few years ago where I was trying to explain the concept to a couple of people who thought gravitational potential energy was, in their words, “a cheat”; a fudge introduced into the system to make conservation of energy work.

That’s not it at all, of course. The best way to think of gravitational potential energy is this: if you have an object high above a gravitational source, and there are no other forces acting on this object, and you let the object go, the object will be pulled towards the gravitational source. It will accelerate slowly but constantly, and as it approaches the gravitational source this acceleration will increase as the gravitational force exerted on it gets stronger. Eventually it will collide with the source, and the kinetic energy it is travelling with when it does so can be expressed as (groan)

So you put your object mass and its velocity into that equation, and you crunch the numbers, and you get a number for the object’s kinetic energy after it has finished falling towards the gravitational source. This amount of kinetic energy is *exactly the same* as the amount of gravitational potential energy the object had before it started falling; by falling all the way to the source, the object has converted its gravitational potential energy entirely to kinetic energy.

Hopefully from this example you can see that having an object a certain height above a gravity source doesn’t magically imbue it with some extra amount of energy. The object is just the same as it ever was. The energy comes from the gravity source; it is the *potential* amount of energy the object would have were it allowed to freefall towards it. This would be obvious if we actually looked at the proper form of the goddamn equation. You start with

“Work done” is equivalent to energy; it is the product of the amount of force exerted on an object multiplied by the distance through which it acts. Our force here is gravitational force, so

where M is the mass of the source, m is the mass of the object, G is the gravitational constant (6.67 × 10^{-11}) and R is the distance from centre of gravity at which the gravitational force acts.

This is a much better equation because it includes the mass of the source and the gravitational constant, which is where this potential energy is actually coming from! GM/R^{2} is equivalent to g for an object near the surface of the Earth, which is why the approximation GPE = mgh is used. Unfortunately this does mean that our new equation is also an approximation that is only valid in situations where d is small enough that the product of GM/R^{2} will not change significantly. If you wanted a truly robust equation that could be used to calculate the gravitational potential for any object no matter how far away from the source it is and no matter how big d is, you’d have to integrate the equation for gravitational force with respect to R. This is what we physicists technically refer to as “a fun time”; note that in this case, R no longer represents some physical stop point like the surface of the Earth, but is instead simply the distance separating the two centres of gravity:

where C is one of those irritating constants of integration that we can just throw out of the window because it suits us to do so. Of course, this particular form of the equation ain’t taught in schools for obvious reasons, but it does a much better job of illustrating the point, which is that gravitational potential energy is work done by gravity, not some weird kind of energy physically locked away inside something.

” a couple of people who thought gravitational potential energy was, in their words, “a cheat”; a fudge introduced into the system to make conservation of energy work.”

I was one of those people until now. Thanks for explaining it!

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