illuminating science

Here’s a neat conceptual problem that I first puzzled over in first year, and encountered again doing a bit of tutoring recently. I like it, because it forces you to think about the assumptions and connections between a couple of different things. This is probably only really interesting if you’ve done high school physics, though you really just need to know (and understand) three things:

1. That gravitational potential energy is given by E = mgh
2. The concept of work, basically being the energy added to or removed from a system (usually, by mechanical means)
3. That the work done on an object is given by W = Fs, that is, the work is the net force on an object multiplied by the distance (displacement) over which that force acted.

So, if I push a smooth block along some ice with a force of 10 Newtons, over a distance 5 metres, it gains 50 Joules of energy from my 50 Joules of work. If the ice is flat, this will be kinetic energy, and you can calculate its resulting speed the usual way.

Now, let’s say I’m holding a mass on my palm. I raise it up very slowly from my waist to my head, starting and ending at rest. What is the work done on the object? Well, if we neglect the initial and final accelerations (I said “very slowly”, so these are small) the block will move with a constant speed all the way up. Hence, there’s no net force, and the work done is zero. But, it now has more gravitational potential energy, because it’s higher up - despite there being no net external force to give it any energy. It seems we’ve created energy from nothing.

What, if any, is the flaw in my reasoning?