To g or not to g?

It might sometimes seem unnecessary to list what looks like two values of g separately, as often seen in lists of physical constants, like this:

Acceleration due to gravity:  g = 9.81 ms⁻²

Gravitational field strength at Earth's surface: g = 9.81 N kg⁻¹

Although it's obviously unnecessary from a numerical point of view, and it's not difficult to show that the units ms⁻² and Nkg⁻¹ are exactly equivalent, it is nevertheless entirely physically correct to distinguish between the two, and the reason is more subtle than you might think.

An unspoken assumption in most, if not all, school level physics is that gravitational and inertial mass are the same. Inertial mass is the property of an object expressed as m in the equation F = ma, namely its reluctance to be accelerated. Gravitational mass, on the other hand, is the tendency of objects to attract each other, expressed as the m in the equation W = mg (or in Newton's law of gravitation). We're so used to treating them as interchangeable that it might take a moment to realise that there's no inherent reason why the one phenomenon should physically or logically entail the other and, therefore, that the ms in those equations don't have to be the same.

As it happens, we have no way of proving that an object's gravitational and inertial mass are the same. Even Einstein had to assume that inertial and gravitational mass were exactly equal to each other - or at least completely indistinguishable - as expressed in the Weak Equivalence Principle in General Relativity. The best we can do is try to measure a difference between the two, usually by trying to detect differences in the accelerations of objects in freefall (as David Scott did relatively informally on the Moon, or Brian Cox did in the world's largest vacuum chamber). Such experiments are sometimes called Eötvös experiments, and one lab whose website is worth a look is the Eöt-Wash group in America. So far, we have not detected any difference, to an accuracy of one part in well over a billion. Of course, no experiment can ever prove anything to an infinite degree of accuracy, but this is where we currently stand.

With this in mind, I feel it's therefore important that, as a teacher, you describe g to your pupils as either acceleration due to gravity or gravitational field strength depending on the situation you're using it in. So, for example, in W = mg, GPE = mgh and ρghg is standing for gravitational field strength. By contrast, when using constant acceleration formulas, g is standing for acceleration due to gravity. To what extent you explain this distinction to your pupils is a separate decision, of course. It might seem picky, but I would say that it's something a good physics teacher should aim to get right.