Acceleration due to gravity or the gravitational field

You see this all the time in textbooks:

This is often described as "the acceleration due to gravity". Is this really the best thing to call this? No. A better name would be "the local gravitational field" and list it in units of:

Where does g come from? The usual source is from the weight, or gravitational force:

This is the best way to write it (in vector notation) but you will often see it written as:

which looses the vector nature. In the vector notion, the vector g is a vector pointing towards the center of the Earth.

Why not acceleration due to gravity?

Take the following case of a block sitting on a table. Here is the free body diagram. So what's wrong with this? Nothing. The problem is that if you call g the acceleration due to gravity, its just plain weird. The box is NOT accelerating. I think this leads to the confusion that acceleration itself is a force.

Gravitational Field

If g is the gravitational field, things work out better. First using the units of Newtons per kilogram, one can see that the gravitational field gives a force on a mass. The great thing about this is that it is extremely similar to the electric field in units of Newtons per Coloumb (an idea that students seem to have some trouble with). Note that I am not the first to promote this terminology for g, but I think its a great idea.

When is it the acceleration due to gravity?

Take the case of an object with only the force of gravity on it:

In this case, the only force is gravity and it depends on mass. Using Newton's second law, I can write:

So, in free fall the acceleration has the same magnitude as the gravitational field (and same direction). This is actually something strange. The strange thing is that there are essentially two masses. There is inertial mass (the resistance to change in motion by a force) and there is gravitational mass that is proportional to the gravitational force. To me, it is odd that these two masses are equivalent. Sure there is a way to explain this equivalence in general relativity, but in that area I am no expert (not even close).

The units:

Just as a check it is important to see that gravitational field and acceleration have equivalent units.

So, what about the kinematics equation for free fall? In free fall, the acceleration IS g, so one could write the vertical position as the function:

so this g could be called the acceleration due to gravity. HEY WAIT! How come there is a negative sign there? This is not a vector equation, this is a one-dimensional equation so that the negative sign is needed. It usually bothers students that we use g as a positive number. This is because g is the magnitude of the vector g so it has to be positive.