Centripetal force on a person on the Earth

[kuruman]

Do you think that diagram would be improved by ignoring the definition of g, making $F_g$ larger and adding another arrow in the same direction as $F_{normal}$? How would you label that arrow: $F_{centrifgugal}$? $F_{reaction\ force\ to\ the\ centripetal\ force}$?

The simple answer to your question is "No". If the diagram needs improvement it's not along the lines you suggest in my opinion. FBDs of this sort are used for solving problems almost exclusively in introductory physics courses. The underlying approximation in these is that the Earth is an inertial frame. Some textbooks mention the approximation explicitly but most don't bother. In this approximation, a plumb bob points in the direction of $\vec g$ which is towards the center of a perfectly spherical Earth.

It is a good approximation meant to bring to the forefront the application of Newton's 2nd law to solutions of dynamics problems. Ignoring air resistance serves a similar purpose in projectile motion problems. So the label F_g = W = mg in the diagram expresses the approximation and the definition of weight: the force with which the Earth attracts the object is the same as the weight which is the same as the mass multiplied by the magnitude of the local acceleration of gravity.

If we are going to refine the approximation because we are thinking of the non-inertial frame of a rotating Earth, we are moving into the realm of an intermediate mechanics course. However, why consider only the Earth's rotation and not add other sources that affect $\vec g$, e.g. the gravitation effects of the Moon and the Sun which cause the observable effects of ocean tides?

Also, in my opinion, the egregious error in this FBD is the label F_normal = - W which confuses the magnitude of a vector with its component. We have $\vec W=|\vec W|(-\hat y)$ which implies that $W_y=-|W|$. The convention is that labels in FBDs are magnitudes of vectors whilst the direction of the vector is indicated by the direction of the arrow. Thus, the label on the arrow pointing straight up sets a magnitude equal to a negative number. It's this sort of thing that students see and ask "is $g$ positive or negative?"

 

[jbriggs444]

How does one draw a free-body diagram of a weightless apple?

One includes all forces on the apple other than gravity -- which you have apparently decided to transform away based on calling the apple "weightless".

If for instance, the apple is sitting on a table, it is subject to a real force equal to mg and is experiencing an upward proper acceleration of g in the freely falling frame that you have selected.

 

[vela]

As the Wikipedia article notes, "$\dots~$the famous apple falling from the tree, on its way to meet the ground near Isaac Newton, would be weightless." How does one draw a free-body diagram of a weightless apple? More to the point, what are the rules for drawing FBDs when the system's weight depends on its state of motion?

The FBD would be the same as before: a single force directed downward with magnitude $mg$. You just don't refer to that force as the weight. In other words, you still represent all of the forces acting on a body in the FBD. Whether you call any of those forces the weight of the object is irrelevant.

The more common example is the person standing on a scale in an elevator. As the elevator accelerates upward or downward, the scale's reading changes. Some would say the person's weight is changing; the rest of us say the person's apparent weight changes. In either case, the FBD would generally have two forces acting on the person (as long as $a<g$): $mg$ downward and the normal force upward.