What is a force? What are its dimensions? Is it a vector
(e.g., does it have direction)?
What is gravity? What are its dimensions? Is it a vector
(e.g., does it have direction, and if so, what direction)?
What is weight? What are its dimensions? How does weight
differ from mass?
What is the nature of the force associated with pressure? (That is, what produces that force?) What are the dimensions of pressure? Is it a vector
(e.g., does it have direction)? What about the force associated with pressure?
Consider the three-dimensional rectangular volume shown in the accompanying Figure 1, which could be positioned in the atmosphere, in an ocean, lake, or stream, or inside the solid earth. (There's nothing special physically about the 3-D rectangular shape—it just
makes it easier for us to think about some questions and to express things
mathematically. The same reasoning that we apply to this shape would apply to
other shapes, but the problem might be harder to visualize and to express
Quantities worth identifying and naming in this situation include:
the area of the top and of the bottom of the 3-D rectangle ≡ A
the elevation or altitude of the bottom of the 3-D rectangle relative to some reference level (e.g.,
sea level) ≡ z
the elevation or altitude of the top of the 3-D rectangle ≡ z + Δz
the pressure at the bottom of the 3-D rectangle ≡ p(z)
the pressure at the top of the 3-D rectangle ≡ p(z+Δz)
the average density of the material in the 3-D rectangle ≡ ρ
the magnitude of the force of gravity acting on any object per unit mass of the object
What is the mass of material in this volume, expressed in terms of
quantities identified above?
As shorthands, let's write the force of gravity acting on the material in the space defined by the 3-D rectangle ≡ Fg and the mass of material inside the 3-D rectangular volume ≡ m. In what direction does the force
of gravity pull on the material in the volume? How can we express Fg in terms of (or relate it to) m and quantity(ies) identified on the list above, taking into account the direction of Fg?
Note: We can write vectors in one spatial dimension as a scalar that can be positive or negative, where the sign communicates the direction in which the one-dimensional vector points. In particular, we define a (one-dimensional) coordinate axis in which the coordinate position increases in one direction (the "positive coordinate direction") and decreases in the other direction (the "negative coordinate direction"). In our case we're calling the coordinate z, which we define here to be the distance above a reference elevation or altitude (such as sea level), which increases upward and decreases downward. Hence, in our case, one-dimensional vectorquantities directed upward are positive scalars, and those directed downward must be negative scalars.
As a shorthand, let's write Fp(z) ≡ the force exerted on the bottom of the material in the 3-D rectangular volume due to the pressure
exerted on it by the air or water or rock/sand/soil/ground just outside the volume (where z is the vertical coordinate position of the bottom of the 3-D rectangle). How can we express Fp(z) in terms of (or relate it to) quantities identified on the list above, taking into account its direction?
As a shorthand, we can write Fp(z+Δz) ≡ the force exerted on the top of the material in the 3-D rectangular volume due to the pressure
exerted on it by the air or water or rock/sand/soil just outside the volume. How can we express Fp(z+Δz) in terms of (or relate it to) quantities identified on the list above, taking into account its direction?
What is the net force on this parcel (that is, the sum of the forces) in the vertical direction
due to pressure at the top and bottom of the parcel?
Building on your result in (5)(b), how can we express the force of gravity on each unit of mass of the material in the 3-D rectangular volume (force of gravity per unit mass) in terms of (or relate it to) quantities identified on the list above, taking into account its direction?
How can we express the net force due to pressure in the vertical direction acting on each
unit of mass (net force due to pressure per unit mass) in terms of (or relate it to) quantities identified on the list above? (Note: Take advantage of your result in (5)(a), the definition of density, and the relation between (1) the volume of a 3-D rectangle and (2) its height and the area of its horizontal faces.)
When you're done with your expression in (5)(g), with a bit more algebraic work you should be able to rewrite it as something proportional to the ratio of two coupled differences, a mathematical form that should look familiar to you by now. What mathematical form does the net force/mass due to pressure take as the volume of the 3-D rectangle becomes infinitesimally small? (Note: We intend here to consider a smaller and smaller 3-D rectangular region "carved out of" space, not a fixed mass of material that we imagine compressing. A smaller and smaller 3-D rectangular region in space will, of course, contain less and less material the smaller it gets.)
Suppose that the force of gravity and the net force due to pressure were
the only forces acting on this parcel in the vertical direction. Moreover,
suppose that the parcel is not accelerating vertically.
What relationship would the two forces per unit mass have to each other
in this situation? (What principle would you invoke to establish this
How must the pressure vary with increasing altitude in this case?