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dp/dz = - ρg
We'll start with the simplest interpretation: since the right-hand side of the equation is always negative (g is a constant, and there's no such thing as negative density), the left-hand side must always be negative, too. Recall the definition of dp/dz: the change in pressure over a tiny vertical distance divided by the length of that distance. This change (or any derivative, for that matter) must always be measured in the direction in which z (or whatever variable is in the denominator) increases. So if dp/dz is to be negative, p must decrease as z increases. All of this is just a fancy way of saying that the higher you go, the smaller the pressure. But you probably knew that by now.
Second, you may remember from high school physics that weight is mass times the acceleration of gravity. So ρ g turns out to be the weight of air per unit volume. Thus, the hydrostatic equation states that the vertical change in air pressure is equal to the weight of the air.
Imagine a layer of air 100 m thick. According to the hydrostatic equation, if we know the average weight or mass of the air per unit volume, we can calculate the change in air pressure between the top and the bottom.
A typical value for the density of air near sea level is 1.3 kilograms per cubic meter. (To me, that's surprisingly heavy!) With that number, you have everything you need to know to compute the change in air pressure over 100m using the hydrostatic equation. Take a few moments now to try it on your own, and then click here for the solution or click here for a hint.
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