# Special Cases

The purpose of physics is to describe as much of nature as possible in as few equations as possible. Scientist dream about reducing all phenomena to a single master equation, preferably one that fits on the back of an index card. (This is not as crazy as it sounds: Maxwell’s Equations, which tell you everything you need to know about electricity and magnetism, can be condensed into a single tensor expression). For this reason, it bothers me a little when some laws, which are really just special cases of other, more general laws, are taught as separate concepts for no compelling reason (besides historical accident). Usually, these specific laws have names based on who discovered them first (or, at least, discovered them later but ended up with the credit anyway). For example, Pascal’s law for fluids says that the pressure of an enclosed incompressible fluid of uniform density is the same for all points at the same depth (and increases linearly as the depth increases). As we have seen, Blaise Pascal, was especially talented at getting things named for him. Along with his triangle, he also has a wager and a theorem. But in the case of the fluids, Pascal’s law is nothing more than a special case of Bernoulli’s principle, where you require the velocity of the fluid to be zero (hydrostatic). Even worse is Torricelli’s law,  where you just set the outside pressures to be constant.

What is interesting about both Pascal’s and Torricelli’s discoveries is that fluids can be used to transmit energy. In the case of Pascal, pressure can be transmitted. This is the basis for hydraulic lifts:

For Torricelli, the velocity of the water coming out of the spigot is equal to the velocity the water would have if you let it fall the same height as from the top to the spigot. Indeed, even though it is not the same water molecules, since the end result is the same (the water level drops and water flies out a certain distance below, conservation of energy requires that the velocity be the same).