Fluids: Liquids and Gases




P = F/A





P = ρgh


Einstein, Newton and Pascal are playing hide and seek. Its Einstein’s turn to count so he covers his eyes and starts counting to ten.


Pascal runs off and hides. Newton draws a one meter by one meter square on the ground in front of Einstein, then stands in the middle of it.


Einstein reaches ten and uncovers his eyes. He sees Newton immediately and exclaims, “Newton! I found you! You’re it!


Newton smiles and says, “You didn’t find me, you found a newton over a square meter. You found Pascal!”


Pressure in Liquids



Because fluids have weight they exert pressure on whatever is under them or in them.


 Pressure = force / area P = F/A


If force is in newtons, and area in square meters the units are pascals (Pa), after the 15th century French scientist and philosopher Blaise Pascal who studied fluids and pressure. One newton/m2 is one pascal.


It is difficult to measure force in liquids, so F/A can be transformed into a more convenient formula:


Remember F = m*a and when gravity is the source of acceleration then g replaces a, so


P = (mg)/A, but remember density (the Greek letter rho-p) = m/v, so m=pv.

P = (ρVg)/A


Now also remember that, for a cylinder, volume = area*height (A*h) so:


P = (ρgAh)/A,
and the A's cancel, leaving

 Pressure = density x gravity x height (or depth) P = ρgh


This is pronounced, pressure equals rho g h.


The density, ρ, of water is 1000 kg/m3, g  = 9.8 m/s2 and h (or actually depth) in meters is often easy to measure.


Here is an example: what is the pressure at the bottom of a 10 m deep lake?


P = ρgh = 1000 kg/m3 * 9.8 m/s2 * 10 m


P = 98,000 kg/ms2


Now convert from these strange units to Newtons and then Pascals.


Remember 1N = 1kg m/s2, so divide the answer above by this – which is identical to multiplying it by the reciprocal s2/kg m


P = 98,000 kg/ms2 * s2/kg m


The kg and s2 cancel out leaving


P = 98,000 N/m2  = 98,000 Pa



(But from now on, if you use kg, m and s for units in Equ. 4.2 you know the answers automatically work out to be Pascals. Hallelujah!)


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