Pressure is a scalar quantity. Positive pressure means that adjacent fluid elements are pushing each other F dF P= = , F = PA A dA The pushing force acts through the interface and is proportional to the area of the interface. 2x area => 2x force. “Fluid exerts the same pressure in all directions” means that the force transmitted through an interface near a given point does not depend on the orientation of the interface. point of interest Hydrostatic equilibrium (no motion, no net forces) Condition for hydrostatic equilibrium – constant pressure throughout the fluid volume. Variation of pressure creates net force in the direction of decreasing pressure. Hydrostatic equilibrium with external forces: Gravity Force from above (pushing down) Force from below (pushing up) PA ( P + dP) A Gravitational force (pulling down) dFg = mg = !gV = !gA dh Balance (equilibrium) equation PA + !gA dh = ( P + dP) A !gA dh = AdP dP = !g dh Hydrostatic equilibrium with external forces: Gravity dP = !g dh We have got a differential equation… Where do we go from here? We need to integrate it. Can we? If both ρ and g are constant, certainly yes. P = !gh + P0 What is P0? And BTW, what is h? P0 is the constant of integration – the value of pressure at h = 0. It is natural to count h downwards from the fluid surface (h = depth). Then P0 is the pressure at the surface – the atmospheric pressure. If the atmospheric pressure changes, does the pressure at a given depth change? Yep! Hydrostatic equilibrium with external forces: Gravity dP = !g dh We have got a differential equation… Where do we go from here? We need to integrate it. Can we? If both ρ and g are constant, certainly yes. P = !gh + P0 What is P0? And BTW, what is h? Pascal’s law: a pressure change anywhere in a fluid is felt throughout the fluid. Hydrostatic equilibrium with external forces: Gravity dP = !g dh The density, ρ, is only constant in liquids. Gasses are compressible, and one must assume variable ρ(h). Earth atmosphere: " = " 0 exp(! h / h0 ) P = P0 exp(! h / h0 ) P = !gh + P0 Pressure at the surface of water or at 0 altitude is the usual good candidate for P0 P0 = Patm Ocean (or cup). Atmosphere P = Patm + ! w gh P = Patm " ! air gh Here h is the depth. Pressure grows by about 1 atm (105 Pa) every 10 m Here h is the altitude. Pressure drops by about 120 Pa every 10 m Drinking through a straw. When you get a tooth extracted you are warned not to drink through a straw. Why so? The pressure of liquid at the top of the straw is h Ptop = Patm " !gh Patm For you to drink, the pressure inside your mouth should drop below Ptop. This implies a substantial negative pressure difference, ρgh, between the blood vessels in your gums and your mouth, which may open the wound. The total force acting on the bottom of the vessel is F = P! A The most remarkable thing about the expression for pressure is what it does not include. The expression for hydrostatic pressure is easy to see for the straight, unobstructed column, but not obvious for more contorted geometries. The force exerted on the bottom may be strikingly different from the weight of the liquid! P = !gh' Hydraulic lift A multiplication of force can be achieved by the application of fluid pressure according to Pascal's principle, which for the two pistons implies P1 = P2 This allows the lifting of a heavy load with a small force, as in an auto hydraulic lift… but of course there can be no multiplication of work, so in an ideal case with no frictional loss: Win = Wout Hydraulic lift is very similar to a lever…. “GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH” Archimedes A vacuum cleaner Does the vacuum suck the dust? Strictly speaking the pump of the cleaner creates lower pressure inside it and the air and dust are driven into the hose by the pressure of the atmosphere. evacuated volume A barometer Pressure of the air, Patm, drives mercury into the hollow tube. There is no air and no pressure inside the tube. h Therefore Patm is balanced by hydrostatic pressure of the mercury. Patm = ρgh ρ = 13600 kg/m3 Ptop = Patm ! "gh = 0 - density of mercury " Patm = !gh A manometer – pressure gauge Now there is a fluid (gas) under pressure in the reservoir. The difference between the pressure inside the reservoir, Pres, and Patm is now balanced by hydrostatic pressure of the mercury. Pres " Patm = !gh – gauge pressure Pres = Patm + !gh – absolute pressure A manometer is measuring gauge pressure Buoyant force Archimede’s principle: The buoyant force on an object is equal to the weight of the fluid displaced by the object. Fp = mw g = ! wVg The buoyant force is applied to the center of gravity of the fluid (center of the submerged volume of the body). Archimedes principle: The buoyant force on an object is equal to the weight of the fluid displaced by the object, Vdisp ρfluid g. Ship loaded with 50 ton of iron. Ship empty. Ship loaded with 50 ton of styrofoam. Vsub ! water g = mship g Volume of the submerged part of the ship (or any other floating object) is equal to the mass of the ship divided by the density of water: Vsub = mship g ! water g = mship ! water Equal Volumes Feel Equal Buoyant Forces Suppose you had equal sized balls of cork, aluminum and lead, with respective densities of 0.2, 2.7, and 11.3 times the density of water. If the volume of each is 10 cubic centimeters then their masses are 2, 27, and 113 gram. Each would displace 10 grams of water, yielding apparent masses of -8 (the cork would accelerate upward), 17 and 103 grams respectively (and weights of -0.08, 0.17 and 1.03 N). Apparent mass can be defined as apparent weight divided by the gravitational acceleration, g. Center of gravity vs. center of buoyancy Gravitational force is applied at the center of gravity. Buoyancy force is applied to the center of buoyancy. Center of gravity should be below center of buoyancy for stable equilibrium. Is that a necessary condition of equilibrium? There is something wrong with the picture on the left… What? Center of buoyancy is the center of volume of the submerged part of the boat. It cannot possibly be at or above the water-line!