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Fluid mechanics Pressure at depth In a cylindrical column of water, as in any cylinder the volume is the height x cross sectional area The density of the water = mass ÷volume Pressure at depth The mass of water = density x volume = ρ density x height x area The weight of water = mass x gravity = density x height x area x gravity ρhAg (ρ = density) Pressure at depth Pressure = force ÷ area (weight ÷ area) =(density x height x area x gravity) ÷ area = Density x height x gravity (area cancels out) Pressure = pgh Pressure at depth p Pressure = pgh Since the increases uniformly with depth The average pressure h = pgh/2 or pgh A pgh (h = mean depth) Pressure = force ÷ area Pressure at depth Force is pressure x area F = pgAh p This is the hydrostatic thrust acting on the side of the cylinder h F F A pgh The point at which it acts Is 2/3 the depth from the surface of the water (1/3 from the bottom) Archimedes F1 If a mass is suspended By a cable in air the tension force in the cable is equal to the weight of the mass F1 = mg = (ρVg)body m mg where ρ is the density of the body V is the volume of the body g = gravity Archimedes If a body is totally or partially submersed in water (or other liquid) it will displace some of the water (the water level will rise). The volume of water displaced is the same as the volume of the water F1 m Fmg Fup Archimedes The body will experience an apparent loss in weight which is equal to the weight of water (or other liquid displaced) This apparent loss in weight is equal to the up-thrust force of the liquid on the body F2 m Fmg Fup Archimedes F2 = Fmg – Fup F2 = (ρVg)body – (ρVg)water where ρ is the density of the body V is the volume of the body g = gravity and ρ is the density of the water (liquid) V is the volume of the water(liquid) F2 m Fmg Fup Flow through a tapered pipe Volume flow = Volume/time (m3/s) Flow Mass flow = Mass /time (kg/s) Volumetric flow symbol V V = Av (m3/s) m = ρ V Av (kg/s) A = area v= velocity Ρ = density Flow through a tapered pipe Volume flow = Volume/time (m3/s) Flow Mass flow = Mass /time (kg/s) Volumetric flow symbol V V = Av (m3/s) m = ρ V Av (kg/s) A = area v= velocity Ρ = density Flow through a tapered pipe Rearranging for v Flow v= A1 A2 V ρA Since V is the same at inlet area A1 and area A2 and ρ is the same we can calculate the velocity of flow at each by using the two different areas in the equation A = area v= velocity Ρ = density Flow through a tapered pipe Rearranging for v Flow v= v1 = A A2 A1 V A1 V v2 = V A2 A = area v= velocity Ρ = density Area = πr2