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Transcript
Chapter 10 LECTURE NOTES FLUIDS The four phases of matter are solid, liquid, gas, and plasma. Liquids and gasses are known as fluids. Define density to be the mass per unit volume of a substance measured in units of kg/m3. Thus = m/V. The density of water at 4C is 1g/cm3 = 1000 kg/m3. Define specific gravity SG as the ratio of the density of a substance to the density of water. This quantity is dimensionless and is numerically equal to the density of the substance g/cm3. Thus SG = substance/ water. Define P as the perpendicular component of the force exerted by a fluid per unit area. The units of pressure are pascals (Pa) where 1 Pa = 1 N/m2. Pressures in the British system are often given in lb/ft2 or lb/in2 (psi). Note 1psi = 6900Pa. We write P = F/A. For fluids at rest, we observe that at any point in a fluid the pressure is the same in all directions. The force due to fluid pressure acts perpendicularly to any area in contact with it. The pressure at depth h in a fluid of constant density is P = gh. This is due to the weight of the overlying fluid. The pressure is the same at all points of the same depth. Also the change in pressure is given by P = g h for changes in depth h. Both formulas assume constant density which is true for liquids (which are nearly incompressible) but not gasses. Note: For every 1m difference in height, the change in pressure (for water) is P = (1000kg/m3) (9.80 m/s2) 1m = 9,800 Pa. Air Pressure Air is a fluid and the weight of the atmosphere above us exerts air pressure PA = 101,300 Pa = 14.7 lb/in2 = 1 atm. Atmospheric pressure is also measured in bars where 1 bar = 100,000 Pa. When we measure pressure with a gauge we really are measuring the pressure above or over atmospheric pressure. The absolute or total pressure P is the sum of atmospheric pressure P A and the gauge pressure PG. If a tire gauge says your tires are inflated to 29.4 psi = 202.6 kPa, the absolute pressure P is PA (14.7psi) + PG (29.4psi) = 44.1psi (or 3 atm). A barometer can demonstrate the pressure of air. Air pressure will push a fluid like water up a tube if there is less than atmospheric pressure at the top of the tube (straws work like this). Chapter 10 LECTURE NOTES P=0 If there is a vacuum at the top of the tube, then the pressure of the air balances the pressure of the water column when PA = fluid gh . For water, h = 10.3m or 34 ft. PA How can trees taller than this draw up water to their leaves? Pascal's Principle Pressure applied to a confined fluid increases the pressure throughout by the same amount. This is how a hydraulic lift operates. A small force pushes a piston into a small cylinder; this produces pressure P in the fluid which is transmitted to the lift where a larger area cylinder lifts a great weight. A F Fin out or Fout out A in A out A in where MA is the mechanical advantage of this machine. Since by Pascal's Principal Pin = Pout we have Fin = MA • Fin Measuring Pressure A manometer is a U-shaped tube filled with a fluid like mercury. One end is open to the air -the other connects to the source whose pressure is being measured. The difference in fluid levels in the two arms of the U is a measure of the gauge pressure. We write P = PA + gh; if the h is lower on the side of the tube open to air, it is negative; if higher, positive. In many cases, we simply report the height of the fluid h in mm of mercury. 1mm-Hg = 133Pa using Hg = 13.6 g/cm3. The previously mentioned barometer was invented by Toricelli. The torr = 1mm-Hg is named in his honor. A column of mercury 76cm = 760mm high is supported by 1 atm pressure so 1 atm = 760 torr. Buoyancy Fluids exerts an upward buoyant force . In water, this force almost balances the weight of fish or swimmers. The buoyant force is due to pressure increasing with depth; the greater pressure on the lower surface of an object results in a net upward force. In a fluid of density , an object with upper surface at depth h1 and lower surface at depth h2 feels a buoyant force equal to the weight of the fluid it displaces. Pupper surface = g h1 and the Chapter 10 LECTURE NOTES force this exerts is F1 = g h2 A. Thus Fbuoyant = F2 - F1 = A g (h2 - h1) = • g • volume = mass of fluid displaced by volume of object • g. This is known as Archimedes' principle: The buoyant force on a body immensed in fluid is equal to the weight of the fluid displaced by the body. This principle also applies to floating objects. A solid object floats if object < fluid. The buoyant force upward balances the gravitational force downward of fluid Vdisplaced g = Vdisplaced object = SG of object if the fluid is water. objectVobject g which gives Vobject fluid Look up the density of ice; what part of an ice cube will be under water when dropped in a glass full of water? Note: A hydrometer indicates SG of a liquid by how far it sinks. Note: In a helium filled balloon, the buoyant force equals the weight of displaced air. The balloon floats if this is greater than the weight of the balloon, helium, and payload. Fluids in Motion We study streamline or laminar flow (smooth flow at low speed in a less viscous fluid) versus turbulent flow (characterized by eddies, occurring at higher speeds or more viscous fluids). The equation of continuity states that the mass flowing into a system must equal the mass flowing out (assuming no sources or sink) per unit time. This reduces to the equation A1 v1 = A2 v2 where A1 and A2 are the cross sectional areas through which the fluid enters and leaves at speeds v1 and v2. Note: When the area decreases, the velocity increases. Putting your finger partly over the end of a garden hose decreases the area and increases the speed of the water. Bernoulli's Equation Daniel Bernoulli, member of a celebrated Swiss family, discovered that a fluid's velocity is high when its pressure is low. If a fluid at one point has pressure P1 velocity v1, and is at height h1, at 1 1 another point P2, v2, y2, then P1 + v 12 gy 1 P2 v 22 gy 2 2 Example A hole is punched a distance h beneath the surface of a can, open to the air above and filled with water. How fast will the water exit the hole? Chapter 10 LECTURE NOTES Air pressure is above the open top of the can and the hole. The velocity v1 is the fluid in the can 1 is low. Pair + 0 + gy top Pair v 22 gy hole and v 2 2gy top y hole 2gh , just as if the 2 water fell through the height h. Question: If holes are punched at different heights beneath the top of the can, which jet of water leaking out of the can will travel farthest?