Today s Discussion: Fluids Pressure and Pascal s principle Bouyancy, Archimedes principle Bernoulli s equation

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2 Physics 213 Waves, Fluids and Thermal Physics Summer 2007 Lecturer: Mike Kagan 322 Whitmore) Today s Discussion: Fluids Pressure and Pascal s principle Bouyancy, Archimedes principle Bernoulli s equation 2

3 Fluids. Description. So far we have only considered motion of point particles. Fluid = too many particles (e.g.( ~10 e.g. ~10 19 molecules in 1cm 3 of air) Need new collective description, new physical quantities But! We shall use the same physical laws: Newton s s Laws, Conservation Laws etc. 3

4 Fluids. Parameters. Density Space Densities (kg/m 3 ) (for homogeneous fluid) Lab vacuum Sample problem: What is the mass of the air in this room? Compare with the mass of people in this room. Air Ice Water Mercury Gold Platinum Uranium nucleus Neutron star Black hole (solar mass)

5 Fluids. Parameters. Pressure - similar to stress, BUT! Different physical mechanism (collisions of molecules will learn in 2 weeks) Stress: something pooling/stretching Pressure: something pushing out (in other words: stress = -pressure) Pressure is a scalar (F is the force magnitude) Vector force perpendicular to given plane, NO shear stress in (non-viscous) fluids! 5

6 Fluids. Parameters. Density Space Densities (kg/m 3 ) (for homogeneous fluid) Lab vacuum Sample problem: What is the mass of the air in this room? Compare it with the mass of people in this room. Air Ice Water Mercury Gold Platinum Uranium nucleus Neutron star Black hole (solar mass)

7 SI: Pascal, 1 Pa = 1 N/m 2 Pressure. Units. Torr: named after Evangelista Torricelli (Galileo s apprentice, first to measure the atmospheric pressure) Equal to the millimeter of mercury, or mmhg American: pounds/in 2 or psi Conversions: 1 atm = 1.01 x 105 Pa = 760 Torr = 14.7psi (where 1 atm is the average pressure at sea level due to the large fluid mass of the atmosphere above and around us) Sample problem: You inflate the front tires on your car to 28 psi. Later you measure your blood pressure, obtaining a reading of 120/80, the reading s being in mm Hg. In kilopascals, what are (a) your tire pressure and (b) Your blood pressure? 7

8 Pressure. Observed phenomena. Fluids exert pressure on their surroundings. Need to explain: 1) Pressure increases under water - your years feel this effect 2) Pressure decreases at high altitudes - harder to breathe on mountaintops - ears popping in airplanes 3) 4) Pascal vases (demo) 8

9 Pressure. Pascal s Law. 1) Isolated fluid (no external forces) Free body diagram 2) Fluid in gravitational field p=const h A 9

10 Pascal s Law. Sample problem. Q: In which container is pressure highest at depth h? A: None. They are all the same. 10

11 Pascal s Law. Applications. Hydraulic press Gain in force! 11

12 Archimedes principle. Bouyancy. "any body partially or completely submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body." Indeed, the bouyant force (see previous slide): Block ( ) in water ( ) Free body diagram h A floats sinks Bouyant force = weight of liquid within volume displaced by body 12

13 Archimedes principle. Sample problems. 1. Block of ice is floating in water. What fraction of the block is submerged? 2. Block of ice is floating in water covered with a thick layer of oil (density 700 kg/m 3 ). What fraction of the block is submerged into water? (Water and oil do not mix.) 13

14 Conservation laws. Mass. Cross section changing in the middle: Incoming volume Outgoing volume All that flows in flows out (fluid incompressible density const) Continuity equation the narrower the faster =R rate 14

15 Continuity equation. Sample problem. What is the volume flow rate from the faucet? 15

16 Conservation laws. Energy. Bernoulli equation or work per unit volume kinetic energy per unit volume potential energy per unit volume For constant y, faster flowing fluids have lower pressure than slower flowing fluids 16

17 Bernoulli equation. Example. Not spinning: Not a Curve Ball Top view Spinning: Curve Ball Figure taken from Georgia State University Physics Department website 17

18 Bernoulli equation. Applications. Lift force on airplane wing Curves represent velocity field lines: (analogously to electric field lines) denser lines imply greater speed, hence regions of lower pressure Velocity of air above wing greater than that under wing Pressure difference exerts Lift Force on wing 18

19 Pressure & Pascal s Law What we learned: fluid pressure at all points in a connected body of an incompressible fluid at rest, which are at the same absolute height, are the same Archimedes principle & buoyancy Bouyant force = weight of liquid within volume displaced by body Equation of continuity (the narrower the faster) A 1 v 1 =A 2 v 2 Bernoulli s equation (faster flowing fluids have lower pressure) 19

20 Next Time Sound Beats,, and shock waves 20

21 Creative problems (next Monday s recitation). 1) Find the density of a cork, using a hard (massless) wire and a graduated jar. 2) Explain how the sprinkler works (Fig 1). 3) A tank is filled with water to a height H. A hole is punched in one of the walls at a depth h below the water surface (see figure below). What value of h would maximize the distance x? Check your prediction using a plastic bottle and water. Fig 1 Fig 2 21