Pressure, depth and Pascal s. Law

Transcription

1 Pressure, depth and Pascal s Fluid statics Law What is a fluid? Density Pressure Fluid pressure and depth Pascal s principle Buoyancy Archimedes principle Fluid dynamics Reynolds number Equation of continuity Bernoulli s principle Viscosity and Turbulent flow Poiseuille s equation Lecture 2 Dr Julia Bryant web notes: Fluidslect2.pdf pressure3.pdf pascal.pdf 1

2 From last class: In a static fluid, with uniform density ρ, Pressure at depth h = pressure acting on surface + pressure due to height of liquid p h = p 0 + F/A F = weight of column liquid of cross sectional area A F = mg M = ρ V = ρ Ah p 0 pressure acting on surface F/A = ρ gh p h = p 0 + ρgh h A Weight of column of liquid F Liquid uniform density ρ 2

3 ! The pressure within a uniform stationary fluid is the same at all points in the same horizontal plane. h The pressure exerted by a static fluid depends only upon the depth of the fluid, the density of the fluid, and the acceleration of gravity p h = p 0 + ρ g h Static pressure does not depend upon mass or surface area of liquid and the shape of container due to pressure exerted by walls. DEMO 3

4 Example: Estimate the difference in fluid pressure between the neck and base of a bottle of wine when (a) upright and (b) cellaring (lying down) ρ = 1.08 x 10 3 kg m -3 h = 0.23 m p = ρ g h = 1.08 x 10 3 x 9.8 x 0.23 = 2434Pa = 2.4kPa A 175cm tall person has a difference in blood (density 1.06 x 10 3 kg m -3) pressure of 18179Pa between their head and feet. Work it out. 4

5 Example: Water is in a U-tube. Oil is added to one side until it is a height d above the water, which has risen a distance H. What is the density of the oil? d Pressure at the interface height is Left tube: p L = ρ w g H Right tube: p R = ρ oil g (H+d) p L = p R water oil ρ w g H = ρ oil g (H+d) Interface p oil =p water ρ oil = H ρ w H+d ρ/ ρ w is the specific gravity 5

6 In liquids p=p 0 + ρgh but what about gases?! Gas pressure! pv = N k T pv = n R T p i V i = p f V f! T i p is the gas pressure (Pa), " V is the volume of the gas (m 3 ), " T is the gas temperature (K), " N is the number of molecules and " n is the number of moles of the gas (mol) " Boltzmann constant k = 1.38x10-23 J.K -1" Universal gas constant R = J.mol -1.K -1" k = R / N A R = k N A " T f " Avogadro's constant N A = 6.023x10 23 mol -1" 6

7 Isothermals pv = constant 180 pressure p (kpa) p i V i = p f V f T i T f 100 K 200 K 300 K 400 K volume V (m 3 ) Gas laws (fixed quantity of gas) " Boyle's Law (constant temperature) p = constant / V" Charles Law (constant pressure) V = constant x T (constant volume) p = constant x T " 7

8 Measuring Pressure A barometer or manometer can be used to measure pressure 8

9 Measuring relative pressure Manometer D DEMO p D = atmospheric pressure p 2 = p 1 A p 0 + ρ g y 2 = p A + ρ g y 1 Pressure at base p 1 = p A + ρ g y 1 B C h Pressure at base p 2 = p 0 + ρ g y 2 p A - p 0 = ρ g( y 2 - y 1 ) = ρ g h p is the absolute pressure p - p 0 is the gauge pressure y 1 y 2 B and C are at the same level so p B = p C 9

10 Measuring relative pressure Mercury Barometer measures atmospheric pressure A" P 0 = 0" For example p atm = 760 mmhg" P atm = ρgh" h! ρ = 13.6 x 10 3 kg.m -3 " g = 9.8 m.s -2 " h = 760 mm = 760 x 10-3 m" P atm = (13.6 x 10 3 )(9.8)(760 x 10-3 ) Pa = 1.01 x 10 5 Pa" p atm" ρ 10

11 Why does a brain tumor affect the spinal cord? tumor Increased pressure transmitted down spinal cord 11

12 Pascal's Principle 1653 Blaise Pascal ( ) Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and walls of the containing vessel. force DEMO 12

13 Pascal's Principle 1653 Blaise Pascal ( ) Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and walls of the containing vessel. p h p h p 0ʼ p 0 p 0 2.4kPa + P incease (0,0) h (0,0) h Linear relationship between pressure and depth. If the pressure at the surface increases then the pressure at a depth h also increases by the same amount. 13

14 Tennis Ball Impact on Eye A blow to the eye by a tennis ball can cause more damage than one might expect because of the transmission of the pressure to the back of the eye The cornea on the front of the eye is tough and may feel the pain of the impact but without damage. Pascal s principle means that the pressure is transmitted through the fluid, from the front of the eye, undiminished to all parts of the eye. In this way the retina can be severely damaged or torn. 14

15 How can a person easily lift a car? 15

16 A piston with area A 1 exerts force F 1 on a fluid, which connects a larger piston of area A 2. Based on Pascal s principle, the pressure is the same on both cylinders. F 1 F 1 F 2 P = = A 1 A 2 A 2 F 2 = F 1 >> F 1 A 1 F 1 F 2 pa 1 h 1 oil F 2 h 2 pa 2 A 1 A 2 16

17 F 1 Small piston moves a distance h 1 small force + large distance Volume of fluid displaced V=h 1 A 1 ==> large lifting force Large piston moves h 2 with V=h 2 A over small distance 2 so A h 2 2 = h 1 << h F A 1 F pa 1 h 1 oil F 2 h 2 pa 2 A 1 A 2 17

18 A 2 F 2 = F 1 >> F 1 A 1 A 2 h 2 = h 1 << h 1 A 1 But work done= F 1 h 1 A 1 A 2 = (F 2 x )(h 2 x ) = F 2 h 2 A 2 A 1 Energy is conserved 18

19 BUOYANCY - FLOATING AND SINKING Why do ice cubes float on water? Less dense than water. Yes, but why does something with less density than water float? Why does a hot air balloon rise? 19

20 If I suspend a block on a rope, what force do we need to pull up with on the rope to make the block hang steady? If I immerse the block in water, what happens to the tension in the rope? T=? rope m W 20

21 Buoyancy When a solid object is wholly or partly immersed in a fluid, the fluid molecules are continually striking the submerged surface of the object. " The forces due to these impacts can be combined into a single force the buoyant force which counteracts the weight." T = W - F B F B T W 21

22 If F b > F body floats. g If F b < F body sinks. g A body floats in any liquid with density ρ fluid > ρ body F b F g F b > F g F b < F g 22

23 Thin sack filled with water. F b F g F b F g F b Weight of water mg=f b Weight of object = weight of fluid displaced by object " Volume of displaced water = volume of object Replace sack with stone m s g>f b ===> sinks Weight of object > weight of fluid displaced by object " Volume of displaced water = volume of object Replace sack with wood m w g<f b ==>floats (ρ wood <ρ water ) Weight of object < weight of fluid displaced by object " F g Volume of displaced water < volume of object" Weight of liquid displaced by submerged part " of the object = weight of object 23

24 How high will it float? Wood breaks surface and displaces less water until F b = m wood g=m'g where m' and V' are the mass and volume of the water displaced less dense more dense F b F g ρv'g = ρ wood V wood g V' = ρ wood V wood ρ Fraction of block submerged is ρ wood / ρ 24

25 How high will it float? - What fraction of an iceberg is under water? Water expands on freezing by 10%. Density of ice is 0.9g/cm 3 Fraction of iceberg submerged is ρ ice / ρ water = 0.9/1.0 Therefore 90% of the iceberg is submerged. 25

26 Example problem: The pressure on the surface of a lake is atmospheric pressure, P at. (a) At what depth is the pressure twice atmospheric pressure? (b) If the lake was full of mercury, at what depth is the pressure 2P at? (a) p = p at + ρgh When p=2p at, 2p at = p at + ρgh p at = ρgh h = p at /ρg = 1.01 x 10 5 /(1x 10 3 x 9.8) = 10.3m (b) h Hg = p at /ρ Hg g = 1.01 x 10 5 /(13645 x 9.8) = 75cm 26

27 Another example problem: Blood flows into the aorta through a circular opening of radius 0.9cm. If the blood pressure is 120 torr, how much force must be exerted by the heart? 1 torr = Pa = N.m torr =120 x N.m -2 F = pa F = 120 x x π x (9x10-3 ) 2 N = 4.07 N 27