Phases of Matter. Since liquids and gases are able to flow, they are called fluids. Compressible? Able to Flow? shape?

Transcription

1 Fluids Chapte 3

2 Lectue Sequence. Pessue (Sections -3). Mechanical Popeties (Sections 5, and 7) 3. Gauge Pessue (Sections 4, and 6) 4. Moving Fluids (Sections 8-0)

3 Pessue

4 Phases of Matte Phase Retains its shape? Compessible? Able to Flow? Solid Yes No No Liquid No No Yes Gas No Yes Yes Since liquids and gases ae able to flow, they ae called fluids.

5 Density and Specific Gavity Physics Lingo ) Density () efes to the mass of a substance divided by its volume. The SI unit of density is kg/m 3. ) Specific gavity efes to the density of a substance divided by the density of wate at 4º C. This quantity has no units. m V Specific H gavity 3 O, 4º C kg/m 000 H O,4º C

6 Pessue Physics Lingo Pessue (P) efes to the foce pe unit aea on an object. The SI unit of pessue is the pascal (Pa), and it is a scala quantity. F P Pa atm ba To psi Pa N/m atm p ba dyn/cm To =mmhg psi lb F /in A

7 Pessue Pessue is the poduct of the collisions between the fluid s molecules its containe s walls.

8 ConcepTest 3. Conside what happens when you push both a pin and the blunt end of a pen against you skin with the same foce. What will detemine whethe you skin will be punctued? Too Much Pessue ) the pessue on you skin ) the net applied foce on you skin 3) both pessue and net applied foce ae equivalent 4) neithe pessue no net applied foce ae elevant hee

9 ConcepTest 3. Conside what happens when you push both a pin and the blunt end of a pen against you skin with the same foce. What will detemine whethe you skin will be punctued? Too Much Pessue ) the pessue on you skin ) the net applied foce on you skin 3) both pessue and net applied foce ae equivalent 4) neithe pessue no net applied foce ae elevant hee The net foce is the same in both cases. Howeve, in the case of the pin, that foce is concentated ove a much smalle aea of contact with the skin, such that the pessue is much geate. Because the foce pe unit aea (i.e., pessue) is geate, the pin is moe likely to punctue the skin fo that eason.

10 Pessue in Static Fluids In a static fluid, pessue is the same in evey diection (at a given depth). Question: If we ignoe gavity, what is the net foce on dm? Answe: Zeo! Nada! Zilch! Explanation: Since dm is so small, then the same amount of pessue is pushing against the same aea. Theefoe, F F dm F dm F

11 Pessue in Static Fluids A static fluid exets a foce on the wall of its containe which is pependicula to the wall. Thee is no foce component paallel to the wall. In the containe to the ight: F 0 F 0

12 Deivation: Pessue in Static Fluids Given: A static fluid of constant density, and gavity. Find: The pessue at a depth h below the fluid. Solution: P F A F mg m V V Ah P gh P gha A F gha m Ah

13 Pessue in Static Fluids Deivation: Given: A static fluid of vaiable density, in a gavitational field g. Find: The pessue diffeence between a depth of y and y in the fluid. y y Solution: Even though we cannot ou pevious esult diectly, it can still help us find the pessue, we just need to wok with small enough slices fluid.

14 Pessue in Static Fluids Let us pick a slice of fluid dy thick. This thickness can be small enough that the density acoss the slice is constant. Then, dp gdy Integating both sides ove ou egion of inteest, we get y y dy P gdy y y

15 Pessue Poblems

16 Poblem 3.8 Estimate the pessue needed to aise a column of wate to the same height as a 35-m-tall oak tee. The density of wate is appoximately 000 kg/m 3, theeby, P gh (000 kg/m 3 ) (9.8 m/s ) (35 m) P Pa

17 Poblem 3.9 Chai v s Elephant Estimate the pessue exeted on a floo by (a) one pointed chai leg (66 kg on all fou legs) of aea = 0.00 cm, and (b) a 300-kg elephant standing on one foot (aea = 800 cm ). (a) Assuming each leg suppots the same amount of weight, then each of the legs suppot 6.5 kg. Then, P F A mg A (6.5 kg) (9.8 m/s 0.00 cm 4 ) 0 cm m P chai Pa

18 Book Poblem 3-9 Chai v s Elephant (cont.) (b) 4 F (300 kg) (9.8 m/s ) 0 cm P A 800 cm m P elephant.60 5 Pa P P chai elephant 500!!!

19 Example 3.A Atmospheic Pessue Assume that the density of ai at a cetain depth y below the edge of Eath s atmosphee is given by: 6 y 0, fo 0 y h 6 ( y) h 0, othewise Whee 0 is the atmospheic pessue at sea level, and h is the height of the atmosphee. Find the pessue, P, as a function of depth. Assume the gavitational acceleation emains constant, and that P(0)= P 0

20 Example 3.A Atmospheic Pessue (cont.) Vacuum x Ai h y Gound

21 Example 3.A Atmospheic Pessue (cont.) The fom of can easily be integated, but caeful! The poblem asks fo the pessue as a function of depth, not the diffeence in pessue. P( y) gdy 0 g 6 6 h 0g 6 7h We can find the constant of integation by applying the initial condition given in the poblem. y y 7 dy C

22 Example 3.A Atmospheic Pessue (cont.) P( y 0) g C 6 7h 0 C 0 Using this esult, we get: P g 7h 0 7 ( y) y 6

23 Example 3.A Atmospheic Pessue (cont.) Using 0. kg/m 3, and h = 640 km y/h y/h (y). P(y)

24 Mechanical Popeties

25 Pascal s Pinciple Pascal s Pinciple states that if an extenal pessue is applied to a confined fluid, the pessue at evey point within the fluid inceases by the same amount. P 0 x P(y) y P( y) P0

26 Hydaulic Lifts Hydaulic lifts/jacks seve as leve ams.

27 Hydaulic Lift Pin P out F A in in F A out out Fout A out mechanical advantage F in A in If the fluid is incompessible, then h out can be found by using V out. V in V out A in h in A out h out h out A A in out h in

28 Buoyancy When an object is submeged in a fluid, thee is a net foce on the object because the pessues at the top and bottom of it ae diffeent. This foce is called the buoyant foce, and it is an upwads foce equals to the weight of the displaced fluid:

29 Achimedes Pinciple Achimedes pinciple says that: The buoyant foce on an object immesed in a fluid is equal to the weight of the fluid displaced by that object.

30 Floating If an object is submeged in wate, and its density is less than that of wate, then buoyant foce will be geate than the weight of the object, and it will ise until it is patially out of the wate.

31 Floating The faction of the object that is submeged is given by the atio of the object s density to that of the fluid. OR V F displ g mg V displ m F

32 Example 3-0: Is the Cown Gold? When a cown of mass 4.7 kg is submeged in wate, an accuate scale eads only 3.4 kg. Is the cown made of gold? SG gold 9. 3 w' ˆj wˆj F ˆ Bj m' gˆj mgˆj V gˆ F C j m' m V F C m m' V C F C H m m O H V m m O C ' SG C.3

33 Example 3-: Helium Balloon What volume V of helium is needed if a balloon is to lift a load of 80 kg (including the weight of the empty balloon)? F B ˆj m He ai Vg He Vg gˆj m m V( ) ai He Load Load m gˆj g Load 0 0 V ( m Load ai He ) 3 60 m

34 Gauge Pessue

35 Gauge Pessue Most pessue gauges measue the pessue above the atmospheic pessue this is called the gauge pessue. The absolute pessue is the sum of the atmospheic pessue and the gauge pessue.

36 Pessue Gauges Thee ae a numbe of diffeent types of pessue gauges. This one is an open-tube manomete. The pessue in the open end is atmospheic pessue; the pessue being measued will cause the fluid to ise until the pessues on both sides at the same height ae equal.

37 Pessue Gauges Hee ae two moe devices fo measuing pessue: the aneoid gauge and the tie pessue gauge.

38 Pessue Gauges This is a mecuy baomete, developed by Toicelli to measue atmospheic pessue. The height of the column of mecuy is such that the pessue in the tube at the suface level is atm. Theefoe, pessue is often quoted in millimetes (o inches) of mecuy.

39 Example 3-7: Suction A student suggests suction-cup shoes fo Space Shuttle astonauts woking on the exteio of a spacecaft. What is wong with this plan? Atmospheic pessue is what makes suction-cup wok! P 0 Even a 0-cm diamete suction cup can povide about 80 lbs of suction.

40 Poblem 3.7: Oil Density Wate and oil (which don t mix) ae poued into a U-shaped tube, open at both ends. They come to equilibium as shown on the diagam. What is the density of the oil? Hint: Pessues at point a and b ae equal. Solution: Let ou oigin be at point a, h oil be the height of the oil above a, and h wate be the height of the wate. P P oil oil gh P a P b P 0 wate P 0 oil wate gh wate oil oil h h wate oil wate 683 kg/m 3

41 Moving Fluids

42 Types of Flow If the flow of a fluid is smooth, it is called steamline o lamina flow (a). Above a cetain speed, the flow becomes tubulent (b). Tubulent flow has eddies; the viscosity of the fluid is much geate when eddies ae pesent. (a) (b)

43 Mass Flow Rate The mass flow ate is the mass that passes though a given point pe unit time. If no fluid is being added o taken away, then the flow ates at any two points must be equal. m m t t m V i i A i V i i i A A t t A v Av Equation of Continuity

44 ConcepTest 3.5a: Fluid Flow Wate flows though a -cm diamete pipe connected to a ½-cm diamete pipe. Compaed to the speed of the wate in the -cm pipe, the speed in the ½-cm pipe is: () one-quate () one-half (3) the same (4) double (5) fou times v v

45 ConcepTest 3.5a: Fluid Flow Wate flows though a -cm diamete pipe () one-quate connected to a ½-cm diamete pipe. () one-half Compaed to the speed of the wate in the (3) the same -cm pipe, the speed in the ½-cm pipe is: (4) double (5) fou times v v

46 ConcepTest 3.5a: Fluid Flow The continuity equation tells us that: A v Av Since wate is incompessible, and the pipes ae cylindical, then v v v v Since the st pipe is twice as wide as the nd pipe, then v 4v

47 Question 3.8: Flowing Faucet Why does the steam of wate fom a faucet become naowe as it falls? Answe: Gavitational acceleation makes the wate futhe down move faste than the wate above. A 0 v Av v v gy x y y A gy v0 A gy v 0 y A A gy gy v v 0 0

48 Question 3.8: Flowing Faucet Futhemoe Fo a faucet of adius 0 and wate s initial velocity v 0 : A 0 v 0 Av v v gy v gy v ) ( v gy v y

49 Question 3.8: Flowing Faucet / 0 y (cm)

50 Benoulli s Pinciple Benoulli s pinciple: Whee the velocity of a fluid is high, the pessue is low, and whee the velocity is low, the pessue is high.

51 Benoulli s Equation We can deive Benoulli s equation by using the wokenegy theoem and consevation of enegy. Fist, notice that the fluid in volume V is pushing the fluid in V, and theefoe wok is being done. W W F Δ P A FΔ P A Next, we will need to calculate the wok done by gavity to aise the fluid fom y to y.

52 Benoulli s Equation Accoding to the continuity equation, fo an incompessible fluid: A A t t A A Which educes to: Since the density emains constant, then gavity is doing wok on a fixed mass m. W F mg( y ) G Δy 3 y We do not calculate the wok done on gavity because gavity is an extenal foce, and hence not pat of ou system.

53 Benoulli s Equation The total wok done on ou system is then: W W P A W W 3 P A mg( y y) Recalling that the wok done on a system also equals the change in its kinetic enegy, then mv mv P A y P A mg( y ) Using again the continuity equation, and the definition of density, we see that m A A

54 Benoulli s Equation Combining ou two pevious esults: ( ) y y g A P A P A v A v A Simplifying and eaanging: v gy P v gy P

55 Toicelli s Theoem Using Benoulli s pinciple, we find that the speed of fluid coming fom a spigot on an open tank is: This is called Toicelli s theoem.

56 Lift on Aiplane Wings Lift on an aiplane wing is due to the diffeent ai speeds and pessues on the two sufaces of the wing.

57 Cuved Balls A ball s path will cuve due to its spin, which esults in the ai speeds on the two sides of the ball not being equal; theefoe thee is a pessue diffeence. Fo the ball on the left: va v B PA P B Theefoe, the ball will cuve to the ight.

58 Example 3-B Got Wings? Which of the following wing coss-sections will give you the most lift? (a) (b) Solution: In both cases, the fist thing we need to emembe is that the lifting fluid must take the same amount of time to get acoss the wing though the top and though the bottom sides.

59 Example 3-B Got Wings? (a) Let us assume that ou wing is moving at a speed of v 0 elative to the fluid. Then we can find the length of the top and bottom paths to find the fluid s top and bottom velocities. v / t v 0 v v v0 Also, since the fluid ends at the same height that it began, then the gavity tems in Benoulli s equation will cancel each othe. v

60 Let us ewite Benoulli s equation in the fom: Example 3-B Got Wings? ) ( ) ( v v y y g P P Now we can apply ou pevious esults to obtain the pessue diffeential: ) ) (( 0 0 v v P P 0 4 ) ( v P P Whee

61 Example 3-B Got Wings? (b) Following the same logic as befoe: cos(60) ( 3) sin(60) ) ( v v v 0 ( 3) v 0 P P [( ( 3) v0) v 0 ] P P 3) v0, whee 3 ( 0.866