11/4/2003 PHY Lecture 16 1

Transcription

1 Announcements 1. Exams will be returned at the end of class. You may rework the exam for up to 1 extra credit points. Turn in your old exam and your new work (clearly indicated). Due 11/11/3. You may sign up for presentations if there is sufficient interest. 2. Today s lecture fluids Chapter 15 Static fluids buoyant force Fluid flow 11/4/23 PHY Lecture 16 1

2 The physics of fluids. Fluids include liquids (usually incompressible) and gases (highly compressible ). Fluids obey Newton s equations of motion, but because they move within their containers, the application of Newton s laws to fluids introduces some new forms. Pressure: Pforce/area 1 (N/m 2 ) 1 Pascal Density: ρ mass/volume 1 kg/m 3.1 gm/ml Note: In this chapter P pressure (NOT MOMENTUM) 11/4/23 PHY Lecture 16 2

3 Pressure exerted by air at sea-level 1 atm 1.13x1 5 Pa Example: What is the force exerted by 1 atm of air pressure on a circular area of radius.8m? F PA 1.13x1 5 Pa x π(.8m) 2 24 N Pressure exerted by a fluid acts in all directions. 11/4/23 PHY Lecture 16 3

4 y Density: ρ mass/volume Effects of the weight of a fluid: P(y+ y) ρg y P(y) F( y) F( y + y) + mg F( y) F( y + y) mg + A A A P( y) P( y + y) + ρg y P( y + y) P( y) dp lim y y dy dp dy ρg Note: In this formulation +y is defined to be in the up direction. 11/4/23 PHY Lecture 16 4

5 For an incompressible fluid (such as mercury): dp dy Example: ρ x 1 3 kg/m 3 (constant) ρg P P ρg( y y) h kg/m y y.76 m P ρg 5 3 Pa 9.8 m/s 2 ρ x 1 3 kg/m 3 11/4/23 PHY Lecture 16 5

6 Weather report:.254m 3.17in 3.17in in.766m 11/4/23 PHY Lecture 16 6

7 Question: Consider the same setup, but replace fluid with water (ρ 1 kg/m 3 ). What is h? dp dy ρg P P ρg( y y) h y y Pa 3 1 kg/m 9.8 m/s 1.34m P ρg 2 ρ 1 kg/m 3 11/4/23 PHY Lecture 16 7

8 Buoyant force for fluid acting on a solid: F B ρ fluid V displaced g P( y) F B P( y + y) + ρ Buoyant force : F B fluid g y F bottom { P( y) P( y + y) } A ρ g ya ρ gv fluid fluid submerged F top A mg y F B mg ρ fluid V submerged g ρ solid V solid g V submerged V solid ρ ρ solid fluid 11/4/23 PHY Lecture 16 8

9 Suppose you have a boat which floats in a fresh water lake, with 5% of it submerged below the water. If you float in the same boat in salt water, which of the following would be true? A.More than 5% of the boat will be below the salt water. B.Less than 5% of the boat will be below the salt water. C.The submersion fraction depends upon the boat's total mass and volume. D.The submersion fraction depends upon the barometric pressure. 11/4/23 PHY Lecture 16 9

10 Archimede s method of finding the density of the King s gold crown W water W air W W ρ water air object mg mg ρ ρ water F B object W air ρ V object Wair W object g water V object g ρ water V object g 11/4/23 PHY Lecture 16 1

11 Effects of the weight of a compressible fluid on pressure. dp ρ ρg For a gas, ρ P dy P dp dy ρg P P Solution : ρg ( y y ) y y y y P 8m 5mi P( y) P e P e P e y-y (mi) P (atm) 11/4/23 PHY Lecture 16 11

12 Summary: Application of Newton s second law to fluid (near Earth s surface) dp dy Incompressible fluid : example: ρg P P ρ 1kg/m ρg( y y 3 ) (water) Compressible fluid : P P ρ P e ρg ( y y P g( y y ) ) (for ρg P ( y y ) << 1) example: ρ 1.29kg/m 3 (air) 11/4/23 PHY Lecture 16 12

13 Peer instruction question Suppose that a caterer packed some food in an air tight container with a flexible top at sea-level. This food was loaded on to an airplane with a cruising altitude of ~6 mi above the earth s surface. Assuming that the airplane cabin is imperfectly pressurized, what do you expect the container to look like during the flight? (A) (B) (C) 11/4/23 PHY Lecture 16 13

14 Peer instruction question Suppose that a caterer packed some food in an air tight container with a flexible top at sea-level. This food was loaded on to a submarine which typically submerges at 2m below sea level. Assuming that the submarine cabin is imperfectly pressurized, what do you expect the container to look like during the submersion? (A) (B) (C) 11/4/23 PHY Lecture 16 14

15 Physics of flowing (incompressible) * fluids A 1 v 1 ta 2 v 2 t * Later will generalize to compressible fluids in streamline flow. 11/4/23 PHY Lecture 16 15

16 Energetics of fluids: x 1 v 1 t x 2 v 2 t A 1 x 1 A 2 x 2 m ρ A 1 x 1 K 2 + U 2 K 1 + U 1 +W 12 ½ mv 22 + mgh 2 ½ mv 12 + mgh 1 + (P 1 A 1 x 1 P 2 A 2 x 2 ) P 2 + ½ ρv 22 + ρgh 2 P 1 + ½ ρv 12 + ρgh 1 11/4/23 PHY Lecture 16 16

17 Bernoulli s equation: P 2 + ½ ρv 22 + ρgh 2 P 1 + ½ ρv 12 + ρgh 1 Example: Suppose we know A 1, A 2, ρ, P, y 1, y 2 P + ½ ρv 22 + ρgy 2 P + ½ ρv 12 + ρgy 1 A v A v v 1 2 ( gh ( P P) / ρ) 1 ( A ) 2 1 A 2 11/4/23 PHY Lecture 16 17

18 Streamline flow of air around an airplane wing: F lift (P 2 -P 1 )A P 1 v 1 P 2 + ½ ρv 22 + ρgh 2 P 1 + ½ ρv 12 + ρgh 1 h h P 2 v 2 v 1 1 P 2 > v 2 2 P Example: ρ ( 2 2 v v ) 1 v 1 27 m/s v 2 26 m/s 2 ρ.6 kg/m 3 A 4 m 2 F lift 63,6 N 11/4/23 PHY Lecture 16 18

19 Peer instruction question Suppose you see a tornado coming toward your house. Assuming that you have enough time, which of the following should you do before entering your shelter? (A)Open all of the windows and doors. (B) Make sure that all of the windows and doors are tightly shut. 11/4/23 PHY Lecture 16 19

20 Homework problem: A hypodermic syringe contains a medicine with the density of water. The barrel of the syringe has a cross-sectional area A2.5x1-5 m 2, and the needle has a cross-sectional area a 1.x1-8 m 2. In the absence of a force on the plunger, the pressure everywhere is 1 atm. A force F of magnitude 2 N acts on the plunger, making the medicine squirt horizontally from the needle. Determine the speed of the medicine as leave the needle s tip. 11/4/23 PHY Lecture 16 2