Conceptual Physics Matter Liquids Gases

Transcription

1 Conceptual Physics Matter Liquids Gases Lana Sheridan De Anza College July 19, 2016

2 Last time the atom history of our understanding of the atom solids density

3 Overview elasticity liquids pressure buoyancy and Archimedes Principle Pascal s principle gases the atmosphere Bernoulli s principle

4 Elasticity Elasticity is a property of some solids: they can be deformed and return on their own to their original shape. A spring is a device that makes use of this effect: it is a coil of metal. It can be stretched to a longer length or compressed to a shorter one, but when released will return to its original length.

5 Elasticity The force that the spring exerts to restore itself to its original length is proportional to how much it is compressed or stretched. This is called Hooke s Law: F = kx where x is the distance that the spring is stretched or compressed by and k is a constant that depends on the spring itself. (The spring constant ). If a very large force is put on the spring eventually it will break: it will not return to its original shape. The elastic limit is the maximum distance the spring can be stretched so that it still returns to its original shape.

6 Spring example If a 2 kg painting is hung from a spring, the spring stretches 10 cm. What if instead a 4 kg painting is hung from the spring? How far will it stretch?

7 Spring example If a 2 kg painting is hung from a spring, the spring stretches 10 cm. What if instead a 4 kg painting is hung from the spring? How far will it stretch? We don t know the spring constant, but we can work it out from the information about the first 2 kg painting. The force on the spring is just the weight of the painting. k = F g x = (2 kg)g 0.1 m 200 N/m x = F k = = 0.2 m

8 Spring example If a 2 kg painting is hung from a spring, the spring stretches 10 cm. What if instead a 4 kg painting is hung from the spring? How far will it stretch? We don t know the spring constant, but we can work it out from the information about the first 2 kg painting. The force on the spring is just the weight of the painting. k = F g x = (2 kg)g 0.1 m 200 N/m x = F k = = 0.2 m If you put on twice the force, you stretch the spring twice as far!

9 Spring example If a 2 kg painting is hung from a spring, the spring stretches 10 cm. Now suppose a 6 kg painting is hung from the same spring. How far does it stretch?

10 Spring example If a 2 kg painting is hung from a spring, the spring stretches 10 cm. Now suppose a 6 kg painting is hung from the same spring. How far does it stretch? x = 0.3 m. [Also see prob. 4 from the textbook.]

11 Liquids liquid a state of matter where the constituent particles are free to move around relative to one another, but the substance is nearly incompressible. Volume is fixed, but shape is not. A solid that is heated will melt to form a liquid. If that liquid is further heated it will boil to form a gas. Both liquids and gases can flow, but the density of a substance in its liquid state is close to that in its solid state. For gases the density can vary.

12 Liquids Liquids require a fairly narrow range of temperatures and pressures to exist. Earth has the right temperature and pressure on its surface for water to be liquid. Examples of liquids at room temperature: water oil alcohol mercury

13 Pressure Most liquids maintain nearly constant volume whatever the pressure. Pressure = Force Area P = F A An illustration: a hammer and a nail pounded by the same hammer exert the same force on a board. The pressure at the tip of the nail is much higher.

14 Pressure Pressure is measured in Pascals, Pa. 1 Pa = 1 N/m 2. Atmospheric pressure is kpa, or Pa. Since the Pascal is a small unit (even 1 atmosphere is 10 5 ) sometimes atmospheres are also used as a unit of pressure. 1 atmosphere = Pa

15 Fluid Pressure Pressure is a scalar quantity. We understand that its underlying cause is molecular collisions: 1 Diagram by Brant Carlson, on Wikipedia.

16 Liquid Pressure Liquids also apply pressure onto submerged objects. Liquid pressure depends on depth. liquid pressure = weight density of fluid depth P liq = ρgh where ρ = m/v is the mass density of the fluid and h is the depth. It does not depend on the total amount of water involved, just the depth of water.

17 Liquid Pressure The force exerted downward by the column of water is F = mg = ρvg.

18 Liquid Pressure The force exerted downward by the column of water is F = mg = ρvg. Pressure, P liq = ρvg/a = ρgh.

19 Total Pressure The liquid pressure only expresses the pressure due to the weight of the fluid above. However, this is not the total pressure in most circumstances, eg. diving on earth.

20 Total Pressure The liquid pressure only expresses the pressure due to the weight of the fluid above. However, this is not the total pressure in most circumstances, eg. diving on earth. The total pressure is the sum of the pressure due to the liquid and the pressure due to the atmosphere. P total = ρgh + P atm where P atm = Pa.

21 Pressure varies with Depth

22 Pascal s Barrel

23 Question Plug and Chug # 2, page 243. Calculate the water pressure at the base of the Hoover Dam. The depth of water behind the dam is 220 m.

24 Surface Tension The H 2 O molecules that make up water are slightly polar and attract one another. This means that water droplets tend to arrange themselves to minimize their surface area. cohesion The attraction between like molecules adhesion The attraction between molecules of different substances Water in air behaves as if it is covered with an elastic membrane because the cohesive forces between the water molecules are stronger than the adhesive forces between water and air.

25 Surface Tension

26 Mensicus A meniscus, or curvature of a liquid surface happens when a water droplet is in air, or when water is in a glass beaker. Adhesive forces between water and glass are stronger than cohesive forces within the water, so the water tends to stick to the glass. This effect can also be used to draw water up in a glass tube.

27 Capillary Action

28 Siphoning 1 Figure from Wikipedia.

29 Buoyancy Astronauts training in their spacesuits: The total mass of NASA s EMU (extravehicular mobility unit) is 178 kg. Why does training underwater make maneuvering in the suits easier? 1 Picture from Hubblesite.org.

30 Buoyancy The apparent weight of submerged objects is less than its full weight. For an object that would float, but is held underwater, its apparent weight is negative! There is an upward force on an object in a fluid called the buoyant force.

31 Buoyancy and Archimedes Principle Archimedes Principle The buoyant force on an object is equal to the weight of the fluid that the object displaces. Logically, if a brick falls to the bottom of a pool it must push an amount water equal to its volume up and out of the way.

32 Buoyancy and Archimedes Principle For a fully submerged object the buoyant force is: F buoy = ρ f V obj g where ρ f is the mass density of the fluid and V obj is the volume of the object. ρ f V obj is the mass of the water moved aside by the object.

33 Buoyancy and Archimedes Principle An object that floats will displace less fluid than its entire volume. For a floating object: F buoy = ρ f V sub g where V sub is the volume of the part of the object underneath the fluid level only.

34 Sinking and Floating Will a particular object sink or float in a particular fluid? If the object is less dense than the fluid it will float. If the object is more dense than the fluid it will sink. If the object and the fluid have the same density if will neither float or sink, but drift at equilibrium.

35 Sinking and Floating A floating object displaces a mass of water equal to its own mass! (Equivalently, a weight of water equal to its own weight.) This also means that ρ f V sub = m obj.

36 Questions Military ships are often compared by their displacements, the weight (or mass, depending on context) of water they displace. The USS Enterprise was an aircraft carrier (now decommissioned). Displacement: 94,781 tonnes (metric tons), fully loaded. 1 tonne = 1000 kg What is the mass of the fully loaded USS Enterprise in kgs?

37 Questions Military ships are often compared by their displacements, the weight (or mass, depending on context) of water they displace. The USS Enterprise was an aircraft carrier (now decommissioned). Displacement: 94,781 tonnes (metric tons), fully loaded. 1 tonne = 1000 kg What is the mass of the fully loaded USS Enterprise in kgs? Problem # 8, page246. Your friend of mass 100 kg can just barely float in fresh water. Calculate her approximate volume.

38 Pascal s Principle (Or, how hydraulic systems work.) Liquids are (mostly) incompressible. If you push on a piston at one end of a pipe you will move. Pascal s Principle A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid.

39 Pascal s Principle Since the pressures at the left end and the right end are the same: Since A 2 > A 1, F 2 > F 1. P 1 = P 2 F 1 = F 2 A 1 A 2

40 Pascal s Principle This has applications: 1 Figure from hyperphysics.phy-astr.gsu.edu.

41 Question If a pair of pistons are connected on either end of a hydraulic tube. The first has area 0.2 m 2 and the second has an area of 4 m 2. A force of 30 N is applied the first piston. What is the force exerted by the second piston on a mass that rests on it?

42 Question If a pair of pistons are connected on either end of a hydraulic tube. The first has area 0.2 m 2 and the second has an area of 4 m 2. A force of 30 N is applied the first piston. What is the force exerted by the second piston on a mass that rests on it? If the first piston is depressed a distance of 1 m by the 30 N force, how far does the second piston rise?

43 Gases gas a state of matter where the constituent particles are vastly separated from one another and free to move around. A liquid that is heated will boil to form a gas. A gas heated much further will form a plasma. Both liquids and gases can flow, but the density of gases can vary and they are readily compressible. The large distances between individual molecules or atoms is why most gases are transparent.

44 Elemental Gases hydrogen (H 2 ), nitrogen (N 2 ) oxygen (O 2 ) fluorine (F 2 ) chlorine (Cl 2 ) helium (He) neon (Ne) argon (Ar) krypton (Kr) xenon (Xe) radon (Rn)

45 The Atmosphere We are living in a sea of air. It is held in place by Earth s gravity and made up of: nitrogen, N 2, 78% oxygen, O 2, 21% argon, Ar, 1% carbon dioxide, CO 2, 0.4% It also contains less that 1% water vapor.

46 Layers of the Atmosphere 1 Picture shot from the ISS, courtesy of NASA Earth Observatory.

47 Air Pressure Just like liquids, gases also exert pressure on their surroundings. Atmospheric pressure at sea level is kpa. Atmospheric pressure falls with increasing altitude. Atmospheric pressure fluctuates locally from place to place, day to day.

48 Barometers Barometers are devices for measuring local atmospheric pressure. Typically, simple barometers are filled with mercury, which is very dense. The weight of the mercury in the tube exerts the same pressure as the surrounding atmosphere. On low pressure days, the level of the mercury drops. On high pressure days it rises.

49 Boyle s Law Suppose you seal a fixed mass of gas in a container. If you compress it, you will decrease the volume, but increase the pressure. If you let it expand, you increase the volume but decrease the pressure. For a fixed mass of gas at a constant temperature: P 1 V 1 = P 2 V 2 This is Boyle s Law.

50 Buoyancy in Air Buoyancy in air works the same way as in liquids: F buoy = ρ f V obj g If an object is less dense than air, it will float upwards. However, in the atmosphere, the density of air varies with height.

51 Bernoulli s Principle A law discovered by the 18th-century Swiss scientist, Daniel Bernoulli. Bernoulli s Principle As the speed of a fluid s flow increases, the pressure in the fluid decreases. This leads to a surprising effect: for liquids flowing in pipes, the pressure drops as the pipes get narrower.

52 Bernoulli s Principle Why should this principle hold? Where does it come from? 1 Something similar can be argued for compressible fluids also.

53 Bernoulli s Principle Why should this principle hold? Where does it come from? Actually, it just comes from the conservation of energy, and an assumption that the fluid is incompressible. 1 Consider a fixed volume of fluid, V. In a narrower pipe, this volume flows by a particular point 1 in time t. However, it must push the same volume of fluid past a point 2 in the same time. If the pipe is wider at point 2, it flows more slowly. 1 Something similar can be argued for compressible fluids also.

54 Bernoulli s Principle V = A 1 v 1 t also, V = A 2 v 2 t

55 Bernoulli s Principle V = A 1 v 1 t also, V = A 2 v 2 t This means A 1 v 1 = A 2 v 2 The Continuity equation.

56 Bernoulli s Equation Bernoulli s equation is just the conservation of energy for this fluid. Let the volume of fluid, V, have mass m. Remember: kinetic energy: gravitational potential energy: work:

57 Bernoulli s Equation Bernoulli s equation is just the conservation of energy for this fluid. Let the volume of fluid, V, have mass m. Remember: kinetic energy: 1 2 mv 2 gravitational potential energy: mgh work: W = Fd

58 Bernoulli s Equation Bernoulli s equation is just the conservation of energy for this fluid. The system here is all of the fluid in the pipe shown. Both light blue cylinders of fluid have the same volume, V, and same mass m. We imagine that in a time t, volume V of fluid enters the left end of the pipe, and another V exits the right.

59 Bernoulli s Equation At t 0, fluid in the blue portion is moving past It makes sense that the energy of the S fluid might change: the fluid point 1 at velocity v 1. is moved along, and some is lifted up. A 1 a x 1 S v 1 Point 2 Point 1 x 2 A 2 S v 2 How does it change? Depends on the work done: After a time interval t, the fluid in the blue portion is moving past W = K + U S The path taken by velocity of the particl A set of streamlines particles cannot flow lines would cross one Consider ideal flui ure Let s focus shows the segment at

60 a and 14.18b are equal because the fluid is incompressible.) This work ecause Bernoulli s the force on the Equation segment of fluid is to the left and the displacepoint of application of the force is to the right. Therefore, the net work segment by these forces in the time interval Dt is The pressure at point 1 is P 1. W 5 (P 1 2 P 2 )V The work done is the sum of the work done on each end of the fluid by more fluid that is on either side of it: P 1 A 1 ˆi W = F 1 x 1 F 2 x 2 y 1 x 1 S v 1 Point 2 = P 1 A 1 x 1 P 2 A 2 x 2 a b Point 1 The pressure at point 2 is P 2. x 2 S v 2 y 2 P 2 A 2 ˆi (The environment fluid just to the right of the system fluid does negative work on the system as it must be pushed aside by the system fluid.) 1 Diagram from Serway & Jewett.

61 Bernoulli s Equation Notice that V = A 1 x 1 = A 2 x 2 W = P 1 A 1 x 1 P 2 A 2 x 2 = (P 1 P 2 )V Conservation of energy: W = K + U

62 Bernoulli s Equation Notice that V = A 1 x 1 = A 2 x 2 W = P 1 A 1 x 1 P 2 A 2 x 2 = (P 1 P 2 )V Conservation of energy: W = K + U (P 1 P 2 )V = 1 2 m(v 2 2 v 2 1 ) + mg(h 2 h 1 )

63 Bernoulli s Equation Notice that V = A 1 x 1 = A 2 x 2 W = P 1 A 1 x 1 P 2 A 2 x 2 = (P 1 P 2 )V Conservation of energy: W = K + U (P 1 P 2 )V = 1 2 m(v 2 2 v 2 1 ) + mg(h 2 h 1 ) Dividing by V : P 1 P 2 = 1 2 ρv ρg(h 2 h 1 ) P ρv ρgh 1 = P ρv ρgh 2

64 Bernoulli s Equation P ρv ρgh 1 = P ρv ρgh 2 This expression is true for any two points along a streamline. Therefore, P ρv 2 + ρgh = const is constant along a streamline in the fluid. This is Bernoulli s equation.

65 Bernoulli s Equation P ρv 2 + ρgh = const Even though we derived this expression for the case of an incompressible fluid, this is also true (to first order) for compressible fluids, like air and other gases. The constraint is that the densities should not vary too much from the ambient density ρ.

66 Bernoulli s Principle from Bernoulli s Equation For two different points in the fluid, we have: 1 2 ρv ρgh 1 + P 1 = 1 2 ρv ρgh 2 + P 2

67 Bernoulli s Principle from Bernoulli s Equation For two different points in the fluid, we have: 1 2 ρv ρgh 1 + P 1 = 1 2 ρv ρgh 2 + P 2 Suppose the height of the fluid does not change, so h 1 = h 2 : 1 2 ρv P 1 = 1 2 ρv P 2

68 Bernoulli s Principle from Bernoulli s Equation For two different points in the fluid, we have: 1 2 ρv ρgh 1 + P 1 = 1 2 ρv ρgh 2 + P 2 Suppose the height of the fluid does not change, so h 1 = h 2 : If v 2 > v 1 then P 2 < P ρv P 1 = 1 2 ρv P 2

69 Bernoulli s Principle However, rom the continuity equation A 1 v 1 = A 2 v 2 we can see that if A 2 is smaller than A 1, v 2 is bigger than v 1. So the pressure really does fall as the pipe contracts!

70 Implications of Bernoulli s Principle Bernoulli s principle can help explain why airplanes making use of airfoil-shaped wings can fly. Air must travel further over the top of the wing, reducing pressure there. That means the air beneath the wing pushes upward on the wing more strongly than the air on the top of the wing pushes down. This is called lift.

71 Air Flow over a Wing In fact, the air flows over the wing much faster than under it: not just because it travels a longer distance than over the top. This is the result of circulation of air around the wing. 1 Diagram by John S. Denker, av8n.com.

72 Airflow at different Angles of Attack 1 Diagram by John S. Denker, av8n.com.

73 Implications of Bernoulli s Principle A stall occurs when turbulence behind the wing leads to a sudden loss of lift. The streamlines over the wing detach from the wing surface. This happens when the plane climbs too rapidly and can be dangerous. 1 Photo by user Jaganath, Wikipedia.

74 Implications of Bernoulli s Principle Spoilers on cars reduce lift and promote laminar flow. 1 Photo from

75 Implications of Bernoulli s Principle Wings on racing cars are inverted airfoils that produce downforce at the expense of increased drag. This downforce increases the maximum possible static friction force turns can be taken faster. 1 Photo from

76 Implications of Bernoulli s Principle A curveball pitch in baseball also makes use of Bernoulli s principle. The ball rotates as it moves through the air. Its rotation pulls the air around the ball, so the air moving over one side of the ball moves faster. This causes the ball to deviate from a parabolic trajectory. 1 Diagram by user Gang65, Wikipedia.

77 Implications of Bernoulli s Principle Bernoulli s principle also explains why in a tornado, hurricane, or other extreme weather with high speed winds, windows blow outward on closed buildings.

78 Implications of Bernoulli s Principle Bernoulli s principle also explains why in a tornado, hurricane, or other extreme weather with high speed winds, windows blow outward on closed buildings. The high windspeed outside the building corresponds to low pressure. The pressure inside remains higher, and the pressure difference can break the windows. It can also blow off the roof! It makes sense to allow air a bit of air to flow in or out of a building in extreme weather, so that the pressure equalizes.

79 Summary elasticity liquids pressure buoyancy gases the atmosphere Bernoulli s principle Talks! Aug 2-3. Homework Hewitt, Ch 12, onward from page 225. Problem: 3 Ch 13, onward from page 243. Plug and Chug: 1, 3; Exercises: 11, 15, 19, 25; Problems: 1, 3, 5, 7 Ch 14, onward from page 262. Ranking 1; Exercises: 9, 17, 19, 40, 53; Problems: 1