# Chapter 14 - Fluids. -Archimedes, On Floating Bodies. David J. Starling Penn State Hazleton PHYS 213. Chapter 14 - Fluids. Objectives (Ch 14)

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1 Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced. -Archimedes, On Floating Bodies David J. Starling Penn State Hazleton PHYS 213

2 Objectives for Chapter 14 (a) Identify and apply concepts related to pressure, force, mass and density as they relate to fluids. (b) Identify the difference between absolute and gauge pressure and relate these concepts. (c) Describe the use of both a barometer and an open-tube manometer in measuring pressure. (d) Apply Pascal s principle to solve problems involving hydraulic lift. (e) Apply Archimedes s principle to connect the buoyant force to gravity, mass and volume of an object. (f) Describe and apply the concepts related to fluid flow using the equation of continuity. (g) Describe and apply Bernoulli s equation to problems involving fluid flowing in a gravitational field.

3 A penny is laying flat on a desk. When a student blows over the penny, it is observed to be lifted upward and then carried away. Which of the following statements best describes why the penny was lifted up from the desk? (a) The penny was attracted by the force of the wind. (b) The pressure of the moving air above the penny was greater than that of the air between the penny and the desk top. (c) The pressure of the moving air above the penny was less than that of the air between the paper and the desk top. (d) The weight of the penny was reduced by the wind blowing over it. (e) The wind pushed the side of the penny and lifted it upward.

4 A fluid can flow freely and conforms to its container.

5 A fluid can flow freely and conforms to its container. Although a fluid resists compression/expansion, shearing forces cause motion.

6 Fluids include both gases and liquid, excluding solids.

7 Fluids include both gases and liquid, excluding solids. Differnt fluids have different properties, such as viscocity, density, compressibility.

8 Density is the ratio of mass to volume of particular sample of fluid. ρ = m V ρ = m V (1)

9 Density is the ratio of mass to volume of particular sample of fluid. ρ = m V ρ = m V (1) This is a scalar quantity with units kg/m 3.

10 Density is the ratio of mass to volume of particular sample of fluid. ρ = m V ρ = m V (1) This is a scalar quantity with units kg/m 3. If the fluid is incompressible (e.g., water), this density does not change much. For compressible fluids (e.g., air), the density can vary.

11 The pressure experienced by a fluid is defined as the ratio of force to area: p = F A p = F A. (2)

12 The pressure experienced by a fluid is defined as the ratio of force to area: p = F A p = F A. (2) Pressure is a scalar (i.e., the sensor doesn t care about direction).

13 Pressure has a variety of common units in addition the S.I. standard N/m 2 : pascal (1 Pa = 1 N/m 2 ) astmosphere (atm) torr (equivalent to mm Hg) pound/in 2 (psi)

14 Pressure has a variety of common units in addition the S.I. standard N/m 2 : pascal (1 Pa = 1 N/m 2 ) astmosphere (atm) torr (equivalent to mm Hg) pound/in 2 (psi) 1 atm = Pa = 760 torr = 14.7 lb/in 2.

15 A swimming pool with a width of 15.0 m and a length of 20.0 m is filled with water. If the total force, which is directed downward, on the bottom of the pool is N, what is the pressure on the bottom of the bottom of the pool? (a) Pa (b) Pa (c) Pa (d) Pa (e) Pa

16 When a fluid is at rest, we can easily describe its properties using forces and density.

17 When a fluid is at rest, we can easily describe its properties using forces and density. These three forces sum together to give us the net force.

18 Using pressure, mass and density, we can derive a very useful relationship: Fi = 0 F 2 F 1 mg = 0

19 Using pressure, mass and density, we can derive a very useful relationship: Fi = 0 F 2 F 1 mg = 0 p 2 A p 1 A (ρa[y 1 y 2 ])g = 0

20 Using pressure, mass and density, we can derive a very useful relationship: Fi = 0 F 2 F 1 mg = 0 p 2 A p 1 A (ρa[y 1 y 2 ])g = 0 p 2 = p 1 + ρg(y 1 y 2 )

21 Using pressure, mass and density, we can derive a very useful relationship: Fi = 0 F 2 F 1 mg = 0 p 2 A p 1 A (ρa[y 1 y 2 ])g = 0 p 2 = p 1 + ρg(y 1 y 2 ) If we use the surface of the water as the reference, we get: p = p 0 + ρgh.

22 p = p 0 + ρgh. h is measured downward

23 h is measured downward p = p 0 + ρgh. For upward: p = p 0 ρ air gh.

24 h is measured downward p = p 0 + ρgh. For upward: p = p 0 ρ air gh. p is known as the absolute pressure

25 h is measured downward p = p 0 + ρgh. For upward: p = p 0 ρ air gh. p is known as the absolute pressure p g = p p 0 = ρgh is known as the gauge pressure

26 h is measured downward p = p 0 + ρgh. For upward: p = p 0 ρ air gh. p is known as the absolute pressure p g = p p 0 = ρgh is known as the gauge pressure Pressure only depends on depth and density, not on shape at all!

27 An above ground water pump is used to extract water from an open well. A pipe extends from the pump to the bottom of the well. What is the maximum depth from which water can be pumped? (a) 19.6 m (b) 39.2 m (c) 10.3 m (d) 101 m (e) With a big enough pump, you can extract water from any depth.

28 To measure pressure, we will use p = p 0 ρgh along with a dense fluid (mercury, for instance).

29 To measure pressure, we will use p = p 0 ρgh along with a dense fluid (mercury, for instance). In this case, p 0 and so p 0 = ρgh. By measuring h, we measure pressure.

30 For mercury barometers, the height difference (mm-hg) is equal to torr if:

31 For mercury barometers, the height difference (mm-hg) is equal to torr if: Gravity g has its standard value (9.8 m/s 2 ) The mercury is held at 0.

32 Alternatively, we can use an open-tube manometer with a gas on both sides.

33 Alternatively, we can use an open-tube manometer with a gas on both sides. Here, we measure the gauge pressure: p g = p p 0 = ρgh.

34 Consider the mercury U-shaped tube manometer shown. Which one of the following choices is equal to the gauge pressure of the gas enclosed in the spherical container? The acceleration due to gravity is g and the density of mercury is ρ. (a) ρgc (b) ρgb (c) ρga (d) p atm + ρgb (e) p atm ρgc

35 For enclosed, incompressible fluids, a change in pressure at one point in the fluid is transmitted undiminished to every other point of the fluid.

36 For enclosed, incompressible fluids, a change in pressure at one point in the fluid is transmitted undiminished to every other point of the fluid. This effect is independent of height: p = p ext

37 This affect is known as Pascal s Principle and is the explanation for hydraulic lever:

38 This affect is known as Pascal s Principle and is the explanation for hydraulic lever: If a small force F i is applied, this increases the pressure within the fluid by p = F i /A i.

39 This affect is known as Pascal s Principle and is the explanation for hydraulic lever: If a small force F i is applied, this increases the pressure within the fluid by p = F i /A i. This increase in pressure propagates through the material to the other side: p = F o /A 0.

40 Together: F i A i = F o A o F o = F i A o A i (3)

41 Together: F i A i = F o A o F o = F i A o A i (3) The bigger the ratio A o /A i, the bigger the output force!

42 Together: F i A i = F o A o F o = F i A o A i (3) The bigger the ratio A o /A i, the bigger the output force! How does this affect distance?

43 The fluid is conserved and uncompressed, so V i = V o, d i A i = d o A o d o = d i A i A o. (4)

44 The fluid is conserved and uncompressed, so V i = V o, d i A i = d o A o d o = d i A i A o. (4) The smaller the ratio A i /A o, the smaller the output motion!

45 When an object is submerged in a fluid, the pressure on the object varies along its surface according to its depth: p = p 0 + ρgh.

46 When an object is submerged in a fluid, the pressure on the object varies along its surface according to its depth: p = p 0 + ρgh. The force on the bottom is greater than the force at the top, resulting in a buoyant force.

47 How big is the buoyant force on an object in a fluid? Archimedes s principle states: F b = m f g (5)

48 How big is the buoyant force on an object in a fluid? Archimedes s principle states: F b = m f g (5) The buoyant force on a (partially) submerged object is upward and equal to the weight of the fluid that has been displaced, m f g.

49 When an object floats (in or on top of the fluid), it is in static equillibrium and F b = F g = m f g.

50 When an object floats (in or on top of the fluid), it is in static equillibrium and F b = F g = m f g. Buoyant forces also make things appear less heavy. This is known as apparent weight: W app = W F b.

51 A solid block of mass m is suspended in a liquid by a thread. The density of the block is greater than that of the liquid. Initially, the fluid level is such that the block is at a depth d and the tension in the thread is T. Then, the fluid level is decreased such that the depth is 0.5d. What is the tension in the thread when the block is at the new depth? (a) 0.25T (b) 0.50T (c) T (d) 2T (e) 4T

52 The motion of real fluids is extremely complex; we will focus on an ideal fluid instead:

53 The motion of real fluids is extremely complex; we will focus on an ideal fluid instead: (a) steady/laminar flow: velocity of fluid is fixed in time for a particular point;

54 The motion of real fluids is extremely complex; we will focus on an ideal fluid instead: (a) steady/laminar flow: velocity of fluid is fixed in time for a particular point; (b) incompressible flow: the density is constant throughout the fluid;

55 The motion of real fluids is extremely complex; we will focus on an ideal fluid instead: (a) steady/laminar flow: velocity of fluid is fixed in time for a particular point; (b) incompressible flow: the density is constant throughout the fluid; (c) nonviscous flow: there is no resistance to motion;

56 The motion of real fluids is extremely complex; we will focus on an ideal fluid instead: (a) steady/laminar flow: velocity of fluid is fixed in time for a particular point; (b) incompressible flow: the density is constant throughout the fluid; (c) nonviscous flow: there is no resistance to motion; (d) irrotational flow: particles placed in fluid can only translate, not rotate.

57 We can visualize the flow of a fluid using streamlines made of the motion of tracers.

58 We can visualize the flow of a fluid using streamlines made of the motion of tracers. Velocity is tangent to the streamlines; no two streamlines intersect, and streamlines are stationary for laminar flow.

59 Since the fluid is incompressible, a fixed volume of fluid will experience different speeds based upon cross-sectional area.

60 Since the fluid is incompressible, a fixed volume of fluid will experience different speeds based upon cross-sectional area. A 1 v 1 = A 2 v 2 (6) Can apply to any tube of flow that follows the streamlines.

61 We can define two flow rates for an ideal fluid: Volume flow rate: R V = Av (m 3 /s) Mass flow rate: R m = ρav (kg/s)

62 We can define two flow rates for an ideal fluid: Volume flow rate: R V = Av (m 3 /s) Mass flow rate: R m = ρav (kg/s) Both R V and R m are constants for ideal fluids.

63 Let s apply conservation of energy to the motion of a volume of ideal fluid, using the work done by the pressure at each side. W = E

64 Let s apply conservation of energy to the motion of a volume of ideal fluid, using the work done by the pressure at each side. W = E W 1 W 2 = E 2 E 1

65 Let s apply conservation of energy to the motion of a volume of ideal fluid, using the work done by the pressure at each side. W = E W 1 W 2 = E 2 E 1 W 1 + E 1 = W 2 + E 2

66 Let s apply conservation of energy to the motion of a volume of ideal fluid, using the work done by the pressure at each side. W = E W 1 W 2 = E 2 E 1 W 1 + E 1 = W 2 + E 2 F 1 d mv2 1 + mgy 1 = (same) 2

67 Let s apply conservation of energy to the motion of a volume of ideal fluid, using the work done by the pressure at each side. W = E W 1 W 2 = E 2 E 1 W 1 + E 1 = W 2 + E 2 F 1 d mv2 1 + mgy 1 = (same) 2 p 1 V (ρv)v2 1 + (ρv)gy 1 = (same) 2

68 Let s apply conservation of energy to the motion of a volume of ideal fluid, using the work done by the pressure at each side. W = E W 1 W 2 = E 2 E 1 W 1 + E 1 = W 2 + E 2 F 1 d mv2 1 + mgy 1 = (same) 2 p 1 V (ρv)v2 1 + (ρv)gy 1 = (same) 2 p ρv2 1 + ρgy 1 = (same) 2

69 Bernoulli s equation states that fluid flow has this particular quantity conserved: p ρv2 + ρgy = constant (7)

70 Bernoulli s equation states that fluid flow has this particular quantity conserved: p ρv2 + ρgy = constant (7) For stationary fluid: p + ρgy = constant

71 Bernoulli s equation states that fluid flow has this particular quantity conserved: p ρv2 + ρgy = constant (7) For stationary fluid: p + ρgy = constant For horizontal fluid flow: p ρv2 1 = p ρv2 2

72 Fluid is flowing from left to right through the pipe shown in the drawing. Rank the pressures at the three locations in order from lowest to highest? (a) p A > p B > p C (b) p B > p A = p C (c) p C > p B > p A (d) p B > p A and p B > p C (e) p C > p A and p C > p B

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