1. Chapter 9 Fluids

2. Ch09b - Fluids-Revised: 3/6/2010
Chapter 9: Fluids
Introduction to Fluids
Pressure
Measurement of Pressure
Pascal’s Principle
Gravity and Fluid Pressure
Archimedes’ Principle
Continuity Equation
Bernoulli’s Equation
Viscosity and Viscous Drag
Surface Tension
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

3. Ch09b - Fluids-Revised: 3/6/2010
Pressure
Pressure arises from the collisions between the particles of a fluid with another object (container walls for example).
There is a momentum change (impulse) that is away from the container walls. There must be a force exerted on the particle by the wall.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

4. Ch09b - Fluids-Revised: 3/6/2010
By Newton’s 3rd Law, there is a force on the wall due to the particle.
Pressure is defined as
The units of pressure are N/m2 and are called Pascals (Pa).
Note: 1 atmosphere (atm) = kPa
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

5. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.1): Someone steps on your toe, exerting a force of 500 N on an area of 1.0 cm2. What is the average pressure on that area in atmospheres?
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

6. Ch09b - Fluids-Revised: 3/6/2010
Measuring Pressure
A manometer is a U-shaped tube that is partially filled with liquid.
Both ends of the tube are open to the atmosphere.
MFMcGraw
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7. Ch09b - Fluids-Revised: 3/6/2010
A container of gas is connected to one end of the U-tube
If there is a pressure difference between the gas and the atmosphere, a force will be exerted on the fluid in the U-tube. This changes the equilibrium position of the fluid in the tube.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

8. Ch09b - Fluids-Revised: 3/6/2010
From the figure:
At point C
Also
The pressure at point B is the pressure of the gas.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

9. Ch09b - Fluids-Revised: 3/6/2010
A Barometer
The atmosphere pushes on the container of mercury which forces mercury up the closed, inverted tube. The distance d is called the barometric pressure.
MFMcGraw
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10. Ch09b - Fluids-Revised: 3/6/2010
From the figure
and
Atmospheric pressure is equivalent to a column of mercury 76.0 cm tall.
MFMcGraw
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11. Ch09b - Fluids-Revised: 3/6/2010
Pascal’s Principle
A change in pressure at any point in a confined fluid is transmitted everywhere throughout the fluid. (This is useful in making a hydraulic lift.)
MFMcGraw
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12. Ch09b - Fluids-Revised: 3/6/2010
The applied force is transmitted to the piston of cross-sectional area A2 here.
Apply a force F1 here to a piston of cross-sectional area A1.
MFMcGraw
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13. Ch09b - Fluids-Revised: 3/6/2010
Mathematically,
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

14. Ch09b - Fluids-Revised: 3/6/2010
Example: Assume that a force of 500 N (about 110 lbs) is applied to the smaller piston in the previous figure. For each case, compute the force on the larger piston if the ratio of the piston areas (A2/A1) are 1, 10, and 100.
Using Pascal’s Principle:
50,000 N
100
5000 N 10
500 N 1 F2
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

15. Gravity’s Effect on Fluid Pressure
FBD for the fluid cylinder
A cylinder of fluid x y
P1A
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 9, Questions 4 and 5. w
P2A
MFMcGraw
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16. Ch09b - Fluids-Revised: 3/6/2010
Apply Newton’s 2nd Law to the fluid cylinder:
If P1 (the pressure at the top of the cylinder) is known, then the above expression can be used to find the variation of pressure with depth in a fluid.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

17. Ch09b - Fluids-Revised: 3/6/2010
If the top of the fluid column is placed at the surface of the fluid, then P1 = Patm if the container is open.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

18. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.19): At the surface of a freshwater lake, the pressure is 105 kPa. (a) What is the pressure increase in going 35.0 m below the surface?
Part b of this problem is not included.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

19. Ch09b - Fluids-Revised: 3/6/2010
Example: The surface pressure on the planet Venus is 95 atm. How far below the surface of the ocean on Earth do you need to be to experience the same pressure? The density of seawater is 1025 kg/m3.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

20. Archimedes’ Principlew F2 F1 x y
An FBD for an object floating submerged in a fluid.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 9, Questions 6, 7, 14, 18, and 20.
The total force on the block due to the fluid is called the buoyant force.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

21. Ch09b - Fluids-Revised: 3/6/2010
The magnitude of the buoyant force is:
From before:
The result is
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

22. Ch09b - Fluids-Revised: 3/6/2010
Archimedes’ Principle: A fluid exerts an upward buoyant force on a submerged object equal in magnitude to the weight of the volume of fluid displaced by the object.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

23. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.28): A flat-bottomed barge loaded with coal has a mass of 3.0105 kg. The barge is 20.0 m long and 10.0 m wide. It floats in fresh water. What is the depth of the barge below the waterline? x y w FB
Apply Newton’s 2nd Law to the barge:
FBD for the barge
Symbols defined: mw is the mass of water displaced, Vw is the volume of water displaced by the barge, w is the density of fresh water, and mb is the mass of the barge.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

24. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.40): A piece of metal is released under water. The volume of the metal is 50.0 cm3 and its specific gravity is 5.0. What is its initial acceleration? (Note: when v = 0, there is no drag force.) x y w FB
Apply Newton’s 2nd Law to the piece of metal:
FBD for the metal
The magnitude of the buoyant force equals the weight of the fluid displaced by the metal.
Solve for a:
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

25. Ch09b - Fluids-Revised: 3/6/2010
Example continued:
Since the object is completely submerged V=Vobject.
where water = 1000 kg/m3 is the density of water at 4 °C.
Given
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

26. Ch09b - Fluids-Revised: 3/6/2010
Fluid Flow
A moving fluid will exert forces parallel to the surface over which it moves, unlike a static fluid. This gives rise to a viscous force that impedes the forward motion of the fluid.
A steady flow is one where the velocity at a given point in a fluid is constant.
V1 = constant
V2 = constant
v1v2
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 9, Questions 8 and 11.
MFMcGraw
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27. Ch09b - Fluids-Revised: 3/6/2010
Steady flow is laminar; the fluid flows in layers. The path that the fluid in these layers takes is called a streamline. Streamlines do not cross.
An ideal fluid is incompressible, undergoes laminar flow, and has no viscosity.
MFMcGraw
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28. Ch09b - Fluids-Revised: 3/6/2010
The continuity equation—Conservation of mass.
The amount of mass that flows though the cross-sectional area A1 is the same as the mass that flows through cross-sectional area A2.
MFMcGraw
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29. Ch09b - Fluids-Revised: 3/6/2010
is the mass flow rate (units kg/s)
is the volume flow rate (units m3/s)
The continuity equation is
If the fluid is incompressible, then 1= 2.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

30. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.41): A garden hose of inner radius 1.0 cm carries water at 2.0 m/s. The nozzle at the end has radius 0.20 cm. How fast does the water move through the constriction?
MFMcGraw
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31. Ch09b - Fluids-Revised: 3/6/2010
Bernoulli’s Equation
Bernoulli’s equation is a statement of energy conservation.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 9, Questions 9, 10, and 19.
MFMcGraw
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32. Ch09b - Fluids-Revised: 3/6/2010
Work per unit volume done by the fluid
Potential energy per unit volume
Points 1 and 2 must be on the same streamline
Kinetic energy per unit volume
MFMcGraw
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33. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.49): A nozzle is connected to a horizontal hose. The nozzle shoots out water moving at 25 m/s. What is the gauge pressure of the water in the hose? Neglect viscosity and assume that the diameter of the nozzle is much smaller than the inner diameter of the hose.
Let point 1 be inside the hose and point 2 be outside the nozzle.
The hose is horizontal so y1 = y2. Also P2 = Patm.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

34. Ch09b - Fluids-Revised: 3/6/2010
Example continued:
Substituting:
v2 = 25 m/s and v1 is unknown. Use the continuity equation.
Since d2<<d1 it is true that v1<<v2.
MFMcGraw
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35. Ch09b - Fluids-Revised: 3/6/2010
Example continued:
MFMcGraw
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36. Ch09b - Fluids-Revised: 3/6/2010
Viscosity
A real fluid has viscosity (fluid friction). This implies a pressure difference needs to be maintained across the ends of a pipe for fluid to flow.
MFMcGraw
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37. Ch09b - Fluids-Revised: 3/6/2010
Viscosity also causes the existence of a velocity gradient across a pipe. A fluid flows more rapidly in the center of the pipe and more slowly closer to the walls of the pipe.
The volume flow rate for laminar flow of a viscous fluid is given by Poiseuille’s Law.
where  is the viscosity
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

38. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.55): A hypodermic syringe attached to a needle has an internal radius of mm and a length of 3.00 cm. The needle is filled with a solution of viscosity 2.0010-3 Pa sec; it is injected into a vein at a gauge pressure of 16.0 mm Hg. Neglect the extra pressure required to accelerate the fluid from the syringe into the entrance needle.
(a) What must the pressure of the fluid in the syringe be in order to inject the solution at a rate of mL/sec?
Solve Poiseuille’s Law for the pressure difference:
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

39. Ch09b - Fluids-Revised: 3/6/2010
Example continued:
This pressure difference is between the fluid in the syringe and the fluid in the vein. The pressure in the syringe is
Ps is the pressure in the syringe and Pv is the pressure in the vein.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

40. Ch09b - Fluids-Revised: 3/6/2010
Example continued:
(b) What force must be applied to the plunger, which has an area of 1.00 cm2?
The result of (a) gives the force per unit area on the plunger so the force is just F = PA = N.
MFMcGraw
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41. Ch09b - Fluids-Revised: 3/6/2010
Viscous Drag
The viscous drag force on a sphere is given by Stokes’ law.
Where  is the viscosity of the fluid, r is the radius of the sphere, and v is the velocity of the sphere.
MFMcGraw
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42. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.62): A sphere of radius 1.0 cm is dropped into a glass cylinder filled with a viscous liquid. The mass of the sphere is 12.0 g and the density of the liquid is 1200 kg/m3. The sphere reaches a terminal speed of 0.15 m/s. What is the viscosity of the liquid? FD w x y FB
Apply Newton’s Second Law to the sphere
FBD for sphere
FD is the drag force as given by Stokes’ law and FB is the buoyant force as given by Archimedes principle.
MFMcGraw
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43. Ch09b - Fluids-Revised: 3/6/2010
Example continued:
When v = vterminal, a = 0 and
Solving for 
Symbol definitions: vt is the terminal velocity, ml is the mass of liquid displaced by the sphere, Vl is the volume of liquid displaced by the sphere and is equal to the volume of the sphere Vs, ms is the mass of the sphere, and l is the density of the liquid.
MFMcGraw
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44. Ch09b - Fluids-Revised: 3/6/2010
Surface Tension
The surface of a fluid acts like a a stretched membrane (imagine standing on a trampoline). There is a force along the surface of the fluid.
The surface tension is a force per unit length.
Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site ( Instructor Resources: CPS by eInstruction, Chapter 9, Question 16.
MFMcGraw
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45. Ch09b - Fluids-Revised: 3/6/2010
Example (text problem 9.70): Assume a water strider has a roughly circular foot of radius 0.02 mm. The water strider has 6 legs.
(a) What is the maximum possible upward force on the foot due to the surface tension of the water?
The water strider will be able to walk on water if the net upward force exerted by the water equals the weight of the insect. The upward force is supplied by the water’s surface tension.
From the text =0.070 N/m.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

46. Ch09b - Fluids-Revised: 3/6/2010
Example continued:
(b) What is the maximum mass of this water strider so that it can keep from breaking through the water surface?
To be in equilibrium, each leg must support one-sixth the weight of the insect.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

47. Ch09b - Fluids-Revised: 3/6/2010
Summary
Pressure and its Variation with Depth
Pascal’s Principle
Archimedes Principle
Continuity Equation (conservation of mass)
Bernoulli’s Equation (conservation of energy)
Viscosity and Viscous Drag
Surface Tension
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

48. Quick Questions

49. Ch09b - Fluids-Revised: 3/6/2010
Consider a boat loaded with scrap iron in a swimming pool. If the iron is thrown overboard into the pool, will the water level at the edge of the pool rise, fall, or remain unchanged?
1. Rise 2. Fall 3. Remain unchanged
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

50. 1. Rise 2. Fall 3. Remain unchanged
Consider a boat loaded with scrap iron in a swimming pool. If the iron is thrown overboard into the pool, will the water level at the edge of the pool rise, fall, or remain unchanged?
1. Rise 2. Fall 3. Remain unchanged
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

51. Ch09b - Fluids-Revised: 3/6/2010
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

52. Ch09b - Fluids-Revised: 3/6/2010
In the presence of air, the small iron ball and large plastic ball balance each other. When air is evacuated from the container, the larger ball
1. rises. 2. falls. 3. remains in place.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

53. Ch09b - Fluids-Revised: 3/6/2010
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

54. Ch09b - Fluids-Revised: 3/6/2010
The weight of the stand and suspended solid iron ball is equal to the weight of the container of water as shown above. When the ball is lowered into the water the balance is upset. The amount of weight that must be added to the left side to restore balance, compared to the weight of water displaced by the ball, would be
1. half. 2. the same. 3. twice. 4. more than twice.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

55. Ch09b - Fluids-Revised: 3/6/2010
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

56. Ch09b - Fluids-Revised: 3/6/2010
A pair of identical balloons are inflated with air and suspended on the ends of a stick that is horizontally balanced. When the balloon on the left is punctured, the balance of the stick is
1. upset and the stick rotates clockwise. 2. upset and the stick rotates counter-clockwise. 3. unchanged.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

57. Ch09b - Fluids-Revised: 3/6/2010
MFMcGraw
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58. Ch09b - Fluids-Revised: 3/6/2010
Consider an air-filled balloon weighted so that it is on the verge of sinking—that is, its overall density just equals that of water. Now if you push it beneath the surface, it will
1. sink. 2. return to the surface. 3. stay at the depth to which it is pushed.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

59. Ch09b - Fluids-Revised: 3/6/2010
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

60. Extras

61. The density of the block of wood floating in water is
1. greater than the density of water. 2. equal to the density of water. 3. less than half that of water. 4. more than half the density of water. 5. … not enough information is given.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

62. Ch09b - Fluids-Revised: 3/6/2010
The density of the block of wood floating in water is
1. greater than the density of water. 2. equal to the density of water. 3. less than half that of water. 4. more than half the density of water. 5. … not enough information is given.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

63. 1. rises. 2. falls. 3. remains in place.
In the presence of air, the small iron ball and large plastic ball balance each other. When air is evacuated from the container, the larger ball
1. rises. 2. falls. 3. remains in place.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

64. 1. half. 2. the same. 3. twice. 4. more than twice.
The weight of the stand and suspended solid iron ball is equal to the weight of the container of water as shown above. When the ball is lowered into the water the balance is upset. The amount of weight that must be added to the left side to restore balance, compared to the weight of water displaced by the ball, would be
1. half. 2. the same. 3. twice. 4. more than twice.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

65. Ch09b - Fluids-Revised: 3/6/2010
A pair of identical balloons are inflated with air and suspended on the ends of a stick that is horizontally balanced. When the balloon on the left is punctured, the balance of the stick is
1. upset and the stick rotates clockwise. 2. upset and the stick rotates counter-clockwise. 3. unchanged.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010

66. Ch09b - Fluids-Revised: 3/6/2010
Consider an air-filled balloon weighted so that it is on the verge of sinking—that is, its overall density just equals that of water. Now if you push it beneath the surface, it will
1. sink. 2. return to the surface. 3. stay at the depth to which it is pushed.
MFMcGraw
Ch09b - Fluids-Revised: 3/6/2010