Mastering Fluid Mechanics: Density, Pressure, Buoyancy Explained

1. Chapter 5: FLUID MECHANICS

2. This lecture will help you understand: • Density • Pressure • Buoyancy in a Liquid • Archimedes’ Principle • Pressure in a Gas • Atmospheric Pressure • Pascal’s Principle • Buoyancy in a Gas • Bernoulli’s Principle

3. Density Density • Importantproperty of materials (solids, liquids, gases) • Measure of compactness of how much mass an object occupies • “lightness” or “heaviness” of materials of the same size

4. Density • Equation : • Units of: • mass in grams or kilograms • volume in cm3 or m3 • density in kg/m3 or g/cm3 Example: The density of mercury is 13.6 g/cm3, so mercury has 13.6 times as much mass as an equal volume of water (density 1 g/cm3).

5. Density Weight density • in equation form: often expressed in pounds per cubic foot example:density of salt water is 64 lb/ft3, more dense than fresh water (density 62.4 lb/ft3)

6. Density CHECK YOUR NEIGHBOR Which of these has the greatest density? A. 100 kg of lead • 100 kg of water • Both are the same • None of the above

7. Density CHECK YOUR ANSWER Which of these has the greatest density? A. 100 kg of lead • 100 kg of water • Both are the same • None of the above Explanation: They have the same mass and weight, but different volumes. Any amount of lead is more dense than any amount of water.

8. Pressure • force per unit area that one object exerts on another • equation: • depends on area over which force is distributed • units in lb/ft2, N/m2, or Pa (Pascals)

9. Pressure Example:The teacher between beds of nails is unharmed because the applied force is spread over many nails. Combined surface area of the nails results in a tolerable pressure that does not puncture the skin.

10. Pressure in a Liquid • Force per unit area that a liquid exerts on something • Depth dependent and not volume dependent Example:Swim twice as deep and the pressure due to the weight of water above you is twice as much. (For total pressure, add to this the atmospheric pressure acting on the water surface.)

11. Pressure in a Liquid Effects of water pressure • acts perpendicular to surfaces of a container • liquid spurts at right angles from a hole in the surface curving downward • The greater the depth, the greater the exiting speed

12. Pressure in a Liquid • Acts equally in all directions Examples: • your ears feel the same amount of pressure under water no matter how you tip your head • bottom of a boat is pushed upward by water pressure • pressure acts upward when pushing a beach ball under water

13. Pressure in a Liquid • Independent of shape of container whatever the shape of a container, pressure at any particular depth is the same • Equation:

14. Force of gravity acting on the water in a tall tower produces pressure in pipes below that supply many homes with reliable water pressure. Water Tower

15. Pressure CHECK YOUR NEIGHBOR Suppose water from a tall tower supplies a nearby home. If water faucets upstairs and downstairs are turned fully on, will more water per second flow from the downstairs or the upstairs faucet? Or will water flow in each be the same? • A. Downstairs. • Upstairs. • Same. • Not enough information in problem.

16. Pressure CHECK YOUR ANSWER Suppose water from a tall tower supplies a nearby home. If water faucets upstairs and downstairs are turned fully on, will more water per second flow from the downstairs or the upstairs faucet? Or will water flow in each be the same? • A. Downstairs • Upstairs • Same • Not enough information in problem. • Explanation: • Water pressure depends on the depth below the free surface. Downstairs faucets are simply “deeper” and receive greater pressure, which means greater rate of water flow.

17. Pressure CHECK YOUR NEIGHBOR Does a 3-meter deep lake or a 6-meter deep small pond exert more pressure on a dam? A. The three-meter deep lake. • The six-meter deep small pond. • Same amount of pressure is exerted (atmospheric) so same force. • Not enough information given in the question.

18. Pressure CHECK YOUR ANSWER Does a 3-meter deep lake or a 6-meter deep small pond exert more pressure on a dam? A. The three-meter deep lake. • The six-meter deep small pond. • Same amount of pressure is exerted (atmospheric) so same force. • Not enough information given in the question.

19. Buoyancy in a Liquid Buoyancy • apparent loss of weight of a submerged object • amount equals the weight of water displaced

20. Archimedes’ Principle Archimedes’ Principle • discovered by Greek scientist Archimedes • relates buoyancy to displaced liquid • states that an immersed body (completely or partially) is buoyed up by a force equal to the weight of the fluid it displaces • applies to gases and liquids

21. Archimedes’ Principle Apparent weight of a submerged object • weight out of water – buoyant force Example:if a 3-kg block submerged in water apparently “weighs” 1 kg, then the buoyant force or weight of water displaced is 2 kg (BF = wt out of water – apparent wt = 3 kg – 1 kg = 2 kg)

22. Archimedes’ Principle • Displacement rule: A completely submerged object always displaces a volume of liquid equal to its own volume. Example: Place a stone in a container that is brim-full of water, and the amount of water overflow equals the volume of the stone

23. Buoyant force is equal to the weight of fluid displaced. It can also be understood by pressure differences. The greater pressure against the bottom of the box, minus the pressure on the top, results in an upward force—the buoyant force. Archimedes’ Principle

24. Archimedes’ Principle Buoyant Force • Buoyant force is equal to the weight of fluid displaced. • Understood by pressure differences greater pressure against the box – pressure on the top of box

25. Archimedes’ Principle CHECK YOUR NEIGHBOR On which of these blocks submerged in water is the buoyant force greatest? A. 1 kg of lead. • 1 kg of aluminum. • 1 kg of uranium. • All the same.

26. Archimedes’ Principle CHECK YOUR ANSWER On which of these blocks submerged in water is the buoyant force greatest? A. 1 kg of lead. • 1 kg of aluminum. • 1 kg of uranium. • All the same. Explanation: The largest block is the aluminum one. It displaces more water and therefore experiences the greatest buoyant force.

27. Archimedes’ Principle Flotation • Principle of flotation • A floating object displaces a weight of fluid equal to its own weight Example: A solid iron 1-ton block may displace 1/8 ton of water and sink. The same 1 ton of iron in a bowl shape displaces a greater volume of water—the greater buoyant force allows it to float

28. Archimedes’ Principle CHECK YOUR NEIGHBOR The reason a person finds it easier to float in salt water, compared with fresh water, is that in salt water A. the buoyant force is greater. • a person feels less heavy. • a smaller volume of water is displaced. • None of the above.

29. Archimedes’ Principle CHECK YOUR ANSWER The reason a person finds it easier to float in salt water, compared with fresh water, is that in salt water A. the buoyant force is greater. • a person feels less heavy. • a smaller volume of water is displaced. • None of the above. Explanation: A floating person has the same buoyant force whatever the density of water. A person floats higher because a smaller volume of the denser salt water is displaced.

30. Archimedes’ Principle CHECK YOUR NEIGHBOR On a boat ride, the skipper gives you a life preserver filled with lead pellets. When he sees the skeptical look on your face, he says that you’ll experience a greater buoyant force if you fall overboard than your friends who wear Styrofoam-filled preservers. A. He apparently doesn’t know his physics. • He is correct.

31. Archimedes’ Principle CHECK YOUR ANSWER On a boat ride, the skipper gives you a life preserver filled with lead pellets. When he sees the skeptical look on your face, he says that you’ll experience a greater buoyant force if you fall overboard than your friends who wear Styrofoam-filled preservers. A. He apparently doesn’t know his physics. • He is correct. Explanation: He’s correct, but what he doesn’t tell you is you’ll drown! Your life preserver will submerge and displace more water than those of your friends who float at the surface. Although the buoyant force on you will be greater, the net force downward is greater still!

32. Gas pressure is a measure of the amount of force per area that a gas exerts against containing walls. Here the force is exerted by the motion of molecules bouncing around. Temperature is a measure of the KE per molecules of the gas. Pressure in a Gas

33. Pressure in a Gas Relationship between pressure and density • Gas pressure is proportional to density Example: • Air pressure and air density inside an inflated tire are greater than the atmospheric pressure and density outside • Twice as many molecules in the same volume  air density doubled • For molecules moving at the same speed (same temperature), collisions are doubled  pressure doubled

34. Pressure in a Gas Double density of air by • Doubling the amount of air • Decreasing the volume to half

35. Pressure in a Gas Boyle’s Law • Relationship between pressure and volume for ideal gases • An ideal gas is one in which intermolecular forces play no role • States that pressure volume is a constant for a given mass of confined gas regardless of changes in pressure or volume (with temperature remaining unchanged) • pressure volume = constant means that P1V1 = P2V2

36. Pressure in a Gas CHECK YOUR NEIGHBOR When you squeeze a party balloon to 0.8 its volume, the pressure in the balloon A. is 0.8 its former pressure. • remains the same if you squeeze it slowly. • is 1.25 times greater. • is 8 times greater.

37. Pressure in a Gas CHECK YOUR ANSWER When you squeeze a party balloon to 0.8 its volume, the pressure in the balloon A. is 0.8 its former pressure. • remains the same if you squeeze it slowly. • is 1.25 times greater. • is 8 times greater. Explanation: Boyle’s law, sweet and simple: P(1.0 V) = 1.25 P(0.8 V).

38. Earth’s Atmosphere Atmosphere • ocean of air • exerts pressure The Magdeburg-hemispheres demonstration in 1654 by Otto von Guericke showed the large magnitude of atmosphere’s pressure.

39. Atmospheric Pressure Atmospheric pressure • Caused by weight of air • Varies from one locality to another • Not uniform • Measurements are used to predict weather conditions

40. Atmospheric Pressure • Pressure exerted against bodies immersed in the atmosphere result from the weight of air pressing from above • At sea level is 101 kilopascals(101 kPa) • Weight of air pressing down on 1 m2 at sea level ~ 100,000 N, so atmosphericpressure is ~ 105 N/m2

41. Atmospheric Pressure • Pressure at the bottom of a column of air reaching to the top of the atmosphere is the same as the pressure at the bottom of a column of water 10.3 m high. • Consequence: the highest the atmosphere can push water up into a vacuum pump is 10.3 m • Mechanical pumps that don’t depend on atmospheric pressure don’t have the 10.3-m limit

42. When the piston is lifted, the intake valve opens and air moves in to fill the empty space. When the piston is moved downward, the outlet valve opens and the air is pushed out. Mechanical Pump

43. Barometers Barometer • Device to measure atmospheric pressure • Also determines elevation Aneroid barometer • Small portable instrument that measures atmospheric pressure • Calibrated for altitude, then an altimeter

44. Atmospheric Pressure CHECK YOUR NEIGHBOR Atmospheric pressure is caused by the A. density of Earth’s atmosphere. • weight of Earth’s atmosphere. • temperature of the atmosphere. • effect of the Sun’s energy on the atmosphere.

45. Atmospheric Pressure CHECK YOUR ANSWER Atmospheric pressure is caused by the A. density of Earth’s atmosphere. • weight of Earth’s atmosphere. • temperature of the atmosphere. • effect of the Sun’s energy on the atmosphere.

46. Atmospheric Pressure CHECK YOUR NEIGHBOR Two people are drinking soda using straws. Do they suck the soda up? Could they drink a soda this way on the Moon? A. Yes and yes. • No, they suck the air out and the atmospheric pressure pushes the soda up. Yes, they could do the same thing on the Moon. • No, they reduce air pressure in the straw and the atmospheric pressure pushes the soda up. No, they could not do the same thing on the Moon. • Yes. No, they could not do the same thing on the Moon.

47. Atmospheric Pressure CHECK YOUR ANSWER Two people are drinking soda using straws. Do they suck the soda up? Could they drink a soda this way on the moon? A. Yes and yes. • No, they suck the air out and the atmospheric pressure pushes the soda up. Yes, they could do the same thing on the Moon. • No, they reduce air pressure in the straw and the atmospheric pressure pushes the soda up. No, they could not do the same thing on the Moon. • Yes. No, they could not do the same thing on the Moon. The Moon does not have an atmosphere.

48. Pascal’s Principle Pascal’s principle • Discovered by Blaise Pascal, a scientist and theologian in the 17th century • States that a change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid • Applies to all fluids—gasesand liquids

49. Pascal’s Principle • Application in hydraulic press Example: • Pressure applied to the left piston is transmitted to the right piston • A 10-kg load on small piston (left) lifts a load of 500 kg on large piston (right)

50. Pascal’s Principle CHECK YOUR NEIGHBOR A 10-kg load on the left piston will support a 500-kg load on the right piston. How does the pressure of fluid against the lower part of the left piston compare with the pressure against the lower right piston? A. More pressure on the left piston. • More pressure on the right piston. • Same pressure on each. • Same force on each.