Understanding Fluids in Physics: Density, Pressure, and Behaviors

1. Physics 350 Chapter 10 Fluids

2. Fluids A fluidisa gas or a liquid. A gas expands to fill any container A liquid (at fixed pressure and temperature), has a fixed volume, but deforms to the shape of its container.

3. The atoms in a liquid are closely packed while those in a gas are separated by much larger distances. Gas have a density ~ 1/1000 x liquid density

4. Density and Pressure • The density of a substance of uniform composition is defined as its mass per unit volume: • Units are kg/m3 (SI) or g/cm3 (cgs) • 1 g/cm3 = 1000 kg/m3

5. Density, cont. • The densities of most liquids and solids vary slightly with changes in temperature and pressure • Densities of gases vary greatly with changes in temperature and pressure

6. Density • Density = Mass/Volume •  = M/V • SI unit: [kg/m3] • Densities of some common things (kg/m3) • Water 1000 • ice 917 (floats on water) • blood 1060 (sinks in water) • lead 11,300 • Copper 8890 • Mercury 13,600 • Aluminum 2700 • Wood 550 • air 1.29 • Helium 0.18

7. Specific Gravity • The specific gravity of a substance is the ratio of its density to the density of water at 4° C • The density of water at 4° C is 1000 kg/m3 • Specific gravity is a unitless ratio

8. A1 F1 F2 A2 Pressure Pressure P is the amount of force F per unit area A: By the Action-Reaction principle, Pressure is the inward force per unit area that the container exerts on the fluid. Pressure is the outward force per unit area that the fluid exerts on its container.

9. Pressure The force exerted by a fluid on a submerged object at any point if perpendicular to the surface of the object

10. A woman’s high heels sink into the soft ground, but the larger shoes of the much bigger man do not. Pressure = force/area

11. The pressure exerted on the piston extends uniformly throughout the fluid, causing it to push outward with equal force per unit area on the walls and bottom of the cylinder.

12. Measuring Pressure The spring is calibrated by a known force The force the fluid exerts on the piston is then measured

13. ConcepTest 10.3 On a Frozen Lake You are walking out on a frozen lake and you begin to hear the ice cracking beneath you. What is your best strategy for getting off the ice safely? 1) stand absolutely still and don’t move a muscle 2) jump up and down to lessen your contact time with the ice 3) try to leap in one bound to the bank of the lake 4) shuffle your feet (without lifting them) to move towards shore 5) lie down flat on the ice and crawl toward shore

14. ConcepTest 10.3 On a Frozen Lake You are walking out on a frozen lake and you begin to hear the ice cracking beneath you. What is your best strategy for getting off the ice safely? 1) stand absolutely still and don’t move a muscle 2) jump up and down to lessen your contact time with the ice 3) try to leap in one bound to the bank of the lake 4) shuffle your feet (without lifting them) to move towards shore 5) lie down flat on the ice and crawl toward shore As long as you are on the ice, your weight is pushing down. What is important is not the net force on the ice, but the force exerted on a given small area of ice (i.e., the pressure!). By lying down flat, you distribute your weight over the widest possible area, thus reducing the force per unit area.

15. Atmospheric Pressure Atmospheric pressure comes from the weight of the column of air above us. At sea level, atmospheric pressure is: Pat = 1.01  105 N/m2 = 1.01  105Pa1 Pascal= 1 N/m2 = 14.7 lb/in2(psi) = 1 bar (tire pressure gauges in Europe read 1, 2,..bar) Hurricane Rita 2005: P = 882 millibar = 0.882 bar F=Mg F=PA

16. Pressure examples • Estimate the force of the atmosphere on the top of your head. • A = (10cm)(15cm)=0.015m2 • F=PA = [1.01  105 N/m2 ][0.015 m2] = 1.5 kN • A = (4in)(6in)=24 in2 • F=PA = [15 lb/in2][24in2] = 360 lb. • Is atmospheric pressure on top of a mountain greater or less than at sea level? • Less. At higher altitude, there is less mass above.

17. Pressure • Example

18. Variation of Pressure with Depth • If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium • All points at the same depth must be at the same pressure • Otherwise, the fluid would not be in equilibrium • The fluid would flow from the higher pressure region to the lower pressure region

19. Pressure and Depth Examine the darker region, assumed to be a fluid It has a cross-sectional area A Extends to a depth h below the surface Three external forces act on the region

20. Pressure and Depth equation P = Po + ρgh Po is normal atmospheric pressure 1.013 x 105 Pa = 14.7 lb/in2 = 1 atm The pressure does not depend upon the shape of the container

21. Pressure in a Fluid Pressure in a fluid depends only on the depth h below the surface. P = Pat + rghr = density of fluid Weight/Area of fluid Weight/Area of atmosphere above fluid IFthe density of the fluid is constant and it has atmospheric pressure (Pat) at its surface. Mass of fluid above depth h is (density)(volume) = rhA Force of gravity on fluid above depth h: W=rghA

22. Pressure under water To what depth in water must you dive to double the pressure exerted on your body? P = Pat + rgh rgh = Pat , h= Pat /rg Start to feel strong pressure at 3m

23. Pressure variation in fluid The variation in pressure at two different depths is given by: P2 = P1 + rgh

24. p1=0 h p2=patm Pressure and DepthBarometer: a way to measure atmospheric pressure • p2 = p1 + gh • patm = gh • Measure h, determine patm • example--Mercury •  = 13,600 kg/m3 • patm = 1.05 x 105 Pa •  h = 0.757 m = 757 mm = 29.80” (for 1 atm)

25. Pressure Measurements • Absolute vs. Gauge Pressure • The pressure P is called the absolute pressure • Remember, P = Po + rgh • P – Po = rgh is the gauge pressure

26. Pressure Measurements:Manometer One end of the U-shaped tube is open to the atmosphere The other end is connected to the pressure to be measured Pressure at B is Po+ρgh

27. Pressure Values in Various Units • One atmosphere of pressure is defined as the pressure equivalent to a column of mercury exactly 0.76 m tall at 0o C where g = 9.806 65 m/s2 • One atmosphere (1 atm) = • 76.0 cm of mercury (760mm = 1 torr) • 1.013 x 105 Pa • 14.7 lb/in2

28. Pressure Example:

29. Pascal’s Principle • A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. • First recognized by Blaise Pascal, a French scientist (1623 – 1662)

30. Pascal’s Principle, cont The hydraulic press is an important application of Pascal’s Principle Also used in hydraulic brakes, forklifts, car lifts, etc.

31. A small force F1 applied to a piston with a small area produces a much larger force F2 on the larger piston. This allows a hydraulic jack to lift heavy objects.

32. Pascal’s Principle, Force • A external pressure P applied to any area of a fluid is transmitted unchanged to all points in or on the fluid. • This is just an application of the Action-Reaction principle. • Hydraulic Lift A Force F1 is applied to area A1, displacing the fluid by a distance d1. The pressure increase in the fluid is P=F1/A1. The Pressure F1/A1 creates a force on the car F2= A2 (F1/A1) = F1 (A2 /A1). A small force acting on a small area creates a big force acting over a large area!

33. Archimedes’ Principle: The buoyant force acting on an object fully or partially submerged in a fluid is equal to the weight of the fluid displaced by the object.

34. The weight of a column of water is proportional to the volume of the column. The volume V is equal to the area A times the height h. Equilibrium…

35. Buoyant Force The upward force is called the buoyant force The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object

36. Buoyant Force, cont. The magnitude of the buoyant force always equals the weight of the displaced fluid The buoyant force is the same for a totally submerged object of any size, shape, or density

37. Buoyant Force, final The buoyant force is exerted by the fluid Whether an object sinks or floats depends on the relationship between the buoyant force and the weight

38. Archimedes’ Principle:Totally Submerged Object The upward buoyant force is B=ρfluidgVobj The downward gravitational force is w=mg=ρobjgVobj The net force is B-w=(ρfluid-ρobj)gVobj

39. Totally Submerged Object The object is less dense than the fluid The object experiences a net upward force

40. Totally Submerged Object, 2 The object is more dense than the fluid The net force is downward The object accelerates downward Question: How do steel ships float if steel is roughly 6 times more dense than water?

41. Archimedes’ Principle:Floating Object The object is in static equilibrium The upward buoyant force is balanced by the downward force of gravity Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level

42. Archimedes’ Principle:Floating Object, cont The forces balance

43. Archimedes’s Principle Archimedes’ Principle: The buoyant force on an object equals the weight of the fluid it displaces. Weight of water displaced = Buoyant force = Weight of ice When ice melts it will turn into water of same volume Suppose you float a large ice-cube in a glass of water, and that after you place the ice in the glass the level of the water is at the very brim. When the ice melts, the level of the water in the glass will: 1. Go up causing the water to spill. 2. Go down. 3. Stay the same.

44. Example 9.9 A raft is constructed of wood having a density of 6.00 x 102 kg/m3. Its surface area is 5.70m2, and volume is 0.60m3. When the raft is placed in fresh water, what depth h is the bottom of the raft submerged?

45. CORRECT Tub of water Overflowed water Tub of water + ship Concept Question Which weighs more: 1. A large bathtub filled to the brim with water. 2. A large bathtub filled to the brim with water with a battle-ship floating in it. 3. They will weigh the same. Weight of ship = Buoyant force = Weight of displaced water

46. Fluids in Motion • Streamline flow • Every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier • Also called laminar flow • Streamline is the path • Different streamlines cannot cross each other • The streamline at any point coincides with the direction of fluid velocity at that point

47. Streamline Flow, Example Streamline flow shown around an auto in a wind tunnel

48. Fluids in Motion:Turbulent Flow • The flow becomes irregular • exceeds a certain velocity • any condition that causes abrupt changes in velocity • Eddy currents are a characteristic of turbulent flow

49. Turbulent Flow, Example The rotating blade (dark area) forms a vortex in heated air The wick of the burner is at the bottom Turbulent air flow occurs on both sides of the blade

50. Fluid Flow: Viscosity • Viscosity is the degree of internal friction in the fluid • Measure of a fluid's ability to resist gradual deformation by shear or tensile stresses • The internal friction is associated with the resistance between two adjacent layers of the fluid moving relative to each other