Chapter 15: Fluids. Volume Density Pressure Buoyancy Floating Archimedes Principle Pascal s Principle Fluid Flow Bernoulli s Principle.

Transcription

1 Chapter 15: Fluids Statics Dynamics Volume Density Pressure Buoyancy Floating Archimedes Principle Pascal s Principle Fluid Flow Bernoulli s Principle

2 States of Matter Solid Liquid Gas What is the most common state of matter in the Universe?

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4 Fluids: Liquids & Gases Fluids are substances that are free to flow. Atoms and molecules are free to move. They take the shape of their containers. Cannot withstand or exert shearing forces. Liquids: Incompressible (density constant) Gases: Compressible (density depends on pressure) Parameters to describe Fluids: Density: = mass/volume Pressure: P = Force/Area [P] = N/m 2 = 1 Pascal (Pa)

5 Liquid Units There are 1000 liters in 1 cubic meter! 1 liter = 10-3 m 3 = 10 3 cm 3 1 liter of water has a mass of 1 kg and a weight of 9.8N. 1kg 1000kg H2 0 liter m 3

6 Density Density of C: water = 1g/cm 3 = 1000 kg/m 3 = 1kg/liter Density of 0 C: m V Air = 1.29x10-3 g/cm 3 = 1.29 kg/m 3 Density depends on temperature! Most substances EXPAND upon heating. How does that change their densities? m V m V REDUCES DENSITY!

7 Water: The Exception C: water =1000 kg/m 3 0 C: ice = 917 kg/m 3

8 Pressure in a fluid is due to the weight P Force Area of a fluid. P mg ( V) g A A ( Ah) g A gh Pressure depends on Depth!

9 Pressure Acts ONLY Perpendicularly to the Surface Pressure depends on depth.

10 Pressure IN a Fluid Is due to the weight of the fluid above you Depends on Depth and Density Only Does NOT depend on how much water is present Acts perpendicular to surfaces (no shearing) Pressure s add At a particular depth, pressure is exerted equally in ALL directions including sideways (empirical fact)

11 Pressure Problem What is the water pressure 15 m below the surface of the lake? Assume it is pure water. P gh 1000 kg / m 3 (9.8 m/ s 2 )15m x10 N / m 147kPa 1 Pascal 1 N/ m 2

12 Pressure Units What is the water pressure 150 m below the surface of the lake? Assume it is pure water. P gh 1000 kg / m 3 (9.8 m/ s 2 )150 m 1.47 N/ m 2

13 Pressure ON a Fluid Liquids cannot be compressed to a smaller volume. Liquids are incompressible. Gases can be compressed to a smaller volume. Gases are compressible.

14 The Atmosphere At sea level, the atmosphere has a density of about 1.29 kg/m 3. The average density up to 120 km is about 8.59 x10-2 kg/m 3.

15 The Atmosphere A square meter extending up through the atmosphere has a mass of about 10,000 kg and a weight of about 100,000 N. 1 N/m 2 is a Pascal atm x Pa psi

16 Measuring Pressure 5 1atm 1.013x10 Pa Why is the pressure at X equal to atmospheric pressure? P gh Because if it didn t, the mercury would be pushed out of the dish! P mercury gh h h P mercury g 2 101,300 N/ m 13, 600 kg / m x9.8 m / s 3 2 mercury water 13.6 water 1000 kg / m 3 h 760mm

17 Measuring Pressure Can a barometer be made with Water instead of Mercury? P h water P water gh g h 2 101,300 N/ m 1000 kg / m x9.8 m / s 3 2 mercury water 13.6 water 1000 kg / m 3 h 10.3m (Notice: 10.3m is just 13.6 x 760mm!)

18 Barometers Measuring Air Pressure Fluid in the tube adjusts until the weight of the fluid column balances the atmospheric force exerted on the reservoir. 10.3m Not to Scale!!! Mercury Barometer atm x Pa Water Barometer 760mm

19 Absolute vs. Gauge Pressure Absolute Pressure: P P gh 0 Guage Pressure: P0 gh Guage pressure is what you measure in your tires Absoulte pressure is the pressure at B and is what is used in PV = nrt

20 Measuring Pressure GUAGE PRESSURE is relative to atmospheric pressure. It is found by measuring h with a manometer. P guage gh ABSOLUTE PRESSURE is the total pressure. Pressures ADD! P P P absolute guage atmosphere Water seeks its own level unless there is pressure!

21 Tower Power How high above the ground must the water level be in the tower to create a gauge pressure of 3 x 10 5 Pa (~ 3 atm) in your house below? h P h water P water gh g 5 2 3x10 N / m 1000 kg / m x9.8 m / s h 30.6m 3 2 It takes approximately 4.5 psi (.3atm) for a oneinch diameter pipe to push a column of water ten feet in the air, the standard distance between the floors of a house.

22 Zero Pressure: Making A Vacuum Mechanical Vacuum Pump Minimum pressure produced by mechanical pump:~1pa Minimum pressure produced by hi tech: Pa Zero pressure not allowed by Quantum Uncertainty! Absolute pressure cannot be negative: Pressure pushes not pulls! Gauge pressure can be negative because it is a relative pressure.

23 Nature abhors a Vacuum. -Aristotle

24 Accelerating or Decelerating? A little girl holding a helium-filled balloon travels in a car moving with constant velocity. The windows are up and the car is relatively airtight. The driver then changes speed. From the picture did the driver hit the gas or the break? Explain.

25 Question: Accelerating or Decelerating? Answer: The car accelerates. The girl s inertia causes her to jerk back as the car accelerates forward under her. Similarly, the air in the car jerks back, creating a low pressure area in the front of the car so the helium balloon is buoyed forward.

26 Pascal s Principle: Liquids & Gases A pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid: P = P everywhere. F A F A

27 Pascal s Principle: Liquids & Gases How much pressure is transmitted in this hydraulic lever? F A F A kg 9.8 m / s 2 1m 2 98Pa A = 1m 2

28 P A PA Buoyant Force The difference in pressures at different depths produces a net upward or Buoyant Force: BF F F 2 1 gh Agh A 2 1 ga h h 2 1 V m fluid displaced ga h h 2 1 V m fluid displaced gv displaced mfluid g Weight of fluid displaced by object!

29 Archimedes Principle The buoyant force on an object equals the weight of the fluid it displaces. BF W fluid Note: Wfluid Vg For the same volume of displaced fluid denser fluids exert a greater buoyant force. This is why ships float higher in salt water than fresh water.

30 Archimedes Principle The wood block is floating in the water and has a mass of.08 kg. What is the volume of the displaced water? BF W fluid displaced W fluid displaced BF W block W block V g m g fluid fluid block V m / fluid block fluid Since object floats, BF (weight of fluid displaced) equals the weight of the object. So the masses of the object and fluid displaced are the same. m g m g fluid m fluid m object object kg /1000 kg / m V 8.0x10 m fluid

31 Archimedes Principle The buoyant force on an object equals the weight of the fluid it displaces. BF W fluid W object < BF W object > BF W object = BF Object will Float Object will Sink Object will Be Neutral

32 Archimedes Principle Object will Float if: W BF W object fluid dispalced V g V g object object fluid displaced But the Volume of the object is the same as the Volume of the fluid displaced: V g V g object displaced fluid displaced object fluid An object will float if its density is less than the fluid it is in!

33 Archimedes Principle object fluid Float object fluid Sink object fluid Neutral

34 Archimedes Principle Buoyant force also depends on SHAPE which determines the VOLUME of displaced fluid. This is how Iron Ships can float!! They have a shape that displaces enough water to produce a Buoyant Force to balance the weight!

35 Buoyant Force Question A boulder is thrown into a deep lake. As it sinks deeper and deeper into the water, does the buoyant force on it increase? NO! The buoyant force does not change as the boulder sinks because the boulder displaces the same amount of fluid no matter how deep it is in the water!

36 Question A piece of iron placed on a block of wood makes it float lower in the water. If the iron were instead suspended beneath the wood, would it float as low, lower or higher?

37 Question A piece of iron placed on a block of wood makes it float lower in the water. If the iron were instead suspended beneath the wood, would it float as low, lower or higher? Answer Higher. When the iron is on top, its whole weight pushes the wood into the water. But when the iron is submerged, buoyancy on it reduces its effective weight and less of the block is pulled beneath the water line.

38 Archimedes Principle Fraction of Object Floating V submerged V object V m m displaced V object fluid object object fluid / / fluid object Since object floats, BF (weight of fluid displaced) equals the weight of the object. Thus the masses of the object and fluid displaced are the same. m g m g fluid m fluid m object object

39 Archimedes Principle Fraction of Object Floating V submerged V object object fluid If the above object is floating in water, what is it s density? What is the fraction submerged? 1/2 What is the density in terms of water? 1/2

40 Archimedes Principle Fraction of Object Floating V submerged V object object fluid What is the density of the object? What fraction is submerged? ~1/3 Density is 1/3 that of water.

41 Why does the water stop when the top is closed? Pressure is greater in the fluid at the spout due to weight of water so water flows. Hand covers top and water keeps flowing until the pressure is reduced to 1 atm by increasing volume of air above the fluid just like with a closed barometer!

42 How High?

43 Flow Rate The volume of fluid per second that flows through an area A. V F Av t

44 Continuity Equation Incompressible Fluid ( doesn t change) Av A v v v 1 2

45 Continuity Equation v 1 v 2 v v 1 2

46 Fluid Flow A swelling in a blood vessel doubles its cross sectional area. By how much will the blood flow change as it moves through the enlarged area? The speed of blood (density = 1060 kg/m 3 ) through a normal part of the vessel is v =.5 m/s. A 1 A v A v v v 0.5 m / s 0.25 m / s A2

47 Continuity Equation Calculate the velocity to which water can be squirted with the hose with a constant flow rate of liters per second if it : a) emerges from the hose (r = 0.900cm) without the thumb b) emerges with the thumb reducing the radius to r = cm. c)what is the maximum height the water can reach if it is shot straight up with the thumb?

48 Bernoulli s Bubbles Water with air bubbles flows through a pipe that gets narrower. In the narrow region the water gains speed and the bubbles are a) larger Lower pressure!! b) smaller WHY? c) the same size

49 Bernoulli s Principle Where the speed of a fluid increases, internal pressure in the fluid decreases.

50 Bernoulli s Ball Why does this happen?

51 Bernoulli s Principle

52 Bernoulli s Equation Conservation of energy for a smooth steady fluid flow. W KE PE constant 1 2 Fd1 mv1 mgh1 2 constant Fd1 1 m 2 m v1 gh1 constant V 2 V V 1 2 P1 v1 gh1 constant 2

53 Efflux Speed A large tank is filled with water up to a depth of 15 m above ground. A spout located 10.0 m above the bottom of the tank is then opened. With what speed will water emerge from the spout? The tank is so large that the liquid level at the top (open to air) changes very slowly. The pressures are both at 1 atm. 1 1 P v gh P v gh gh1 v2 gh2 2 2 v 2 g( h h ) m/ s (5 m) v2 9.9 m/ s

54 Why does the water stop when the top is closed? Pressure is greater in the fluid at the spout due to weight of water so water flows. Hand covers top and water keeps flowing until the pressure is reduced to 1 atm by increasing volume of air above the fluid.

55 Bernoulli s Principle: LIFT

56 Sailing Problem Calculate the approximate force on a square meter of sail, given the horizontal velocity of the wind is 6.00 m/s parallel to its front surface and 3.50 m/s along its back surface. Air 1.29 / F ( P P ) A Front Back kg m 3

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58 Question Here is a glass of ice water with an ice cube floating in it. What fraction of the ice cube is submerged in the water? ice 917 kg / m, 1000 kg / m 3 3 water

59 Question Here is a glass of ice water with an ice cube floating in it. What fraction of the ice cube is submerged in the water? ice 917 kg / m, 1000 kg / m 3 3 water V submerged V object object fluid object fluid 917 kg / m 1000 kg / m %

60 Question The glass is full to the brim with ice water and ice cubes. When the ice melts, will the water level decrease? Overflow? Remain the same? WHY?

61 Question Here is a glass of ice water with an ice cube floating in it. When the ice cube melts, will the water level rise, drop or remain the same? Assume constant temperature. BF Answer W Since the cube floats, the BF equals its weight. BF = W ice Weight of the water displaced equals the weight of the ice cube. m water g = m ice g Mass of the water displaced equals the mass of the ice cube. When the ice melts, it is water and since it has the same mass and density as the displaced water, it has the same volume, filling up the space where of the displaced water. The level remains unchanged.

62 Volume above and below How does the volume of the billions of hexagonal open spaces in the structures of ice crystals in a piece of ice compare to the portion of ice that floats above the water line? water =1000 kg/m 3 ice = 917 kg/m 3 Answer: The volume is the same! When the ice melts, the open spaces are filled in by the amount of ice that extends above the water level. This is also why the water level doesn t rise when ice melts.

63 If you release a ball inside a freely-falling elevator, it stays in front of you instead of "falling to the floor" because you, the ball, and the elevator are all accelerating downward at the same acceleration, g. If you similarly release a helium-filled balloon, the balloon will a) also stay in front of you b) press against the ceiling c) press against the floor

64 Question If you release a ball inside a freely-falling elevator, it stays in front of you instead of "falling to the floor" because you, the ball, and the elevator are all accelerating downward at the same acceleration, g. If you similarly release a helium-filled balloon, the balloon will a) also stay in front of you b) press against the ceiling c) press against the floor Answer: a) The balloon stays in front of you because it, along with you and the air and the elevator, is in free fall. When the air is in free fall it has no weight, therefore it cannot provide a buoyant force. Therefore, the balloon falls with everything else.