Ch 11 Fluids

Learning Objectives FLUID MECHANICS AND THERMAL PHYSICS Fluid Mechanics Hydrostatic pressure Students should understand the concept of pressure as it applies to fluids, so they can: Apply the relationship between pressure, force, and area. Apply the principle that a fluid exerts pressure in all directions. Apply the principle that a fluid at rest exerts pressure perpendicular to any surface that it contacts. Determine locations of equal pressure in a fluid. Determine the values of absolute and gauge pressure for a particular situation. Apply the relationship between pressure and depth in a liquid,  P =  g  h

Learning Objectives Buoyancy Students should understand the concept of buoyancy, so they can: Determine the forces on an object immersed partly or completely in a liquid. Apply Archimedes’ principle to determine buoyant forces and densities of solids and liquids. Fluid flow continuity Students should understand the equation of continuity so that they can apply it to fluids in motion. Bernoulli’s equation Students should understand Bernoulli’s equation so that they can apply it to fluids in motion.

Table of Contents Mass Density Pressure Pressure & Depth in a Static Fluid Pressure Gauges Pascal’s Principle Archimedes’ Principle Fluids in Motion The Equations of Continuity Bernoulli’s Equation Applications of Bernoulli’s Equation Viscous Flow (AP?)

Chapter 11: Fluids Section 1: Mass Density

Fluids Fluids are substances that can flow , such as liquids and gases, and even some solids In Physics B, we will limit our discussion of fluids to substances that can easily flow, such as liquids and gases

DEFINITION OF MASS DENSITY The mass density of a substance is the mass of a substance divided by its volume: SI Unit of Mass Density: kg/m 3

Example 1 Blood as a Fraction of Body Weight The body of a man whose weight is 690 N contains about 5.2x10 -3 m 3 of blood. (a) Find the blood’s weight and (b) express it as a percentage of the body weight. (b) (a)

11.1.1. Which one of the following objects has the largest mass? a) a gold solid cube with each side of length r b) a brass solid sphere of radius r a silver solid cylinder of height r and radius r a lead solid cube with each side of length r a concrete solid sphere of radius r

11.1.2. Liquid A has a mass density of 850.0 kg/m 3 and Liquid B has a mass density of 1060.0 kg/m 3 . Seventy-five grams of each liquid is mixed uniformly. What is the specific gravity of the mixture? a) 0.955 b) 0.943 c) 0.878 d) 0.651 e) 0.472

11.1.3. What volume of helium has the same mass as 5.0 m 3 of nitrogen? a) 35 m 3 b) 27 m 3 c) 7.0 m 3 d) 3.5 m 3 e) 0.72 m 3

Chapter 11: Fluids Section 2: Pressure

Pressure SI Unit of Pressure: 1 N/m 2 = 1Pa pascal The force on a surface caused by pressure is always normal (or perpendicular) to the surface. This means that the pressure of a fluid is exerted in all directions, and is perpendicular to the surface at every location

Example 2 The Force on a Swimmer Suppose the pressure acting on the back of a swimmer’s hand is 1.2x10 5 Pa. The surface area of the back of the hand is 8.4x10 -3 m 2 . Determine the magnitude of the force that acts on it. (b) Discuss the direction of the force. Since the water pushes perpendicularly against the back of the hand, the force is directed downward in the drawing.

Atmospheric Pressure Atmospheric pressure is normally about 100,000 Pa Differences in atmospheric pressure cause winds to blow Low atmospheric pressure inside a hurricane’s eye contributes to the severe winds and the development of the storm surge

Atmospheric Pressure at Sea Level: 1.013x10 5 Pa = 1 atmosphere

Problem Calculate the net force on an airplane window if cabin pressure is 90% of the pressure at sea level, and the external pressure is only 50% of that tat sea level. Assume the window is 0.43 m tall and 0.30 m wide and atmospheric pressure at sea level is 100,000 Pa

11.2.1. A swimmer is swimming underwater in a large pool. The force on the back of the swimmer’s hand is about one thousand newtons. The swimmer doesn’t notice this force. Why not? a) This force is actually smaller than the force exerted by the atmosphere. b) The force is large, but the pressure on the back of the hand is small. c) The force is exerted on all sides equally. d) The swimmer is pushing on the water with the same force. e) I do not know, but I’m sure I would feel that kind of force.

11.2.2. In a classroom demonstration, a physics professor lies on a “bed of nails.” The bed consists of a large number of evenly spaced, relatively sharp nails mounted in a board so that the points extend vertically outward from the board. While the professor is lying down, nearly one thousand nails make contact with his body. Which one of the following choices provides the best explanation as to why the professor is not harmed by the bed of nails? a) The nails are not as sharp as nails typically used in construction. b) The professor is wearing special clothes that are not easily penetrated by nails. c) The professor’s skin has been caliced after years of doing the demonstration, so nails no longer penetrate the skin. d) The professor’s weight is distributed over all of the nails in contact with the professor’s body, so the pressure exerted a nail at any location is too small to penetrate the skin. e) The force due to gravity on the professor is balanced by the upward force of the nails, as explained by Newton’s third law of motion, so the professor doesn’t accelerate downward.

11.2.3. Amanda fills the two tires of her bicycle to the pressure specified on the side wall of the tires. She then gets onto her bicycle and notices that the bottoms of the tires look flatter than before she mounted the bicycle. What happens to the pressure in the tires when she is on the bicycle compared to when she was off the bicycle? a) The pressure inside the tire increases. b) The pressure inside the tire decreases. c) The pressure inside the tire has the same value.

11.2.4. In snowy regions of the world, the local people may wear snow shoes below their normal shoes or boots. These snow shoes have a much larger area than a regular shoe or boot. How does a snow shoe improve a hiker’s ability to walk across a snowy region? a) The hiker’s weight is distributed over the area of the snow shoes, which reduces the pressure on the snow below and minimizes sinking into the snow. b) The snow shoes increase the normal force of the snow on the hiker. c) The snow shoes increase the upward pressure of the snow on the hiker. d) The snow shoes compact the snow making it harder to sink into it. e) The hiker’s weight is reduced by wearing large area snow shoes.

11.2.5. Helium gas is confined within a chamber that has a moveable piston. The mass of the piston is 8.7 kg; and its radius is 0.013 m. If the system is in equilibrium, what is the pressure exerted on the piston by the gas? a) 1.639 × 10 4 Pa b) 8.491 × 10 4 Pa c) 1.013 × 10 5 Pa d) 1.606 × 10 5 Pa e) 2.619 × 10 5 Pa

11.2.6. Two identical balloons, filled with unequal amounts of the same gas, are connected via a pipe with a closed valve as shown. One balloon has five times the diameter of the other. Which of the following will occur when the valve is opened? a) Nothing will happen. Both balloons will remain the same size. b) The small balloon will get larger and the large one will get smaller, but they will end up with different sizes. c) The small balloon will get larger and the large one will get smaller. They will end up with equal sizes. d) Most of the air in the smaller balloon will move into the larger balloon. e) Most of the air in the larger balloon will move into the smaller balloon.

11.2.7. In which one of the following cases is the pressure exerted on the ground by the man the largest? a) A man stands with both feet flat on the ground. b) A man stands with one foot flat on the ground. c) A man lies with his back flat on the ground. d) A man kneels with both knees on the ground. e) A man stands with the toes of one foot on the ground.

11.2.8. Carol hangs a piece of stained glass artwork that she has just completed on her window using a suction cup hanger. Which one of the following statements best explains the force that holds the suction cup to the glass window? a) There is a high amount of pressure between the glass window and the suction cup. b) There is a very low pressure between the glass window and the suction cup. c) There is a smaller pressure on the suction cup due to the atmosphere than the pressure between the suction cup and the glass window. d) There is a greater pressure on the suction cup due to the atmosphere than the pressure between the suction cup and the glass window. e) Pushing the suction cup against the glass window causes a very strong chemical bond to form between the glass window and the suction cup.

11.2.9. What is the force that causes liquid to move upward in a drinking straw as a person takes a drink? a) that due to a low pressure region caused by sucking b) that due to the pressure within the liquid c) that due to atmospheric pressure d) that due to the person sucking on the straw e) that due to friction forces within the straw

Chapter 11: Fluids Section 3: Pressure & Depth in a Static Fluid

Conceptual Example 3 The Hoover Dam Lake Mead is the largest wholly artificial reservoir in the United States. The water in the reservoir backs up behind the dam for a considerable distance (120 miles). Suppose that all the water in Lake Mead were removed except a relatively narrow vertical column. Would the Hoover Same still be needed to contain the water, or could a much less massive structure do the job?

Example 4 The Swimming Hole Points A and B are located a distance of 5.50 m beneath the surface of the water. Find the pressure at each of these two locations.

11.3.1. You are vacationing in the Rocky Mountains and decide to ride a ski lift to the top of a mountain. As you go up, your feel your ears make a “pop” sound because of changes in atmospheric pressure. Which way does your ear drum move, if at all, during your ascent up the mountain? a) inward, because the pressure outside the ear is larger than the pressure inside the ear b) inward, because the pressure outside the ear is smaller than the pressure inside the ear c) No movement occurs. This is just the sound of air bubbles bursting inside the ear. d) outward, because the pressure outside the ear is smaller than the pressure inside the ear e) outward, because the pressure outside the ear is larger than the pressure inside the ear

11.3.2. 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  . a)  gc b)  gb c)  ga d) P atm +  gb e) P atm   gc

11.3.3. An above ground water pump is used to extract water from a 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 it from any depth.

11.3.4. An aquarium has a length L , a width W , and a height H . Glass of thickness d is used for the sides of the aquarium. If you were to design a similar aquarium, except that it has a length 5 L and a width 5 W , what thickness of glass would you need to use? a) d b) c) 5 d d) 10 d e) 25 d

11.3.5. A water well is dug to a depth of 15 meters. A pump is used to bring the water upward through a pipe, but the pump can only raise the water about 7.5 meters. By which of the following methods can the water be brought all the way to the surface? a) Connect a second, identical pump in series with the first pump. b) Connect a second, identical pump in parallel with the first pump. c) Replace the pump with a more powerful pump. d) Replace the pipe with a narrower pipe. e) None of these methods will work.

Chapter 11: Fluids Section 4: Pressure Gauges

Gauge Pressure P =  gh P: pressure (Pa)  : density (kg/m 3 ) g: acceleration due to gravity (m/s 2 ) h: height of column (m) This type of pressure is often called gauge pressure: Does not include the effect of atmospheric pressure on top of the fluid If the fluid is water, this is referred to as hydrostatic pressure.

11.4.1. The height of mercury in one barometer is 0.761 m above the reservoir of mercury. If a second barometer is brought to the same location that has a liquid with a density of 2650 kg/m 3 , what is the height of the fluid in the second barometer? a) 0.761 m b) 2.02 m c) 2.87 m d) 5.03 m e) 3.91 m

Chapter 11: Fluids Section 5: Pascal’s Principle

PASCAL’S PRINCIPLE Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and enclosing walls.

Example 7 A Car Lift The input piston has a radius of 0.0120 m and the output plunger has a radius of 0.150 m. The combined weight of the car and the plunger is 20500 N. Suppose that the input piston has a negligible weight and the bottom surfaces of the piston and plunger are at the same level. What is the required input force?

11.5.1. A fluid is completely enclosed in the system shown. As the piston is moved to the right, which one of the following statements is true? (a) Pushing the piston to the right causes the pressure on the left side of the vertical cylinder to be larger than that on the right side. (b) Pushing the piston to the right causes the pressure on the bottom of the vertical cylinder to be larger than that on the top. (c) Pushing the piston to the right causes the pressure throughout the vertical cylinder to increase by the same amount. (d) Pushing the piston to the right causes the pressure in one part of the vertical cylinder and a corresponding decrease in another part of the cylinder. (e) After the piston is pushed by a distance s and held in that position, the pressure will be the same at all points within the fluid.

11.5.2. At an automotive repair shop, a hydraulic lift is used to raise vehicles so that mechanics can work under them. In one lift, compressed air with a maximum gauge pressure of 5.0 × 10 5 Pa is used to raise a piston with a circular cross-section and radius of 0.15 m. What is the maximum vehicular weight that can be raised using this lift? a) 7.5 × 10 4 N b) 6.0 × 10 4 N c) 4.4 × 10 4 N d) 3.5 × 10 4 N e) 1.1 × 10 4 N

11.5.3. A woman stands on a platform with a diameter D 1 that acts as a piston. The combined mass of the woman and the piston is 75 kg. The piston pushes downward on a reservoir of oil that supports a second platform with a diameter D 2 on which 12 women are standing. The combined mass of the second platform and the 12 women is 780 kg. Both platforms are at the same height during this demonstration; and they are at rest. What is the ratio D 2 / D 1 ? a) 2.4 b) 3.2 c) 4.8 d) 10 e) 12

Chapter 11: Fluids Section 6: Archimedes’ Principle

Archimedes’ Principle A body immersed in a fluid is buoyed up by a force that is equal to the weight of the fluid it displaces When an object floats, the upward buoyant force equals the downward pull of gravity This buoyant force can float very heavy objects, and acts upon object in the water whether that are floating, submerged, or even sitting on the bottom of the body of fluid. F B F W

Floaters If the object is floating then the magnitude of the buoyant force is equal to the magnitude of its weight.

Example 9 A Swimming Raft The raft is made of solid square pinewood. Determine whether the raft floats in water and if so, how much of the raft is beneath the surface. It will float, but how much will be submerged?

Conceptual Example 10 How Much Water is Needed to Float a Ship? A ship floating in the ocean is a familiar sight. But is all that water really necessary? Can an ocean vessel float in the amount of water than a swimming pool contains?

11.6.1. 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 0.5 d . What is the tension in the thread when the block is at the new depth? a) 0.25 T b) 0.50 T c) T d) 2 T e) 4 T

Heavy Objects can be moved with water Check out these photo from the aftermath of Hurricane Katrina!

11.6.2. A fisherman rows his boat out to his favorite spot on a pond, where he decides to drop a heavy anchor from his boat. When he drops the anchor into the pond, what happens to the level of the pond? a) The level of the pond decreases. b) The level of the pond stays the same. c) The level of the pond increases.

11.6.3. In the distant future, a base has been constructed on the moon. Inside the base, a ball is dropped into a container of water. Given that the acceleration due to gravity is about one sixth the value on the Earth’s surface, which of the following statements correctly describes the buoyant force on the ball? a) The buoyant force is equal to 1/6 the weight of the displaced water. b) The buoyant force is equal to 6 times the weight of the displaced water. c) The buoyant force is equal to the weight of the displaced water. d) The buoyant force is equal to 1/6 the weight of the ball. e) The buoyant force is equal to the weight of the ball.

11.6.4. A coin is dropped into a lake. As the coin sinks, how does the buoyant force on the coin change? a) The buoyant force decreases as the depth increases. b) The buoyant force increases as the depth increases. c) The buoyant force decreases as the speed increases. d) The buoyant force decreases as the depth decreases. e) The buoyant force has a constant value.

11.6.5. Three fourths the volume of a glass is filled with water. Ice is then added until it reaches the top of the glass. The water level at that point is h . Which one of the following statements concerning the level of water after the ice melts is true? a) The final level of water will be lower than h . b) The final level of water will be at h . c) The final level of water will be higher than h . d) This cannot be answered without knowing the mass of ice added. e) This cannot be answered without knowing the volume of ice added.

11.6.6. A boy is playing in the bathtub with a toy boat floating on the surface of the water. His favorite rock is the cargo on the boat. If the boy takes the rock from the boat and drops it into the water, how will the water level of the bathtub change, if at all? a) The water level will rise. b) The water level will fall. c) The water level will remain the same.

Chapter 11: Fluids Section 7: Fluids in Motion

Types of flowing fluids: In steady flow the velocity of the fluid particles at any point is constant as time passes. Unsteady flow exists whenever the velocity of the fluid particles at a point changes as time passes. Turbulent flow is an extreme kind of unsteady flow in which the velocity of the fluid particles at a point change erratically in both magnitude and direction.

More types of fluid flow Fluid flow can be compressible or incompressible . Most liquids are nearly incompressible. Fluid flow can be viscous or nonviscous . An incompressible, nonviscous fluid is called an ideal fluid .

When the flow is steady, streamlines are often used to represent the trajectories of the fluid particles.

Making streamlines with dye and smoke.

Chapter 11: Fluids Section 8: The Equation of Continuity

When a fluid flows… … mass is conserved Provided there are no inlets or outlets in a stream of flowing fluid The same mass per unit time must flow everywhere in the stream

The mass of fluid per second that flows through a tube is called the mass flow rate .

EQUATION OF CONTINUITY The mass flow rate has the same value at every position along a tube that has a single entry and a single exit for fluid flow. SI Unit of Mass Flow Rate: kg/s

The Equation of Continuity Incompressible fluid:

Example Problem A Pipe of diameter 6.o cm has fluid lowing through is at 1.6 m/s. How fast is the fluid flowing in an area of the pipe in which the diameter is 3.0 com? How much water flows through the pipe per second?

Problem The water in a canal flows 0.10 m/s where the canal is 12 m deep and 10 m across. If the depth of the canal is reduced to 6.5 m at an area where the canal narrows to 5.0 m, how fast will the water be moving through this narrower region?

11.8.1. At the site of a burning building, a firefighter is using a hose with a radius of 0.032 m that has water flowing through it at a rate of 9.95 m/s. At the end of the hose, there is a nozzle with a radius of 0.013 m. First, calculate the speed of the water exiting the nozzle and use that to determine the range, the maximum horizontal distance the water can reach, if it is directed at an angle of 45  with respect to the horizon. a) 6.2 m b) 59 m c) 120 m d) 210 m e) 370 m

11.8.2. A child has left an outdoor faucet open. An adult walks up and notices that the diameter of the stream is 0.25 d at the bottom and d at the top as it falls vertically from the faucet. What is the physical explanation for this narrowing of the stream? a) The flow rate at the top of the stream is not sufficient to maintain a constant cross-sectional area. b) Atmospheric pressure is greater than the pressure within the stream of water. c) The water has left the pipe that carried it and no longer maintains the shape of the pipe. d) The water accelerates as it falls and the cross-sectional area must decrease to maintain a constant flow rate. e) The stream of water is experiencing friction with the air as it falls and part of it slows down while part of it falls at a constant speed.

11.8.3. The drawing shows a section of a pipe system in which an incompressible fluid can either flow in or flow out various channels. All of the pipes have the same diameter. The direction and flow rates are indicated on the drawing. What is the mass flow rate and direction of flow for the unknown pipe? a) 0.2 m 3 /s b) 0.4 m 3 /s c) 0.6 m 3 /s d) 0.8 m 3 /s e) 1.0 m 3 /s

Chapter 11: Fluids Section 9: Bernoulli’s Equation

Bernoulli's Theorem The sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any one location in the fluid is equal to the sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any other location in the fluid for a non-viscous, incompressible fluid in streamline flow. All other considerations being equal, when a fluid moves faster, the pressure drops

The fluid accelerates toward the lower pressure regions. According to the pressure-depth relationship, the pressure is lower at higher levels, provided the area of the pipe does not change.

BERNOULLI’S EQUATION In steady flow of a nonviscous, incompressible fluid, the pressure, the fluid speed, and the elevation at two points are related by:

Problem Knowing what you know about Bernoulli’s principle, design an airplane wing that you think will keep an airplane aloft. Draw a cross section of the wing

Problem An above ground swimming pool has a hole of radius 0.10 cm in the side1.0 m below the surface of the water. How fast is the water flowing out of the hole?

Problem An above ground swimming pool has a hole of radius 0.10 cm in the side1.0 m below the surface of the water. How much water flows out each second?

Problem An above ground swimming pool has a hole of radius 0.10 cm in the side 1.0 m below the surface of the water. How far does the water land from the side of the pool if the hole is 1.0 m above the ground?

Problem Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out of a nozzle whose diameter is 2.5 cm. What is the speed of the water coming out of the nozzle?

Problem Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out of a nozzle whose diameter is 2.5 cm. If the pressure in the hose is 350 kPa, what is the pressure in the nozzle in kPa?

11.9.1. Many tall buildings have reservoirs of water on the roofs. The surface of these reservoirs are at atmospheric pressure. If all of the water in one tall building were delivered from such a reservoir, would it be a good idea to deliver water to all of the floors of the building from a single pipe connected to the reservoir? a) Yes, that would ensure a steady flow of water at constant pressure on all floors. b) Yes, that would ensure a steady flow of water at a slightly varying pressure on all floors. c) No, the pressure would be too large on the lower floors and exit faucets at dangerous speeds. d) No, the pressure would be too small on the lower floors and would not exit faucets.

11.9.2. A curtain hangs straight down in front of an open window. A sudden gust of wind blows past the window; and the curtain is pulled out of the window. Which law, principle, or equation can be used to explain this movement of the curtain? a) the equation of continuity b) Pascal's principle c) Bernoulli's equation d) Archimedes' principle e) Poiseuille's law

11.9.3. Fluid is flowing from left to right through the pipe shown in the drawing. Points A and B are at the same height, but the cross-sectional areas of the pipe are different at the two locations. Points B and C are at two different heights, but the cross-sectional areas of the pipe are the same at these two locations. 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

11.9.4. An air blower is attached to a funnel that has a light-weight ball inside as shown. How will the ball behave when the blower is turned on? a) The ball will roll around inside the funnel in random directions. b) The ball will bounce around inside the funnel in random directions. c) The ball will be drawn upward to the top of the funnel and remain there. d) The ball will remain stationary, unaffected by the flowing air. e) The ball will spin rapidly as it moves slowly around inside the funnel.

11.9.5. A piece of paper is laying flat on a desk near a closed window. When the window is opened, air rushes over the top of the desk and the paper is observed to be lifted upward and then carried away by the wind. Which of the following statements best describes why the paper was lifted from the desk? a) The paper was attracted by the force of the wind. b) The pressure of the moving air above the paper was greater than that of the air between the paper and the desk top. c) The pressure of the moving air above the paper was less than that of the air between the paper and the desk top. d) The weight of the paper was reduced by the wind blowing over it and the normal force of the desk was then greater than the weight. The normal force pushes the paper upward. e) The wind pushed the side of the paper and lifted it upward.

Chapter 11: Fluids Section 10: Applications of Bernoulli’s Equation

Conceptual Example 14 Tarpaulins and Bernoulli’s Equation When the truck is stationary, the tarpaulin lies flat, but it bulges outward when the truck is speeding down the highway. Account for this behavior.

Example 16 Efflux Speed The tank is open to the atmosphere at the top. Find an expression for the speed of the liquid leaving the pipe at the bottom.

Chapter 11: Fluids Section 11: Viscous Flow

Flow of an ideal fluid. Flow of a viscous fluid.

FORCE NEEDED TO MOVE A LAYER OF VISCOUS FLUID WITH CONSTANT VELOCITY The magnitude of the tangential force required to move a fluid layer at a constant speed is given by: coefficient of viscosity SI Unit of Viscosity: Pa ·s Common Unit of Viscosity: poise (P) 1 poise (P) = 0.1 Pa·s

POISEUILLE’S LAW The volume flow rate is given by:

Example 17 Giving and Injection A syringe is filled with a solution whose viscosity is 1.5x10 -3 Pa ·s. The internal radius of the needle is 4.0x10 -4 m. The gauge pressure in the vein is 1900 Pa. What force must be applied to the plunger, so that 1.0x10 -6 m 3 of fluid can be injected in 3.0 s?

Ch 11 Fluids

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Ch 11 Fluids