Chapter 7: Fluids — Review and Exercises

Glossary

Archimedes’ principle

the buoyant force on an object equals the weight of the fluid it displaces

Bernoulli’s equation

the conservation-of-energy relation along a streamline for steady flow of an incompressible fluid with negligible viscosity: the sum of pressure, kinetic energy per unit volume, and gravitational potential energy per unit volume is constant

Bernoulli’s principle

for steady, incompressible flow with negligible viscosity along a streamline at constant height, pressure is lower where speed is higher

buoyant force

the net upward force due to the pressure of a fluid at rest on an immersed object

density

the mass per unit volume of a substance or object

flow rate

abbreviated Q, it is the volume V that flows past a particular point during a time t, or Q = V/t

fluids

liquids and gases; a fluid is a state of matter that yields to shearing forces

liter

a unit of volume, equal to 10⁻³ m³

pressure

normal force per unit area on a surface

pressure due to the weight of fluid

pressure at a depth below a fluid surface due to its weight; given by

P = ρ g h

specific gravity

the ratio of the density of an object to a fluid (usually water)

Section Summary

7.2: What Is a Fluid?

A fluid is a state of matter that yields to sideways or shearing forces. Liquids and gases are both fluids. Fluid statics is the physics of stationary fluids.

7.3: Density

Density is the mass per unit volume of a substance or object. In equation form, density is defined as
$ρ = \frac{m}{V}$ .
(7.3.1)
The SI unit of density is $kg / m^{3}$ .

7.4: Pressure

Pressure is the force per unit perpendicular area over which the force is applied. In equation form, pressure is defined as
$P = \frac{F}{A}$ .
(7.4.1)
The SI unit of pressure is pascal and $1 Pa = 1 N / m^{2}$ .

7.5: Pressure Due to the Weight of Fluid

Pressure is the weight of the fluid $m g$ divided by the area $A$ supporting it (the area of the bottom of the container):
$P = \frac{m g}{A}$ .
Pressure due to the weight of a liquid is given by
$P = ρ g h$ ,
(7.5.1)
where $P$ is the pressure, $ρ$ is the density of the liquid, $g$ is the acceleration due to gravity, and $h$ is the height of the liquid.

7.6: Archimedes’ Principle

Buoyant force is the net upward force on an object immersed in a fluid at rest. If the buoyant force is greater than the object’s weight, the object will rise to the surface and float. If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force equals the object’s weight, the object will remain suspended at that depth. The buoyant force is always present whether the object floats, sinks, or is suspended in a fluid.
Archimedes’ principle states that the buoyant force on an object equals the weight of the fluid it displaces.
Specific gravity is the ratio of the density of an object to a fluid (usually water).

7.7: Flow Rate and Its Relation to Velocity

Flow rate $Q$ is defined to be the volume $V$ flowing past a point in time $t$ , or $Q = \frac{V}{t}$ where $V$ is volume and $t$ is time.
The SI unit of volume is $m^{3}$ .
Another common unit is the liter (L), which is $10^{- 3} m^{3}$ .
Flow rate and velocity are related by $Q = A \bar{v}$ where $A$ is the cross-sectional area of the flow and $\bar{v}$ is its average velocity.
For incompressible fluids, flow rate at various points is constant. That is,
$\begin{matrix} Q_{1} = Q_{2} \\ A_{1} {\bar{v}}_{1} = A_{2} {\bar{v}}_{2} \\ n_{1} A_{1} {\bar{v}}_{1} = n_{2} A_{2} {\bar{v}}_{2} \end{matrix}}$ .

7.8: Bernoulli’s Equation

Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
$P_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = P_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$ .
(7.8.2)
Bernoulli’s principle is Bernoulli’s equation applied to situations in which depth is constant. The terms involving depth (or height h ) subtract out, yielding
$P_{1} + \frac{1}{2} ρ v_{1}^{2} = P_{2} + \frac{1}{2} ρ v_{2}^{2}$ .
(7.8.4)
Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

Questions and Exercises

7.2: What Is a Fluid?

Exercise 1

What physical characteristic distinguishes a fluid from a solid?

Exercise 2

Which of the following substances are fluids at room temperature: air, mercury, water, glass?

Exercise 3

Why are gases easier to compress than liquids and solids?

Exercise 4

How do gases differ from liquids?

7.3: Density

Exercise 5

A uniform piece of aluminum is cut into two unequal pieces. How do their masses, volumes, and densities compare? Explain.

Exercise 6

Water has a density of approximately

1 g / {cm}^{3}

. (a) What is the mass of

250 {cm}^{3}

of water? (b) Explain why expressing its density as

1000 kg / m^{3}

describes the same property.

Exercise 7

Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?

Exercise 8

Approximately how does the density of air vary with altitude?

7.4: Pressure

Exercise 9

The same uniform pressure acts on two surfaces, one with twice the area of the other. Compare the forces on them. Explain why equal pressure does not necessarily mean equal force.

Exercise 10

Why is force exerted by static fluids always perpendicular to a surface?

Exercise 11

The outward force on one end of an air tank was calculated in Example 7.4.1. How is this force balanced? (The tank does not accelerate, so the force must be balanced.)

Exercise 12

How is pressure related to the sharpness of a knife and its ability to cut?

Exercise 13

Why does a dull hypodermic needle hurt more than a sharp one?

Exercise 14

Toe dancing (as in ballet) is much harder on toes than normal dancing or walking. Explain in terms of pressure.

Exercise 15

How do jogging on soft ground and wearing padded shoes reduce the pressures to which the feet and legs are subjected?

7.5: Pressure Due to the Weight of Fluid

Exercise 16

Two open containers hold water to the same depth, but one is much wider. Both have flat, horizontal bottoms and are exposed to the same atmospheric pressure. Compare (a) the pressure at their bottoms and (b) the total force the water exerts on each bottom. Explain the distinction.

Exercise 17

Figure 7.E.1 shows how sandbags placed around a leak outside a river levee can effectively stop the flow of water under the levee. Explain how the small amount of water inside the column formed by the sandbags is able to balance the much larger body of water behind the levee.

The figure shows a flooding river on the extreme right, with a levee set up on its left, and sandbags are stacked on the left of the levee. The height of the levee and that of the stacked sandbags is greater than the water level of the flooding river, so the water does not flow over their tops, but a leak under the levee allows some water to flow under it and reach the sandbags. — **Figure 7.E.1.** Because the river level is very high, it has started to leak under the levee. Sandbags are placed around the leak, and the water held by them rises until it is the same level as the river, at which point the water there stops rising.

Exercise 18

A plug seals a bathtub drain, with water above it and air at atmospheric pressure below it. Why does increasing the water depth make the plug harder to lift? Explain in terms of the pressures on its upper and lower surfaces.

Exercise 19

A dam holds back a reservoir on one side, while its opposite face is exposed to air. The same atmospheric pressure acts on the reservoir’s surface and on the exposed face. Does atmospheric pressure contribute to the net horizontal force on the dam? Explain.

Exercise 20

Atmospheric pressure exerts a large force (equal to the weight of the atmosphere above your body—about 10 tons) on the top of your body when you are lying on the beach sunbathing. Why are you able to get up?

Exercise 21

Pressure is sometimes expressed as the height of a column of mercury or water. How could you use the fluid’s density and the column’s height to calculate this pressure in pascals?

Exercise 22

What are two reasons why mercury rather than water is used in barometers?

Exercise 23

Does atmospheric pressure add to the gas pressure in a rigid tank? In a toy balloon? When, in general, does atmospheric pressure not affect the total pressure in a fluid?

7.6: Archimedes’ Principle

Exercise 24

Two solid objects have equal volumes but different masses. Both are denser than water and are suspended from thin strings, completely submerged without touching the container. (a) Compare their buoyant forces. (b) Compare the tensions in the strings. Explain.

Exercise 25

Do fluids exert buoyant forces in a “weightless” environment, such as in the space shuttle? Explain your answer.

Exercise 26

Marbles sink to the flat bottom of a container with vertical walls, raising the water level without causing overflow. Once they are at rest, buoyancy supports part of their weight. Explain how the increased water pressure and the marbles’ direct contact with the bottom together increase the downward force on the bottom by the full weight of the marbles.

Exercise 27

Cargo is added to a floating boat, which settles lower but remains afloat. What happens to the buoyant force and the volume of water displaced? Explain.

Exercise 28

Will the same ship float higher in salt water than in freshwater? Explain your answer.

Exercise 29

Ice floats in freshwater with 91.7% of its volume submerged. Taking water’s density as

1000 kg / m^{3}

, determine the ice’s density. Explain how the balance between weight and buoyancy supports your calculation.

Exercise 30

Figure 7.E.2 shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.

A glass filled to the brim with water and ice cubes. — **Figure 7.E.2.**

Exercise 31

In a remote location near the North Pole, an iceberg floats in a lake. Next to the lake (assume it is not frozen) sits a comparably sized glacier sitting on land. If both chunks of ice should melt due to rising global temperatures (and the melted ice all goes into the lake), which ice chunk would give the greatest increase in the level of the lake water, if any?

7.7: Flow Rate and Its Relation to Velocity

Exercise 32

Identify some substances that are incompressible and some that are not.

Exercise 33

What is the difference between flow rate and fluid velocity? How are they related?

Exercise 34

Many figures in the text show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. (Hint: Consider the relationship between fluid velocity and the cross-sectional area through which it flows.)

Exercise 35

You can squirt water a considerably greater distance by placing your thumb over the end of a garden hose and then releasing, than by leaving it completely uncovered. Explain how this works.

7.8: Bernoulli’s Equation

Exercise 36

Friction in a flowing fluid converts some mechanical energy into thermal energy. Does this violate conservation of energy? Explain why the usual frictionless form of Bernoulli’s equation needs modification.

Exercise 37

Water flows steadily through a horizontal pipe that narrows. Neglect friction. Compare the water’s speed and pressure in the wide and narrow portions. Explain how the pressure difference is connected to the water’s gain in kinetic energy.

Exercise 38

Water is shot nearly vertically upward in a decorative fountain and the stream is observed to broaden as it rises. Conversely, a stream of water falling straight down from a faucet narrows. Explain why, and discuss whether surface tension enhances or reduces the effect in each case.

Exercise 39

Look back to Figure 7.8.1. Answer the following two questions. Why is

P_{o}

less than atmospheric? Why is

P_{o}

greater than

P_{i}

Exercise 40

It is dangerous to stand close to railroad tracks when a rapidly moving commuter train passes. Explain why atmospheric pressure would push you toward the moving train.

Exercise 41

Roofs are sometimes pushed off vertically during a tropical cyclone, and buildings sometimes explode outward when hit by a tornado. Use Bernoulli’s principle to explain these phenomena.

Exercise 42

If you lower the window on a car while moving, an empty plastic bag can sometimes fly out the window. Why does this happen?

Exercise 43

Water pressure inside a hose nozzle can be less than atmospheric pressure due to the Bernoulli effect. Explain in terms of energy how the water can emerge from the nozzle against the opposing atmospheric pressure.

Exercise 44

Many entrainment devices have a constriction, called a Venturi, such as shown in Figure 7.E.3. How does this bolster entrainment?

Figure shows a venturi tube, a cylindrical tube broader at both the ends and narrow in the middle. The narrow part is labeled as venturi constriction. The flow of fluid is shown as horizontal arrows along the length of the tube toward the right. The flow lines are closer in the center and spread apart at both the ends. There is an opening on the top portion of the narrow section for the entrained fluid to enter. — **Figure 7.E.3.** A tube with a narrow segment designed to enhance entrainment is called a Venturi. These are very commonly used in carburetors and aspirators.

Exercise 45

A perfume bottle or atomizer sprays a fluid that is in the bottle. (Figure 7.E.4.) How does the fluid rise up in the vertical tube in the bottle?

A perfume bottle with a spray cap. — **Figure 7.E.4.** Atomizer: perfume bottle with tube to carry perfume up through the bottle. (credit: Antonia Foy, Flickr)

Exercise 46

Give an example of entrainment not mentioned in the text.

Exercise 47

Is there a limit to the height to which an entrainment device can raise a fluid? Explain your answer.

Exercise 48

Some chimney pipes have a T-shape, with a crosspiece on top that helps draw up gases whenever there is even a slight breeze. Explain how this works in terms of Bernoulli’s principle.

Exercise 49

Why is it preferable for airplanes to take off into the wind rather than with the wind?

Source sections: What Is a Fluid?, Density, Pressure, Pressure Due to the Weight of Fluid, Archimedes’ Principle, Flow Rate and Its Relation to Velocity, Bernoulli’s Equation. Credits and licenses appear with each source section.