Fundamentals of Fluid Mechanics

Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Chapter 1

Introduction - all with Video Answers

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Chapter Questions

Problem 1

The force, $F$, of the wind blowing against a building is given by $F=C_{D} \rho V^{2} A / 2,$ where $V$ is the wind speed, $\rho$ the density of the air, $A$ the cross-sectional area cf the building, and $C_{D}$ is a constant termed the drag coefficient. Determine the dimensions of the drag coefficient.

Mahnoor Amin

Mahnoor Amin

Numerade Educator

Problem 2

The Mach number is a dimensionless ratio of the velocity of an object in a fluid to the speed of sound in the fluid. For an airplane flying at velocity $V$ in air at absolute temperature $T,$ the Mach number Ma is
\[
\mathrm{Ma}=\frac{V}{\sqrt{k R T}}
\]
where $k$ is a dimensionless constant and $R$ is the specific gas constant for air. Show that Ma is dimeasionless.

Penny Riley

Numerade Educator

Problem 3

Verify the dimensions, in both the $F L T$ and $M L T$ systems, of the following quantities which appear in Table $1.1:(\mathrm{a})$ volume, $(\mathrm{b})$ acceleration,
(c) mass. (d) moment of inertia (area), and (e) Work.

Mahnoor Amin

Numerade Educator

Problem 4

1.4 Verify the dimensions, in both the FLT and MLT systems, of the following quantities which appear in Table 1.1
(a) angular velocity, (b) energy,
(c) moment of inertia (area),
(d) power, and
(e) pressure.

Mahnoor Amin

Numerade Educator

Problem 5

1.5 Verify the dimensions, in both the FLT system and the MLT system, of the following quantities which appear in Table 1.1
(a) frequency,
(b) stress,
(c) strain.
(d) torque, and
(e) work.

Supratim Pal

Numerade Educator

Problem 6

If $u$ is a velocity, $x$ a length, and $t$ a time, what are the (a) $\partial u / \partial t$ dimensions (in the $M L T$ system) of (b) $\partial^{2} u / \partial x \partial t,$ and (c) $\int(\partial u / \partial t) d x ?$

Mahnoor Amin

Numerade Educator

Problem 7

Verify the dimensions, in both the $F L T$ system and the $M L T$ system, of the following quantities which appear in Table 1.1
(a) acceleration.
(b) stress,
(c) moment of a force.
(d) volume, and
(e) Work.

Mahnoor Amin

Numerade Educator

Problem 8

If $p$ is a pressure, $V$ a velocity, and $\rho$ a fluid density, what are the dimensions (in the $M L T$ system) of
(a) $p / \rho$
(b) $p V \rho,$ and
(c) $p / \rho V^{2} ?$

Mahnoor Amin

Numerade Educator

Problem 9

1.9 If $P$ is a force and $x$ a length, what are the dimensions (in the $F L T \text { system })$ of
(a) $d P / d x$
(b) $d^{3} P / d x^{3},$ and
(c) $\int P d x ?$

Mahnoor Amin

Numerade Educator

Problem 10

If $V$ is a velocity, $\ell$ a length, and $\nu$ a fluid property (the kinematic viscosity) having dimensions of $L^{2} T^{-1},$ which of the following combinations are dimensionless: (a) $V \ell \nu$ (b) $V \ell / \nu$ $(\mathbf{c}) V^{2} \nu$ (d) $V / \ell \nu ?$

Mahnoor Amin

Numerade Educator

Problem 11

The momentum flux (discussed in Chapter 5) is given by the product $\dot{m} V$, where $\dot{m}$ is mass flow rate and $V$ is velocity. If mass flow rate is given in units of mass per unit time, show that the momentum fux can be expressed in units of force.

Mahnoor Amin

Numerade Educator

Problem 12

An equation for the frictional pressure loss $\Delta$ p (inches $\mathrm{H}_{2} \mathrm{O}$ ) in a circular duct of inside diameter $d$ (in.) and length $L$ (ft) for air flowing with velocity $V$ (ft/min) is
\[
\Delta p=0.027\left(\frac{L}{d^{122}}\right)\left(\frac{V}{V_{0}}\right)^{1.82}
\]
where $V_{0}$ is a reference velocity cqual to $1000 \mathrm{ft} / \mathrm{min}$. Find the units of the "constant" 0.027.

Willis James

Numerade Educator

Problem 13

The volume rate of flow, $Q,$ through a pipe containing a slowly moving liquid is given by the equation
\[
Q=\frac{\pi R^{4} \Delta p}{8 \mu \ell}
\]
where $R$ is the pipe radius, $\Delta p$ the pressure drop along the pipe, $\mu$ a fluid property called viscosity $\left(F L^{-2} T\right),$ and $\ell$ the length of pipe. What are the dimensions of the constant $\pi / 8 ?$ Would you classify this equation as a general homogeneous equation? Explain.

Mahnoor Amin

Numerade Educator

Problem 14

Show that each lerm in the following equation has units of Ib/ft $^{3}$. Consider $u$ a velocity, $y$ a length, $x$ a length, $p$ a pressure. and $\mu$ an absolute viscosity.
\[
0=-\frac{\partial p}{\partial x}+\mu \frac{\partial^{2} u}{\partial y^{2}}
\]

Mahnoor Amin

Numerade Educator

Problem 15

The pressure difference, $\Delta p$, across a partial blockage in an artery (called a stenosis) is approximated by the equation
\[
\Delta p=K_{o} \frac{\mu V}{D}+K_{u}\left(\frac{A_{0}}{A_{1}}-1\right)^{2} \rho V^{2}
\]
where $V$ is the blood velocity, $\mu$ the blood viscosity $\left(F L^{-2} T\right), \rho$ the blood density $\left(M L^{-3}\right), D$ the artery diameter, $A_{0}$ the area of the unobstructed artery, and $A_{1}$ the area of the stencsis. Determine the dimensions of the constants $K_{v}$ and $K_{v}$. Would this equation be valid in any system of units?

Penny Riley

Numerade Educator

Problem 16

Assume that the speed of sound, $c$, in a fluid depends on an elastic modulus, $E_{v},$ with dimensions $F L^{-2}$, and the fluid density. $\rho,$ in the form $c=\left(E_{v}\right)^{a}(\rho)^{b}$. If this is to be a dimensionally homogeneous equation, what are the values for $a$ and $b ?$ Is your result consistent with the standard formula for the speed of sound? (See Eq. $1.1 \mathrm{S}$.)

Mahnoor Amin

Numerade Educator

Problem 17

1.17 A formula to estimate the volume rate of flow, $Q$, flowing over a dam of length, $B$, is given by the equation
\[
Q=3.09 B H^{3 / 2}
\]
where $H$ is the depth of the water above the top of the dam (called the head). This formula gives $Q$ in $\mathrm{ft}^{3} / \mathrm{s}$ when $B$ and $H$ are in feet. Is the constant, $3.09,$ dimensionless? Would this equation be valid if units other than feet and seconds were used?

Mahnoor Amin

Numerade Educator

Problem 18

A commercial advertisement shows a pearl falling in a bottle of shampoo. If the diameter $D$ of the pearl is quite small and the shampoo sufficiently viscous, the drag 9 on the pearl is given by Stokes's law,
\[
\mathscr{P}=3 \pi \mu V D
\]
where $V$ is the speed of the pearl and $\mu$ is the fluid viscosity. Show tha: the term on the right side of Stokes's law has units of force.

Mahnoor Amin

Numerade Educator

Problem 19

Cite an example of a restricted homogeneous equation contained in a technical article found in an engineering journal in your field of interest. Define all terms in the equation, explain why it is a restricted equation, and provide a complete journal citation (title, date, etc.).

Banhishikha Sinha

Banhishikha Sinha

Numerade Educator

Problem 20

Express the following quantities in SI units: (a) 10.2 in./min, (b) 4.81 slugs,(c) $3.02 \mathrm{lb}$ (d) $73.1 \mathrm{ft} / \mathrm{s}^{2}$ (e) $0.0234 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$.

Mahnoor Amin

Numerade Educator

Problem 21

Express the following quantities in BG units: (a) $14.2 \mathrm{km}$ (b) $8.14 \mathrm{N} / \mathrm{m}^{3}$ (c) $1.61 \mathrm{kg} / \mathrm{m}^{3},(\mathrm{d}) 0.0320 \mathrm{N} \cdot \mathrm{m} / \mathrm{s},$ (e) $5.67 \mathrm{mm} / \mathrm{hr}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 22

Express the following quantities in SI units: (a) 160 acres. (b) 15 gallons (U.S.). (c) 240 miles, (d) $79.1 \mathrm{hp}$ (e) $60.3^{\circ} \mathrm{F}$.

Mahnoor Amin

Numerade Educator

Problem 23

Water flows from a large drainage pipe at a rate of 1200 gal/min. What is this volume rate of flow in (a) $m^{3} / s$ (b) liters / $\min ,$ and (c) $\mathrm{ft}^{3} / \mathrm{s} ?$.

Mahnoor Amin

Numerade Educator

Problem 24

The universal gas constant $R_{0}$ is equal to $49,700 \mathrm{ft}^{2} /\left(\mathrm{s}^{2} \cdot^{\circ} \mathrm{R}\right)$ or $8310 \mathrm{m}^{2} /\left(\mathrm{s}^{2} \cdot \mathrm{K}\right) .$ Show that these two magnitudes are equal.

Mahnoor Amin

Numerade Educator

Problem 25

Dimensionless combinations of quantities (commonly called dimensionless parameters) play an important role in fluid mechanics. Make up five possible dimensionless parameters by using combinations of some of the quantities listed in Table 1.1.

Mahnoor Amin

Numerade Educator

Problem 26

An important dimensionless parameter in certain types of fluid flow problems is the Froude number defined as $V / \sqrt{g \ell}$ where $V$ is a velocity, $g$ the acceleration of gravity, and $\ell$ a length. Determine the value of the Froude number for $V=10 \mathrm{ft} / \mathrm{s}$ $g=32.2 \mathrm{ft} / \mathrm{s}^{2},$ and $\ell=2 \mathrm{ft} .$ Recalculate the Froude number using SI units for $V, g,$ and $\ell .$ Explain the significance of the results of these calculations.

Mahnoor Amin

Numerade Educator

Problem 27

Obtain a photograph/image of a situation in which the density or specific weight of a fluid is imporiant. Print this photo and write a brief paragraph that describes the situation involved.

Hast Aggarwal

Numerade Educator

Problem 28

A tank contains $500 \mathrm{kg}$ of a liquid whose specific gravity is 2. Determine the volume of the liquid in the tank.

Mahnoor Amin

Numerade Educator

Problem 29

A stick of butter at $35^{\circ} \mathrm{F}$ measures 1.25 in. $\times 1.25$ in. $x$ 4.65 in. and weighs 4 ounces. Find its specific weight.

Mahnoor Amin

Numerade Educator

Problem 30

Clouds can weigh thousands of pcunds due to their liquid water content. Often this content is measured in grams per cubic meter $\left(\mathrm{g} / \mathrm{m}^{3}\right)$. Assume that a cumulus cloud occupies a volume of one cubic kilometer, and its liquid water content is $0.2 \mathrm{g} / \mathrm{m}^{3}$.
(a) What is the volume of this cloud in cubic miles? (b) How much does the water in the cloud weigh in pourds?

Mahnoor Amin

Numerade Educator

Problem 31

A tank of oil has a mass of 25 slugs. (a) Determine its weight in pounds and in newtons at the Earth's surface. (b) What would be its mass (in slugs) and its weight (in pounds) if located on the moon's surface where the gravitational at raction is approximately one-sixth that at the Earth's surface?

Mahnoor Amin

Numerade Educator

Problem 32

A certain object weighs $300 \mathrm{N}$ at the Earth's surface. Determine the mass of the object (in kilograms) and its weight (in newtons) when Iocated on a planet with an acceleration of gravity equal to $4.0 \mathrm{ft} / \mathrm{s}^{2}$.

Mahnoor Amin

Numerade Educator

Problem 33

The density of a certain type of jet fuel is $775 \mathrm{kg} / \mathrm{m}^{3}$. Determine its specific gravity and specific weight.

Mahnoor Amin

Numerade Educator

Problem 34

At $4^{\circ} \mathrm{C}$ a mixture of automobile antifreeze $(50 \%$ water and $50 \%$ ethylene glycol by volume ) has a density of $1064 \mathrm{kg} / \mathrm{m}^{3}$. If the water density is $1000 \mathrm{kg} / \mathrm{m}^{3}$, find the density of the ethylene glycol.

Mahnoor Amin

Numerade Educator

Problem 35

A hydrometer is used to measure the specific gravity of liquids. (See Video V2.8.) For a certain liquid, a hydrometer reading indicates a specific gravity of $1.15 .$ What is the liquid's dersity and specific weight?Express your answer in SI units.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 36

An open, rigid-walled, cylindrical tank contains $4 \mathrm{ft}^{3}$ of water at $40^{\circ} \mathrm{F}$. Over a 24 -hour period of time the water temperature varies from 40 to $90^{\circ} \mathrm{F}$. Make use of the data in Appendix $\mathrm{B}$ to determine how mach the volume of water will change. For a tank diameter of $2 \mathrm{ft}$, would the corresponding change in water depth be very noticeable? Explain.

Anand Jangid

Numerade Educator

Problem 37

Estimate the number of pounds of mercury it would take to fill your bathtub. List all assumptions and show all calculations.

Mahnoor Amin

Numerade Educator

Problem 38

A mountain climber's oxygen tank contains 1 lb of oxygen when he begins his rip at sea level where the acceleration of gravity is $32.174 \mathrm{ft} / \mathrm{s}^{2}$. What is the weight of the oxygen in the tank when he reaches the top of Mt. Everest where the acceleration of gravity is $32.082 \mathrm{fts}^{2} ?$ Assume that no oxygen has been removed from the tank; it will be used on the descent portion of the climb.

Mahnoor Amin

Numerade Educator

Problem 39

The information on a can of pop indicates that the can contains $355 \mathrm{mL}$. The mass of a full can of pop is $0.369 \mathrm{kg}$, while an empty can weighs 0.153 N. Determine the specific weight, density, and specific gravity of the pop and compare your results with the corresponding values for water at $20^{\circ} \mathrm{C}$. Express your results in SI units.

Mahnoor Amin

Numerade Educator

Problem 40

The variation in the density of water, $\rho,$ with temperature, $T$ in the range $20^{\circ} \mathrm{C} \leq T \leq 50^{\circ} \mathrm{C},$ is given in the following table.
Use these data to determine an empirical equation of the form $\rho=c_{1}+c_{2} T+c_{3} T^{2}$ which can be used to predict the density over the range indicated Compare the predicted values with the data given. What is the censity of water at $42.1^{\circ} \mathrm{C} ?$

Mahnoor Amin

Numerade Educator

Problem 41

If 1 cup of cream having a density of $1005 \mathrm{kg} / \mathrm{m}^{3}$ is turned into 3 cups of whipped cream, determine the specific gravity and specific weight of the whipped cream.

Mahnoor Amin

Numerade Educator

Problem 42

With the exception of the 410 bore, the gauge of a shotgun barrel indicates the number of round lead balls, each having the bore diameter of the barrel, that together weigh 1 Ib. For example, a shotgun is called a 12 -gauge shotgun if a $\frac{1}{12}-1 b$ lead ball fits the bore of the barrel. Find the diameter of a 12 -gauge shotgun in inches and millimeters. Lead has a specific weight of $0.411 \mathrm{lb} / \mathrm{in}^{3}$.

Mahnoor Amin

Numerade Educator

Problem 43

The presence of raindrops in the air during a heavy rainstorm increases the average density of the air-water mixture. Estimate by what percent the average air-water density is greater than that of just still air. State all assumptions and show calculations.

Supratim Pal

Numerade Educator

Problem 44

A regulation basketball is initially flat and is then inflated to a pressure of approximately $24 \mathrm{lb} / \mathrm{in}^{2}$ absolute. Consider the air temperature to be constant at $70^{\circ} \mathrm{F}$. Find the mass of air required to inflate the basketball. The basketball's inside radius is 4.67 in.

Mahnoor Amin

Numerade Educator

Problem 45

Nitrogen is compressed to a density of $4 \mathrm{kg} / \mathrm{m}^{3}$ under an absolute pressure of 400 kPa. Determine the temperature in degrees Celsius.

Mahnoor Amin

Numerade Educator

Problem 46

The temperature and pressure at the surface of Mars during a Martian spring day were determined to be $-50^{\circ} \mathrm{C}$ and $900 \mathrm{Pa}$. respectively.
(a) Determine the density of the Martian atmosphere for these conditions if the gas constant for the Martian atmosphere is assumed to be equivalent to that of carbon dioxide.
(b) Compare the answer from part (a) with the density of the Earth's atmosphere during a spring day when the temperature is $18^{\circ} \mathrm{C}$ and the pressure $101.6 \mathrm{kPa}(\mathrm{abs})$.

Mahnoor Amin

Numerade Educator

Problem 47

A closed tank having a volume of $2 \mathrm{ft}^{3}$ is filled with $0.30 \mathrm{lb}$ of a gas. A pressure gage attached to the tank reads 12 psi when the gas temperature is $80^{\circ} \mathrm{F}$. There is some question as to whether the gas in the tank is oxygen or helium. Which do you think it is? Explain how you arrived at your answer.

Anand Jangid

Numerade Educator

Problem 48

Assume that the air volume in a small automobile tire is constant and equal to the volume between two concentric cylinders $13 \mathrm{cm}$ high with diameters of $33 \mathrm{cm}$ and $52 \mathrm{cm}$. The air in the tire is initially at $25^{\circ} \mathrm{C}$ and $202 \mathrm{kPa}$. Immediately after air is pumped into the tire, the temperature is $30^{\circ} \mathrm{C}$ and the pressure is 303 kPa. What mass of air was added to the tire? What would be the air pressure after the air has cooled to a temperature of $0^{\circ} \mathrm{C} ?$

Hast Aggarwal

Numerade Educator

Problem 49

A compressed air tank contains 5 kg of air at a temperature of $80^{\circ} \mathrm{C} .$ A gage on the tank reads 300 kPa. Determine the volume of the tank.

Mahnoor Amin

Numerade Educator

Problem 50

A rigid tank contains air at a pressure of 90 psia and a temperature of $60^{\circ} \mathrm{F}$. By how much will the pressure increase as the temperature is increased to $110^{\circ} \mathrm{F}^{\prime}$.

Mahnoor Amin

Numerade Educator

Problem 51

The density of oxygen contained in a tank is $2.0 \mathrm{kg} / \mathrm{m}^{3}$ when the temperature is $25^{\circ} \mathrm{C}$. Determine the gage pressure of the gas if the atmospheric pressure is $97 \mathrm{kPa}$.

Mahnoor Amin

Numerade Educator

Problem 52

The helium-filled blimp shown in Fig. P1.52 is used at various athletic events. Determine the number of pounds of helium within it if its volume is $68,000 \mathrm{ft}$ and the temperature and pressure are $80^{\circ} \mathrm{F}$ and 14.2 psia, respectively.

Ajay Singhal

Numerade Educator

Problem 53

Develop a computer progrem for calculating the density of an ideal gas when the gas pressure in pascals (abs), the temperature in degrees Celsius, and the gas constant in $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ are specified. Plot the density of helium as a function of temperature from $0^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$ and pressures of $50,100,150,$ and $200 \mathrm{kPa}(\mathrm{abs})$.

Hast Aggarwal

Numerade Educator

Problem 54

Obtain a photograph/image of a situation in which the viscosity of a fluid is important. Priat this photo and write a brief paragraph that describes the situation involved.

Hast Aggarwal

Numerade Educator

Problem 55

For flowing water, what is the magnitude of the velocity gradient needed to produce a shear stress of $1.0 \mathrm{N} / \mathrm{m}^{2} ?$.

Mahnoor Amin

Numerade Educator

Problem 56

Make use of the data in Appendix $B$ to determine the dynamic viscosity of glycerin at $85^{\circ} \mathrm{F}$. Express your answer in both SI and BG units.

Mahnoor Amin

Numerade Educator

Problem 57

One type of capillary-tube viscometer is shown in Video V1.5 and in Fig. P1.57. For this device the liquid to be tested is drawn into the tube to a level above the top etched line. The time is then obtained for the liquid to drain to the bottom etched line.The kinematic viscosity, $\nu,$ in $\mathrm{m}^{2} / \mathrm{s}$ is then obtained from the equation $\nu=K R^{4} t$ where $K$ is a constant, $R$ is the radius cf the capillary tube in $\mathrm{mm}$, and $t$ is the drain time in seconds. When glycerin at $20^{\circ} \mathrm{C}$ is used as a calibration fluid in a particular viscometer, the drain time is 1430 s. When a liquid having a density of $970 \mathrm{kg} / \mathrm{m}^{3}$ is tested in the same viscometer the drain time is 900 s. What is the dynamic viscosity of this liquid?

Mahnoor Amin

Numerade Educator

Problem 58

The viscosity of a soft drink was determined by using a capillary tube viscometer similar to that shown in Fig. $\mathrm{P} 1.58$ and Video $\mathrm{V} 1.5 .$ For this device the kinematic viscosity, $\nu,$ is directly proportional to the time, $t$, that it takes for a given amount of liquid to flow through a small capillary tube. That is, $\nu=K t$. The following data were obtained from regular pop and diet pop. The corresponding measured specific gravities are also given. Based on these data, by what percent is the absolute viscosity, $\mu,$ of regular pop greater than that of diet pop?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 59

The viscosity of a certain fluid is $5 \times 10^{-4}$ pcise. Determine its viscosity in both SI and BG units.

Mahnoor Amin

Numerade Educator

Problem 60

The kinematic viscosity and specific gravity of a liquid are $3.5 \times 10^{-4} \mathrm{m}^{2} / \mathrm{s}$ and $0.79,$ respectively. What is the dynamic viscosity of the liquid in SI units?

Mahnoor Amin

Numerade Educator

Problem 61

A liquid has a specific weight of $59 \mathrm{lb} / \mathrm{ft}^{3}$ and a dynamic viscosity of $2.75 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$. Determine its kinematic viscosity.

Mahnoor Amin

Numerade Educator

Problem 62

The kirematic viscosity of oxygen at $20^{\circ} \mathrm{C}$ ard a pressure of $150 \mathrm{kPa}(\mathrm{abs})$ is 0.104 stokes. Determine the dynamic viscosity of oxygen at this temperature and pressure.

Mahnoor Amin

Numerade Educator

Problem 63

Fluids for which the shearing stress, $\tau,$ is not linearly related to the rate of shearing strain, $\dot{\gamma},$ are designated as nonNewtonian fluids. Such fluids are commonplace and can exhibit unusual behavior, as shown in Video V1.6. Some experimental data obtained for a particular non-Newtonian fluid at $80^{\circ} \mathrm{F}$ are shown below.
Plot these data and fit a second-order polynomial to the data using a suitable graphing program. What is the apparent viscosity of this fluid when the rate of shearing strain is $70 \mathrm{s}^{-1}$ ? Is this apparent viscosity larger or smaller than that for water at the same temperature?

Hast Aggarwal

Numerade Educator

Problem 64

$1.64 \Leftrightarrow$ Water flows near a flat surface and some measurements of the water velocity, $u,$ parallel to the surface, at different heights,
$y,$ sbove the surface are obtained. At the surface $y=0 .$ After an analysis of the data, the lab technician reports that the velocity distribution in the range $0<y<0.1 \mathrm{ft}$ is given by the equation
\[
u=0.81+9.2 y+4.1 \times 10^{3} y^{3}
\]
with $u$ in $\mathrm{ft} / \mathrm{s}$ when $y$ is in $\mathrm{ft}$
(a) Do you think that this equation would be valid in any system of units? Explain.
(b) Do you think this equation is correct? Explain. You may want to look at Video 1.4 to help you arrive at your answer.

Mahnoor Amin

Numerade Educator

Problem 65

Calculate the Reynolds numbers for the flow of water and for air through a 4-mm-diameter tube, if the mean velocity is $3 \mathrm{m} / \mathrm{s}$ and the temperature is $30^{\circ} \mathrm{C}$ in both cases (see Example 1.4 ). Assume the air is at standard atmospheric pressure.

Mahnoor Amin

Numerade Educator

Problem 66

SAE 30 oil at $60^{\circ} \mathrm{F}$ flows through a 2 -in.-diameter pipe with a mean velocity of $5 \mathrm{ft} / \mathrm{s}$. Determine the value of the Reynolds number (see Example 1.4 ).

Ma Ednelyn Lim

Numerade Educator

Problem 67

For air at standard atmospheric pressure the values of the corstants that appear in the Sutherland equation (Eq. 1.10 ) are $C=1.458 \times 10^{-6} \mathrm{kg} /\left(\mathrm{m} \cdot \mathrm{s} \cdot \mathrm{K}^{1 / 2}\right)$ and $S=110.4 \mathrm{K} .$ Use these values to predict the viscosity of air at $10^{\circ} \mathrm{C}$ and $90^{\circ} \mathrm{C}$ and $\mathrm{com}$ pare with values given in Table B.4 in Appendix B.

Mahnoor Amin

Numerade Educator

Problem 68

4.63 Use the values of viscosity of air given in Table $B .4$ at temperatures of $0,20,40,60,80,$ and $100^{\circ} \mathrm{C}$ to determine the constants $C$ and $S$ which appear in the Sutherland equation (Eq. 1.10 ). Compare your results with the values given in Problem 1.67 . (Hint: Rewrite the equation in the form
\[
\frac{T^{3 / 2}}{\mu}=\left(\frac{1}{C}\right) T+\frac{S}{C}
\]
and plot $T^{3 / 2} / \mu$ versus $T .$ From the slope and intercept of this curve, $C$ and $S$ can be obtained.

Hast Aggarwal

Numerade Educator

Problem 69

The viscosity of a fluid plays a very important role in determining how a fluid flows. (See Video $V 1.3$.) The value of the viscosity depends not only on the specific fluid but also on the fluid temperature. Some experiments show that when a liquid, under the action of a constant driving pressure, is forced with a low velocity, $V$, through a small horizontal tube, the velocity is given by the equation $V=K / \mu .$ In this equation $K$ is a constant for a given tube and pressure, and $\mu$ is the dynamic viscosity. For a particular liquid of interest, the viscosity is given by Andrade's equation (Eq. 1.11 ) with $D=5 \times 10^{-7} \mathrm{Ib} \cdot \mathrm{s} / \mathrm{ft}^{2}$ and $B=4 \mathrm{COO}^{\circ} \mathrm{R} . \mathrm{By}$ what percentage will the velocity increase as the liquid temperature is increased from $40^{\circ} \mathrm{F}$ to $100^{\circ} \mathrm{F}$ ? Assume all other factors remain constant.

Mahnoor Amin

Numerade Educator

Problem 70

Use the value of the viscosity of water given in Table $B .2$ at temperatures of $0,20,40,60,80,$ and $100^{\circ} \mathrm{C}$ to determine the constants $D$ and $B$ which appear in Andrade's equation (Eq. 1.11 ).
Calculate the value of the viscosity at $50^{\circ} \mathrm{C}$ and compare with the value given in Table B.2. (Hint: Rewrite the equation in the form
\[
\ln \mu=(B) \frac{1}{T}+\ln D
\]
and plot $\ln \mu$ versus $1 / T .$ From the slope and intercept of this curve, $B$ and $D$ can be obtained. If a nonlinear curve-fitting program is available, the constants can be obtained directly from Eq. 1.11 without rewriting the equation.)

Hast Aggarwal

Numerade Educator

Problem 71

For a certain liquid $\mu=7.1 \times 10^{-5} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$ at $40^{\circ} \mathrm{F}$ and $\mu=$
$1.9 \times 10^{-5} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$ at $150^{\circ} \mathrm{F}$. Make use of these data to determine
the constants $D$ and $B$ which appear in Andrade's equation (Eq. $1.11) .$ What would be the viscosity at $80^{\circ} \mathrm{F}$ ?

Mahnoor Amin

Numerade Educator

Problem 72

For a parallel plate arrangement of the type shown in Fig. 1.5 it is found that when the distance between plates is $2 \mathrm{mm}$, a shearing stress of 150 Pa develops at the upper plate when it is pulled at a velocity of $1 \mathrm{m} / \mathrm{s}$. Determine the viscosity of the fluid between the plates. Express your answer in SI units.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

Two flat plates are oricnted parallel above a fixed lower plate as shown in Fig. P1.73. The top plate, located a distance $b$ above the fixed plate, is pulled along with speed $V$. The other thin plate is located a distance $c b,$ where $0<c<1,$ above the fixed plate. This plate moves with speed $V_{1},$ which is determined by the viscous shear forces imposed on it by the fluids on its top and bottom. The fluid on the top is twice as viscous as that on the bottom. Plot the ratio $V_{1} / V$ as a function of $c$ for $0<c<1$.

Hast Aggarwal

Numerade Educator

Problem 74

Three large plates are separated by thin layers of ethylene glycol and water, as shown in Fig. P1.74. The top plate moves to the right at $2 \mathrm{m} / \mathrm{s}$. At what speed and in what direction must the bottom plate be moved to hold the center plate stationary?

Prashant Bana

Numerade Educator

Problem 75

There are many fluids that exhibit non-Newtonian behavior (see, for example, Video $\vee 1.6$ ). For a given fluid the distinction between Newtonian and non-Newtonian behavior is usually based on measurements of shear stress and rate of shearing strain. Assume that the viscosity of blood is to be determined by measurements of shear stress, $t,$ and rate of shearing strain, $d u / d y,$ obtained from a small blood sample tested in a suitable viscometer. Based on the data given below, determine if the blood

Hast Aggarwal

Numerade Educator

Problem 76

The sled shown in Fig. $P 1.76$ slides along on a thin horizontal layer of water between the ice and the runners. The horizontal force that the water puts on the runners is equal to $1.2 \mathrm{lb}$ when the sled's speed is $50 \mathrm{ft} / \mathrm{s}$. The total area of both runners in contact with the water is $0.08 \mathrm{ft}^{2}$, and the viscosity of the water is $3.5 \times 10^{-5} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$ Determine the thickness of the water layer under the runners. Assume a linear velocity distribution in the water layer.

Mahnoor Amin

Numerade Educator

Problem 77

A 25 -mm-diameter shaft is pulled through a cylindrical bearing as shown in Fig. $P 1.77 .$ The lubricant that fills the $0.3-\mathrm{mm}$ gap between the shaft and bearing is an oil having a kinematic viscosity of $8.0 \times 10^{-4} \mathrm{m}^{2} / \mathrm{s}$ and a specific gravity of $0.91 .$ Determine the force $P$ required to pull the shaft at a velocity of $3 \mathrm{m} / \mathrm{s}$. Assume the velocity distribution in the gap is linear.

Ajay Singhal

Numerade Educator

Problem 78

A hydraulic lift in a service station has a 32.50 -cm-diameter ram that slides in a 32.52 -cm-diameter cylinder. The annular space is filled with SAE 10 oil at $20^{\circ} \mathrm{C}$. The ram is traveling upward at the rate of $0.10 \mathrm{m} / \mathrm{s}$. Find the frictional force when $3.0 \mathrm{m}$ of the ram is engaged in the cylinder.

Mahnoor Amin

Numerade Educator

Problem 79

A piston having a diameter of 5.48 in. and a length of 9.50 in. slides downward with a velocity $V$ through a vertical pipe. The downward motion is resisted by an oil film between the piston and the pipe wall. The film thickness is 0.002 in., and the cylinder weighs 0.5 lb. Estimate $V$ if the oil viscosity is $0.016 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$ Assume the velocity distribution in the gap is linear.

Nicholas Mogoi

Numerade Educator

Problem 80

A 10 -kg block slides down a smooth inclined surface as shown in Fig. $P 1.80 .$ Determine the terminal velocity of the block if the 0.1 -mm gap between the block and the surface contains SAE 30 oil at $60^{\circ} \mathrm{F}$. Assume the velocity distribution in the gap is linear, and the area of the block in contact with the oil is $0.1 \mathrm{m}^{2}$.

Mahnoor Amin

Numerade Educator

Problem 81

A layer of water flows down an inclined fixed surface with the velocity profile shown in Fig. $\mathrm{P} 1.81 .$ Determine the magnitude and direction of the shearing stress that the water exerts on the fixed surface for $U=2 \mathrm{m} / \mathrm{s}$ and $h=0.1 \mathrm{m}$.

Prashant Bana

Numerade Educator

Problem 82

Oil (absolute viscosity $=0.0003 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$, density $=50$ Ibm/ft' $^{3}$ flows in the boundary layer, as shown in Fig. P1.82. The plate is $1 \mathrm{ft}$ wide perpendicular to the paper. Calculate the shear stress at the plate surface.

Mahnoor Amin

Numerade Educator

Problem 83

Standard air flows past a flat surface, and velocity measurements near the surface indicate the following distribution:
The coordinate $y$ is measured normal to the surface and $u$ is the velocity parallel to the surface. (a) Assume the velocity distribution is of the form
\[
u=C_{1} y+C_{2} y^{3}
\]
and use a standard curve-fitting technique to determine the constants $C_{1}$ and $C_{2}$. (b) Make use of the results of pert (a) to determine the magnitude of the shearing stress at the wall $(y=0)$ and
\[
\text { at } y=0.05 \mathrm{fl}
\]

Hast Aggarwal

Numerade Educator

Problem 84

1.84 A new computer drive is proposed to have a disc, as shown in Fig. $P 1.84 .$ The disc is to rotate at $10,000 \mathrm{rpm},$ and the reader head is to be positioned 0.0005 in. above the surface of the disc. Estimate the shearing force on the reader head as a result of the air between the disc and the head.

Mahnoor Amin

Numerade Educator

Problem 85

The space between two 6 -in.-long concentric cylinders is filled with glycerin (viscosity $\left.=8.5 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}\right) .$ The inner cylnder has a radius of 3 in. and the gap width between cylinders is $C .1$ in. Determine the torque and the power required to rotate the inner cylinder at 180 rev / min. The outer cylinder is fixed. Assume the velocity distribution in the gap to be linear.

Mahnoor Amin

Numerade Educator

Problem 86

A pivot bearing used on the shaft o an electrical instrument is shown in Fig. $P 1.86 .$ An oil with a viscosity of $\mu=0.010 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}$ fills the 0.001 -in. gap between the rotating shaft and the stationary base. Determine the frictional torque on the shaft when it rotates at 5000 rpm.

Rashmi Sinha

Numerade Educator

Problem 87

The viscosity of liquids can be measured through the use of a rotating cylinder viscometer of the type illustrated in Fig. $\mathrm{P} 1.87$ In this device the outer cylinder is fixed and the inner cylinder is rotated with an angular velocity, $\omega$. The torque $\mathscr{T}$ required to develop $\omega$ is measured and the viscosity is calculated from these two measurements.
(a) Develop an equation relating $\mu, \omega, \mathscr{I}, \ell, R_{o}$ and $R_{i}$ Neglect end effects and assume the velocity distribution in the gap is linear.
(b) The following torque-angular velocity data were obtained with a rotating cylinder viscometer of the type discussed in part (a).
For this viscometer $R_{o}=2.50$ in., $R_{t}=2.45$ in., and $\ell=5.00$ in. Make use of these data and a standard curve-fitting program to determine the viscosity of the liquid contained in the viscometer.

Hast Aggarwal

Numerade Educator

Problem 88

The concentric cylinder viscometer shown in Fig. $\mathrm{Pl} .88$ has a cylinder height of $10.0 \mathrm{cm},$ a cylinder radius of $3.0 \mathrm{cm},$ and a uniform gap between the cylinder and the container (botiom and sides of $0.10 \mathrm{cm} .$ The pulley has a radius of $3.0 \mathrm{cm} .$ Determine the weight required to produce a constant rotational speed of 30 rpm if the gap is filled with: (a) water, (b) gasoline, (c) glycerin.

Anand Jangid

Numerade Educator

Problem 89

A 12 -in.-diameter circular plate is placed over a fixed bottom plate with a 0.1 -in. gap between the two plates filled with glycerin as shown in Fig. $\mathrm{P} 1.89$. Determine the torque required to rotate the circular plate slowly at 2 rpm. Assume that the velocity distribution in the gap is linear and that the shear stress on the edge of the rotating plate is negligible.

Mahnoor Amin

Numerade Educator

Problem 90

Vehicle shock absorbers damp out oscillations caused by road roughness. Describe how a temperature change may affect the operation of a shock absorber.

Hast Aggarwal

Numerade Educator

Problem 91

Some measurements on a blood sample at $37^{\circ} \mathrm{C}\left(98.6^{\circ} \mathrm{F}\right)$ indicate a shearing stress of $0.52 \mathrm{N} / \mathrm{m}^{2}$ for a corresponding rate of shearing strain of $200 \mathrm{s}^{-1}$. Determine the apparent viscosity of the blood and compare it with the viscosity of water at the same temperature.

Mahnoor Amin

Numerade Educator

Problem 92

Obtain a photograph/image of a situation in which the compressibility of a fluid is important. Print this photo and write a brief paragraph that describes the situation involved.

Hast Aggarwal

Numerade Educator

Problem 93

A sound wave is observed to travel through a liquid with a speed of $1500 \mathrm{m} / \mathrm{s}$. The specific gravity of the liquid is 1.5 Determine the bulk modulus for this fluid.

Vipender Yadav

Numerade Educator

Problem 94

A rigid-walled cubical container is completely filled with water at $40^{\circ} \mathrm{F}$ and sealed. The water is then heated to $100^{\circ} \mathrm{F}$ Determine the pressure that develops in the container when the water reaches this higher temperature. Assume that the volume of the container remains constant and the value of the bulk modulus of the water remains constant and equal to 300,000 psi.

Ajay Singhal

Numerade Educator

Problem 95

Estimate the increase in pressure (in psi) required to decrease a unit volume of mercury by $0.1 \%$.

Mahnoor Amin

Numerade Educator

Problem 96

$\mathrm{A} 1-\mathrm{m}^{3}$ volume of water is contained in a rigid container. Estimate the change in the volume of the water when a piston applies a pressure of 35 MPa.

Mahnoor Amin

Numerade Educator

Problem 97

Determine the speed of sound at $20^{\circ} \mathrm{C}$ in (a) air, (b) helium, and (c) natural gas (methane). Express your answer in $\mathrm{m} / \mathrm{s}$.

Mahnoor Amin

Numerade Educator

Problem 98

Air is enclosed by a rigid cylinder containing a piston. $A$ pressure gage attached to the cylinder indicates an initial reading of 25 psi. Determine the reading on the gage when the piston has compressed the air to one-third its original volume. Assume the compression process to be isothemal and the local atmospheric pressure to be 14.7 psi.

Mahnoor Amin

Numerade Educator

Problem 99

Repeat Problem 1.98 if the compression process takes place without friction and without heat transfer (isentropic process).

Mahnoor Amin

Numerade Educator

Problem 100

Carbon dioxide at $30^{\circ} \mathrm{C}$ and 300 kPa absolute pressure expands isothermally to an absolute pressure of $165 \mathrm{kPa}$ Determine the final density of the gas.

Mahnoor Amin

Numerade Educator

Problem 101

Natural gas at $70^{\circ} \mathrm{F}$ and standard atmospheric pressure of 14.7 psi (abs) is compressed isentropically to a new absolute pressure of 70 psi. Determine the final density and temperature of the gas.

Mahnoor Amin

Numerade Educator

Problem 102

A compressed air tank in a service station has a volume of $10 \mathrm{ft}^{3} .$ It contains air at $70^{\circ} \mathrm{F}$ and 150 psia. How many tubeless tires can it fill to 44.7 psia at $70^{\circ} \mathrm{F}$ if each tire has a volume of 1.5 $\mathrm{ft}^{3}$ and the compressed air tank is not refilled? The tank air temperature remains constant at $70^{\circ} \mathrm{F}$ because of heat transfer through the tank's large surface area.

Anand Jangid

Numerade Educator

Problem 103

Oxygen at $30^{\circ} \mathrm{C}$ and $300 \mathrm{kPa}$ absolute pressure expands isothermally to an absolute pressure of 120 kPa. Determine the final density of the gas.

Mahnoor Amin

Numerade Educator

Problem 104

Compare the isentropic bulk modulus of air at $101 \mathrm{kPa}(\mathrm{abs})$ with that of water at the same pressure.

Mahnoor Amin

Numerade Educator

Problem 105

Develop a computer program for calculating the final gage pressure of gas when the initial gage pressure, initial and final volumes, atmospheric pressure, and the type of procass (isothermal or isentropic are specified. Use $\mathrm{BG}$ units. Check your program against the results obtained for Problem 1.98.

Hast Aggarwal

Numerade Educator

Problem 106

Often the assumption is made that the flow of a certain fluid can be considered as incompressible flow if the density of the fluid changes by less than $2 \%$. If air is flowing through a tube such that the air pressure at one section is 9.0 psi and at a downstream section it is 8,6 psi at the same temperature, do you think that this flow could be considered an incompressible flow? Support your answer with the necessary calculations. Assume standard atmospheric pressire.

Mahnoor Amin

Numerade Educator

Problem 107

An important dimensionless parameter concerned with very high-speed flow is the Mach number, defined as $V / c$, where $V$ is the speed of the object such as an airplane or projectile, and $c$ is the speed of sound in the fluid surrounding the object. For a projectile traveling at 800 mph through air at $50^{\circ} \mathrm{F}$ and standard atmospheric pressure, what is the value of the Mach number?

Mahnoor Amin

Numerade Educator

Problem 108

The "power available in the wind" of velocity $V$ through an area $A$ is
\[
\dot{W}=\frac{1}{2} \rho A V^{3}
\]
where $\rho$ is the air density $\left(0.075 \mathrm{lbm} / \mathrm{ft}^{3}\right) .$ For an 18 -mph wind, find the wind area $A$ that will supply a power of 4 hp.

Mahnoor Amin

Numerade Educator

Problem 109

1.109 Air enters the converging nozzle shown in Fig. P1.72 at $T_{1}=70^{\circ} \mathrm{F}$ ard $V_{1}=50 \mathrm{ft} / \mathrm{s} .$ At the exit of the nozzle, $V_{2}$ is given by
\[
V_{2}=\sqrt{V_{1}^{2}+2 c_{p}\left(T_{1}-T_{2}\right)}
\]
where $c_{p}=187 \mathrm{ft} \cdot \mathrm{lb} / \mathrm{lbm} \cdot^{\circ} \mathrm{F}$ and $T_{2}$ is the air temperature at
the exit of the nozzle, Find the temperature $T_{2}$ for which $V_{2}=$
\[
1000 \mathrm{ftfs}
\]

Mahnoor Amin

Numerade Educator

Problem 110

(See The Wide World of Fluids article titled "This water jet is a blast," Section $1.7 .1 .)$ By what percent is the volume of water decreased if its pressure is increased to an equivalent to 3000 atmospheres $(44,100 \mathrm{psi}) ?$

Mahnoor Amin

Numerade Educator

Problem 111

During a mountain climbing trip it is observed that the water used to cook a meal boils at $90^{\circ} \mathrm{C}$ rather than the standard $100^{\circ} \mathrm{C}$ at sea level. At what altitude are the climbers preparing their meal? (See Tables $B .2$ and $C .2$ for data needed to solve this problem.)

Mahnoor Amin

Numerade Educator

Problem 112

When a fluid flows through a sharp bend, low pressures may develop in localized regions of the bend. Estimate the minimum absolute pressure (in psi) that can develop without causing cavitation if the fluid is water at $160^{\circ} \mathrm{F}$.

Anand Jangid

Numerade Educator

Problem 113

A partially filled closed tank contains ethyl alcohol at $68^{\circ} \mathrm{F}$ If the air above the alcohol is evacuated, what is the minimum absolute pressure that develops in the evazuated space?

Mahnoor Amin

Numerade Educator

Problem 114

Estimate the minimum absolute ressure (in pascals) that can be developed at the inlet of a pump to avoid cavitation if the fluid is carbon tetrachloride at $20^{\circ} \mathrm{C}$.

Mahnoor Amin

Numerade Educator

Problem 115

When water at $70^{\circ} \mathrm{C}$ flows through a converging section of pipe, the pressure decreases in the direction of flow. Estimate the minimum absolute pressure that can develop without causing cavitation. Express your answer in both BG and SI units.

Mahnoor Amin

Numerade Educator

Problem 116

At what atmospheric pressure will water boil at $35^{\circ} \mathrm{C} ?$ Express your answer in both SI and BG units.

Hast Aggarwal

Numerade Educator

Problem 117

Obtain a photograph/image of a situation in which the surface tension of a fluid is important. Print this photo and write a brief paragraph that describes the situation involved.

Surendra Kumar

Numerade Educator

Problem 118

When a 2 -mm-diameter tube is inserted into a liquid in an open tank, the liquid is observed to rise $10 \mathrm{mm}$ above the free surface of the liquid (see Video $V 1.10$ ). The contact angle between the liquid and the tube is zero, and the specific weight of the liquid is $1.2 \times 10^{4} \mathrm{N} / \mathrm{m}^{3}$. Determine the value of the surface tension for this liquid.

Mahnoor Amin

Numerade Educator

Problem 119

A soda straw with an inside diameter of 0.125 in. is inserted into a pan of water at $60^{\circ} \mathrm{F}$. The water in the straw rises to a height of 0.150 in. above the water surface in the pan. Determine the value of $\theta,$ the contact angle of the water with the straw (see Fig. 1.10 ).

Mahnoor Amin

Numerade Educator

Problem 120

Small droplets of carbon tetrachloride at $68^{\circ} \mathrm{F}$ are formed with a spray nozzle. If the average dianeter of the droplets is $200 \mu \mathrm{m},$ what is the difference in pressure between the inside and outside of the droplets?

Mahnoor Amin

Numerade Educator

Problem 121

A 12 -mm-diameter jet of water cischarges vertically into the atmosphere. Due to surface tension the pressure inside the jet will be slightly higher than the surrounding atmospheric pressure. Determine this difference in pressure.

Hast Aggarwal

Numerade Educator

Problem 122

A method used to determine the surface tension of a liquid is to determine the force necessary to raise a wire ring through the air-liquid interface. What is the value of the surface tension if a force of $0.015 \mathrm{N}$ is required to raise a 4 -cm-diameter ring? Consider the ring weightless, as a tensiometer (used to measure the surface tension) "zeroes" out the ring weight.

Hast Aggarwal

Numerade Educator

Problem 123

Calculate the pressure difference between the inside and outside of a spherical water droplet having a diameter of $\frac{1}{32}$ in. and a temperature of $50^{\circ} \mathrm{F}$.

Mahnoor Amin

Numerade Educator

Problem 124

As shown in Video $V 1.9,$ surface tension forces can be strong enough to allow a double-edge steel razor blade to "float" on water, but a single-edge blade will sink. Assume that the surface tension forces act at an angle $\theta$ relative to the water surface as shown in Fig. P1.124. (a) The mass of the double-edge blade is $0.64 \times 10^{-3} \mathrm{kg},$ and the total length of its sides is $206 \mathrm{mm}$. Determine the value of $\theta$ required to maintain equilibrium between the blade waight and the resultant surface tension force. (b) The mass of the single-edge blade is $2.61 \times 10^{-3} \mathrm{kg},$ and the total length of its sides is $154 \mathrm{mm}$

Mahnoor Amin

Numerade Educator

Problem 125

Explain how sweat soldering of copper pipe works from a fluid mechanics viewpoint.

Hast Aggarwal

Numerade Educator

Problem 126

Under the right conditions, it is possible, due to surface tension, to have metal objects float on water. (See Video $\vee 1.9$.) Consider placing a short length of a small diameter steel $(\gamma=490$ $\left.\mathrm{Ib} / \mathrm{ft}^{3}\right)$ rod on a surface of water. What is the maximum diameter that the rod can have before it will sink? Assume that the surface tension forces act vertically upward. Note: A standard peper clip has a diameter of 0.036 in. Partially unfold a paper clip and see if you can get it to fleat on water. Do the results of this experiment support your analysis?

Hast Aggarwal

Numerade Educator

Problem 127

An open, clean glass tube, having a diameter of $3 \mathrm{mm}$ is inserted vertically into a dish of mercury at $20^{\circ} \mathrm{C}$ (see Video V1.10). How far will the column of mercury in the tube be depressed?

Mahnoor Amin

Numerade Educator

Problem 128

An oper, clean glass tube $\left(\theta=0^{\circ}\right)$ is inserted vertically into a pan of water (see Video $\mathrm{V} 1.10$ ). What tube diameter is needed if the water level in the tube is to rise one tube diameter (due to surface tension)?

Mahnoor Amin

Numerade Educator

Problem 129

Determine the height that water at $60^{\circ} \mathrm{F}$ will rise due to capillary action in a clean, $\frac{1}{4}$ -in.-diameter tube (see Video $\vee 1.10$ ). What will be the height if the diameter is reduced to 0.01 in.?

Mahnoor Amin

Numerade Educator

Problem 130

Two vertical, parallel, clean glass plates are spaced a distance of $2 \mathrm{mm}$ apart. If the plates are placed in water, how high will the water rise between the plates due to capillary action?

Hast Aggarwal

Numerade Educator

Problem 131

(See The Wide World of Fluids article titled "Walking on water," Section $1.9 .$ ) (a) The water strider bug shown in Fig. $P 1.131$ is supported on the surface of a pond by surface tension acting along the interface between the water and the bug's legs. Determine the minimum length of this interface needed to support the bug. Assume the bug weighs $10^{-4} \mathrm{N}$ and the surface tension force acts vertically upwards. (b) Repeat part (a) if surface tension were to support a person weighing $750 \mathrm{N}$.

Mahnoor Amin

Numerade Educator