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Fundamentals of Fluid Mechanics

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

Chapter 9

Flow Over Immersed Bodies - all with Video Answers

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Chapter Questions

02:45

Problem 1

Assume that water flowing past the equilateral triangular bar shown in Fig. $\mathrm{P} 9.1$ produces the pressure distributions indicated. Determine the lift and drag on the bar and the corresponding lift and drag coefficients (based on frontal area). Neglect shear forces.

Narayan Hari
Narayan Hari
Numerade Educator
03:19

Problem 2

Fluid flows past the two-dimensional bar shown in Fig. P9.2. The pressures on the ends of the bar are as shown, and the average shear stress on the top and bottom of the bar is $\tau_{\mathrm{avg}}$. Assume that the drag due to pressure is equal to the drag due to viscous cffects.
(a) Determine $\tau_{\text {ave }}$ in terms of the dynamic pressure, $\rho U^{2} / 2$
(b) Determine the drag coefficient for this object.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
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Problem 3

Repeat Problem 9.1 if the object is a cone (made by rotating the equilateral triangle about the horizontal axis through its tip) rather than a triangular bar.

Victor Salazar
Victor Salazar
Numerade Educator
03:43

Problem 4

The average pressure and shear stress acting on the surface of the 1 -m-square flat plate are as indicated in Fig. $P 9.4$. Determine the lift and drag generated. Determine the lift and drag if the shear stress is neglected, Compare these two sets of results.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 5

The pressure distribution on the 1 -m-diameter circular disk in Fig. $\mathrm{P} 9.5$ is given in the table. Determine the drag on the disk.

Victor Salazar
Victor Salazar
Numerade Educator
01:55

Problem 6

A nonspinning ball having a mass of 3 oz. is thrown vertically upward with a velocity of 100 mph and has zero velocity at a height 250 ft above the release point. Assume that the air drag on the ball is constant and find this constant "average" air drag. Neglect the buoyant force of air on the ball.

Narayan Hari
Narayan Hari
Numerade Educator
04:11

Problem 7

When you walk through still air at a rate of $1 \mathrm{m} / \mathrm{s}$, would you expect the character of the airflow around you to be most like that depicted in Fig. $9.6 a, b, \text { or } c ? \text { Explain. (See Video } 9.3 .)$

Averell Hause
Averell Hause
Carnegie Mellon University
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Problem 8

A 0.10 -m-diameter circular cylinder moves through air with a speed $U$. The pressure distribution on the cylinder's surface is approximated by the three straight-line segments shown in Fig.
P9.8. Determine the drag coefficient on the cylinder. Neglect shear forces.

Victor Salazar
Victor Salazar
Numerade Educator
02:01

Problem 9

Typical values of the Reynolds number for various animals moving through air or water are listed below, For which cases is inertia of the fluid important? For which cases do viscous effects dominate? For which cases would the flow be laminar; turbulent? Explain. .
$$\begin{array}{lcc}
\text { Animal } & \text { Speed } & \text { Re } \\
\hline \text { (a) large whale } & 10 \mathrm{m} / \mathrm{s} & 300,000,000 \\
\text { (b) flying duck } & 20 \mathrm{m} / \mathrm{s} & 300,000 \\
\text { (c) large dragonfly } & 7 \mathrm{m} / \mathrm{s} & 30,000 \\
\text { (d) invertebrate larva } & 1 \mathrm{mm} / \mathrm{s} & 0.3 \\
\text { (e) bacterium } & 0.01 \mathrm{mm} / \mathrm{s} & 0.00003
\end{array}$$

Dominador Tan
Dominador Tan
Numerade Educator
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Problem 10

Estimate the Reynolds numbers associated with the follow-
objects moving through water. (a) a kayak, (b) a minnow,
(c) a submarine
(d) a grain of sand settling to the bottom,
(e) you swimming. Are viscous cr inertial effects more dominant for each case?

Victor Salazar
Victor Salazar
Numerade Educator
01:11

Problem 11

Approximately how fast can the wind blow past a 0.25 -in. diameter twig if viscous effects are to be of importance throughout the entire flow field (i.e., $\operatorname{Re}<1$ )? Explain. Repeat for a 0.004 -in. diameter tair and a 6 -fi-diameter smokestack.

Narayan Hari
Narayan Hari
Numerade Educator
01:24

Problem 12

Consider the following cases. (a) A small 0.6 -in.-long fish swims with a speed of 0.8 in/s. Would a boundary layer type flow be developed along the sides of the fish? Explain.
(b) A $12-f t$ -long kayak moves with a speed of $5 \mathrm{ft} / \mathrm{s}$. Would a boundary layer type flow be developed along the sides of the boat? Explain.

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 13

Water flows past a flat plate that is oriented parallel to the flow with an upstream velocity of $0.5 \mathrm{m} / \mathrm{s}$. Determine the approximate location downstream from the leading edge where the boundary layer becomes turbulent. What is the boundary layer thickness at this location?

Narayan Hari
Narayan Hari
Numerade Educator
02:04

Problem 14

A viscous fluid flows past a flat plate such that the boundary layer thickness at a distance $1.3 \mathrm{m}$ from the leading edge is $12 \mathrm{mm}$. Determine the boundary layer thickness at distances of 0.20,2.0 and $20 \mathrm{m}$ from the leading edge. Assume laminar flow.

Narayan Hari
Narayan Hari
Numerade Educator
01:02

Problem 15

If the upstream velocity of the flow in Problem 9.14 is $U=$ $1.5 \mathrm{m} / \mathrm{s},$ determine the kinematic viscosity of the fluid.

Narayan Hari
Narayan Hari
Numerade Educator
03:04

Problem 16

Water flows past a flat plate with an upstream velocity of $U=0.02 \mathrm{m} / \mathrm{s} .$ Determine the water velocity a distance of $10 \mathrm{mm}$ from the plate at distances of $x=1.5 \mathrm{m}$ and $x=15 \mathrm{m}$ from the Ieading edge.

Narayan Hari
Narayan Hari
Numerade Educator
02:21

Problem 17

The typical shape of small cumulus clouds is as indicated in Fig. $P^{\prime}$, $17 .$ Based on boundary layer ideas, explain why it is clear that the wind is blowing from right to left as indicated.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:08

Problem 18

Because of the velocity deficit, $U-u,$ in the boundary layer, the streamlines for flow past a flat plate are not exactly parallel to the plate. This deviation can be determined by use of the displacement thickness, $\delta^{*}$. For air blowing past the flat plate shown in Fig. P9.18, plot the streamline $A-B$ that passes through the edge of the boundary layer $\left(y=\delta_{B} \text { at } x=\ell\right)$ at point $B$. That is, plot $y=y(x)$ for streamline $A-B$. Assume laminar boundary layer flow.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 19

Air enters a square duct through a 1 -ft opening as is shown in Fig $P 9.19 .$ Because the boundary layer displacement thickness increases in the direction of flow, it is necessary to increase the cross-sectional size of the duct if a constant $U=2$ ft/s velocity is to be maintained outside the boundary layer, Plot a graph of the duct size, $d,$ as a function of $x$ for $0 \leq x \leq 10$ ft if $U$ is to remain constant. Assume laminar flow.

Victor Salazar
Victor Salazar
Numerade Educator
02:33

Problem 20

A smooth, flat plate of length $\ell=6 \mathrm{m}$ and width $b=4 \mathrm{m}$ is placed in water with an upstream velocity of $U=0.5 \mathrm{m} / \mathrm{s}$. Determine the boundary layer thickness and the wall shear stress at the center and the trailing edge of the plate. Assume a laminar When layer

Narayan Hari
Narayan Hari
Numerade Educator
01:30

Problem 21

An atmospheric boundary layer is formed when the wind blows over the Earth's surface. Typically, such velocity profiles can be written as a power law: $u=a y^{n},$ where the constants $a$ and $n$ depend on the roughness of the terrain. As is inticated in Fig. P9.21, typical values ere $n=0.40$ for urban areas, $n=0.28$ for woodland or suburban areas, and $n=0.16$ for fla open country (Ref. 23 ). (a) If the velocity is $20 \mathrm{ft} / \mathrm{s}$ at the botton of the sail on your boat $(y=4 \mathrm{ft}),$ what is the velocity at the top of the mast $(y=$ $30 \mathrm{ft}$ ? (b) If the average velocity is $10 \mathrm{mph}$ on the tenth floor of an urban building, what is the average velocity on the sixtieth floor?

Narayan Hari
Narayan Hari
Numerade Educator
02:10

Problem 22

A 30 -story office building (each story is $12 \mathrm{ft}$ tall) is built in
suburban industrial park. Plot the dynamic pressure, $\rho u^{2} / 2$ as
function of elevation if the wind blows at huricane strength $(75 \mathrm{mph})$ at the top of the building. Use the atmospheric boundary layer information of Problem 9.21

Narayan Hari
Narayan Hari
Numerade Educator
00:57

Problem 23

Show that by writing the velocity in terms of the similarity variable $n$ and the function $f(\eta),$ the momentum equation for boundary layer flow on a flat plate (Eq. 9.9b) can be written as the ordinary differential equation given by Eq. 9.14.

James Kiss
James Kiss
Numerade Educator
02:39

Problem 24

Integrate the Blasius equation (Eq, 9.14) aumerically to determine the boundary layer profile for laminar flow past a flat plate. Compare your results with those of Table 9.1

Chai Santi
Chai Santi
Numerade Educator
02:17

Problem 25

An airplane flies at a speed of $400 \mathrm{mph}$ at an altitude of $10,000 \mathrm{ft}$. If the boundary layers on the wing surfaces behave as those on a flat plate, estimate the extent of laminar boundary layer flow along the wing. Assume a transitional Reynelds number of $\mathrm{Re}_{\mathrm{xcr}}=5 \times 10^{5} .$ If the airplane maintains its 400 -nph speed but descends to sea-level elevation, will the portion of the wing covered by a laminar boundary layer increase or decrease compared with its value at $10,000 \mathrm{ft}$ ? Explain.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 26

If the boundary layer on the hood of your car behaves as one on a flat plate, estimate how far from the front edge of the hood the boundary layer becomes turbulent. How thick is the boundary layer at this location?

Victor Salazar
Victor Salazar
Numerade Educator
01:14

Problem 27

A laminar boundary layer velocity profile is approximated by $u / U=[2-(y / \delta)](y / \delta)$ for $y \leq \delta,$ and $u=U$ for $y>\delta$
(a) Show that this parabolic profile satisfies the appropriate boundary conditions.
(b) Use the momentum integral equation to determine the boundary layer thickness, $\delta=\delta(x)$. Compare the result with the exact Blasius solution.

Dominador Tan
Dominador Tan
Numerade Educator
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Problem 28

Choose two of the velocity profiles and corresponding boundary layer parameter formulas presented in Table $9.2 .$ Plot the thickness of the boundary layer and the wall shear stress versus $x$ for values of $x$ from zero to the laminar-turbulent transition point. The fluid is $20^{\circ} \mathrm{C}$ water and the free-stream velocity is $10 \mathrm{m} / \mathrm{s}$. Ask your instructor to specify which velocity profile models you should use.

Victor Salazar
Victor Salazar
Numerade Educator
15:42

Problem 29

Air at 20 psia and $90^{\circ} \mathrm{F}$ flows over a flat plate at $100 \mathrm{ft} / \mathrm{s}$ Find the value of $x$ at which the Reynolds number is $10^{4}$. Choose three velocity profile models from Table 9.2 and plot the velocity profiles (u versus $y$ ) on the same axes. Ask your instructor to specify which velocity profiles to choose.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 30

A laminar boundary layer velocity profile is approximated by the two straight-line segments indicated in Fig. P9.30. Use the momentum integral equation to determine the boundary layer thickness, $\delta=\delta(x),$ and wall shear stress, $\tau_{w}=\tau_{w}(x) .$ Compare these results with those in Table 9.2

Victor Salazar
Victor Salazar
Numerade Educator
01:28

Problem 31

A laminar boundary layer velocity profile is approximated by $u / U=2(y / \delta)-2(y / \delta)^{3}+(y / \delta)^{4}$ for $y \leq \delta,$ and $u=U$ for
$y >\delta .$ (a) Show that this profile satisfies the appropriate boundary conditions.
(b) Use the momentum integral equation to determine the boundary layer thickness, $\delta=\delta(x)$. Compare the result with the exact Blasius solution.

Dominador Tan
Dominador Tan
Numerade Educator
04:12

Problem 32

Consider $20^{\circ} \mathrm{C}$ water flowing over a thin, wide, smooth flat plate aligned with the flow. The approach velocity is 60 sm/hr and the Reynolds number based on the plate length is $2 \times 10^{5}$ Calculate the drag per unit width of the plate for each of the velocity profiles listed in Table $9.2 .$ Next calculate the drag coefficient $C_{D}$ as used in Fig. $9.15 .$ Which velocity profile(s) provide the best estimation of $C_{D} ?$ Why?

Chai Santi
Chai Santi
Numerade Educator
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Problem 33

For a Tuid of specific gravity $S G=0.86$ flowing past a flat plate with an upstream velocity of $U=5 \mathrm{m} / \mathrm{s}$, the wall shear stress on the flat plate was determined to be as indicated in the table below, Use the momentum integral equation to determine the boundary layer momentum thickness, $\Theta=\Theta(x)$. Assume $\Theta=0$ at the leading edge, $x=0$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 34

Should a canoe paddle be made rough to get a "better grip on the water" for paddling purposes? Explain.

Victor Salazar
Victor Salazar
Numerade Educator
02:28

Problem 35

Two different fluids flow over two identical flat plates with the same laminar free-stream velocity. Both fluids have the same viscosity, but one is twice as dense as the other. What is the relationship between the drag forces for these two plates?

Narayan Hari
Narayan Hari
Numerade Educator
01:30

Problem 36

Fluid flows past a flat plate with a drag force $\mathscr{D}_{1}$. If the freestream velocity is doubled, will the new drag force, $\mathscr{P}_{2},$ be larger or smaller than $\mathscr{P}_{1}$ and by what amount?

Narayan Hari
Narayan Hari
Numerade Educator
01:04

Problem 37

A model is placed in an airflow at standard conditions with a given velocity and then placed in water flow at standard conditions with the same velocity. If the drag coefficients are the same between these two cases, how do the drag forces compare between the two fluids?

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 38

The drag coefficient for a newly designed hybrid car is predicted to be 0.21 . The cross-sectional area of the car is $30 \mathrm{ft}^{2}$ Determine the aerodynamic drag on the car when it is driven through still air at $55 \mathrm{mph}$

Narayan Hari
Narayan Hari
Numerade Educator
01:02

Problem 39

A 5 -m-diameter parachute of a new design is to be used to transport a load from flight altitude to the ground with an average vertical speed of $3 \mathrm{m} / \mathrm{s}$. The total weight of the load and parachute is 200 N. Determine the approximate drag coefficient for the parachute.

Narayan Hari
Narayan Hari
Numerade Educator
01:47

Problem 40

A $180-1 b$ man parachutes from a plane using a hemispherical parachute in air at $20^{\circ} \mathrm{F}$ and 14.60 psia. Calculate the parachute diameter required for the man's terminal velocity not to exceed $20 \mathrm{ft} / \mathrm{s}$. Neglect the weight of the parachute.

Narayan Hari
Narayan Hari
Numerade Educator
01:44

Problem 41

A $70-\mathrm{kg}$ soldier on a secret mission has to parachute from an airplane over the desert. It takes five seconds for him to release his parachute after he jumps. Taking the mass of the parachute as $5 \mathrm{kg}$ estimate the force acting on the soldier as he releases the parachute. The diameter of the parachute is $2.0 \mathrm{m}$. Neglect drag on the soldier's body prior to the opening of the parachute. The air is at $20^{\circ} \mathrm{C}$

Narayan Hari
Narayan Hari
Numerade Educator
01:10

Problem 42

The aerodynamic drag on a car depends on the "shape" of the car. For example, the car shown in Fig. $\mathrm{P} 9.42$ has a drag coefficient of 0.35 with the windows and roof closed. With the windows and roof open, the drag coefficient increases to $0.45 .$ With the windows and roof open, at what speed is the amount of power needed to overcome acrodynamic drag the same as it is at 65 mph with the windows and roof closed? Assume the frontal area remains the same. Recall that power is force times velocity.

Narayan Hari
Narayan Hari
Numerade Educator
04:53

Problem 43

An automobile engine has a maximum power output of $70 \mathrm{hp},$ which occurs at an engine speed of $2200 \mathrm{rpm} . \mathrm{A} 10 \%$ power loss occurs through the transmission and differential. The rear wheels have a radius of 15.0 in. and the automobile is to have a maximum speed of 75 mph along a level road. At this speed, the power absorbed by the tires because of their continuous deformation is 27 hp. Find the maximum permissible drag coefficient for the automobile at this speed. The car's frontal area is $24.0 \mathrm{ft}^{2}$

Chai Santi
Chai Santi
Numerade Educator
02:40

Problem 44

A woman is riding a bicycle down an $18 \%$ slope. Her velocity is $25 \mathrm{km} / \mathrm{hr}$ into an oncoming $25-\mathrm{km} / \mathrm{hr}$ wind. The air is at $15^{\circ} \mathrm{C}$ and $101 \mathrm{kPa}$. Assuming that the only forces affecting the speed are the weight and drag, calculate the drag coefficient if the frontal area is $0.6 \mathrm{m}^{2}$ and the combined mass is $54 \mathrm{kg}$. Speculate whether the rider is in the upright or racing position.

Narayan Hari
Narayan Hari
Numerade Educator
01:45

Problem 45

A baseball is thrown by a pitcher at 95 mph through standard air. The diameter of the baseball is 2.82 in. Estimate the drag force on the baseball. Explain how the actual drag force may be different than your estimate.

Narayan Hari
Narayan Hari
Numerade Educator
03:50

Problem 46

A logging boat is towing a log at $1.5 \mathrm{m} / \mathrm{s}$ against a river current that is $1 \mathrm{m} / \mathrm{s}$. The log is $0.5 \mathrm{m}$ in diameter and $2 \mathrm{m}$ long.
(a) Estimate the power required if the axis of the log is parallel to the flow.
(b) Estimate the power required if the axis of the log is perpendicular to the flow, (c) Explain any reason(s) why the actual power required may be different than your estimates.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
03:31

Problem 47

How fast do small water droplets of $0.06-\mu \mathrm{m}\left(6 \times 10^{-8} \mathrm{m}\right)$ diameter fall through the air under standard sea-level conditions? Assume the drops do not evaporate. Repeat the problem for standard conditions at 5000 -m altitude.

Narayan Hari
Narayan Hari
Numerade Educator
04:08

Problem 48

Determine the drag on a small circular disk of $0.01-\mathrm{ft}$ diameter moving $0.01 \mathrm{ft} / \mathrm{s}$ through oil with a specific gravity of 0.87 and a viscosity 10,000 times that of water. The disk is oriented normal to the upstream velocity. By what percent is the drag reduced if the disk is oriented parallel to the flow?

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 49

The square, flat plate shown in Fig. $P 9.49 a$ is cut into four equal-sized pieces and arranged as shown in Fig. $\mathrm{P} 9.45 b$ Determine the ratio of the drag on the original plate [case (a)] to the drag on the plates in the configuration shown in (b). Assume laminar boundary flow. Explain your answer physically.

Victor Salazar
Victor Salazar
Numerade Educator
03:06

Problem 50

Water flows past a triangular flat plate oriented parallel to the free stream as shown in Fig. P9.50. Integrate the wall shear stress over the plate to determine the friction drag on one side of the plate. Assume laminar boundary layer flow.

Narayan Hari
Narayan Hari
Numerade Educator
01:02

Problem 51

For small Reynolds number flows, the drag coefficient of an object is given by a constant divided by the Reynolds number (see Table 9.4 . Thus, as the Reynolds number tends to zero, the drag coefficient becomes infinitely large. Does this mean that for small velocities (hence, small Reynolds numbers) the drag is very large? Explain.

Narayan Hari
Narayan Hari
Numerade Educator
02:48

Problem 52

A rectangular cartop carrier of $1.6-$ ft height, 4.0 -ft length (front to back), and 4.2 -ft width is attached to the top of a car. Estimate the additional power required to drive the car with the carrier at 65 mph through still air compared with the power required to drive only the car at 65 mph.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 53

As shown in Video V9.2 and Fig. P9.53 $a$, a kayak is a relatively streamlined object. As a first approximation in calculating the drag on a kayak, assume that the kayak acts as if it were a smooth, flat plate 17 ft long and $2 \mathrm{ft}$ wide. Determine the drag as a function of speed and compare your results with the measured values given in Fig. P9.50b. Comment on reasons why the two sets of values may differ.

Victor Salazar
Victor Salazar
Numerade Educator
02:32

Problem 54

A coal barge 1000 ft long and 100 ft wide is submerged a depth of 12 ft in $60^{\circ} \mathrm{F}$ water. It is being towed at a speed of $12 \mathrm{mph}$ Estimate the friction drag on the barge.

Narayan Hari
Narayan Hari
Numerade Educator
03:35

Problem 55

A three-bladed helicopter blade rotates at $200 \mathrm{rpm}$. If each blade is $12 \mathrm{ft}$ long and $1.5 \mathrm{ft}$ wide, estimate the torque needed to overcome the friction on the blades if they act as flat plates.

Narayan Hari
Narayan Hari
Numerade Educator
04:10

Problem 56

A thin, smooth sign is attached to the side of a truck as is indicated in Fig. P9.56. Estimate the friction drag on the sign when the truck is driven at 55 mph.

Prashant Bana
Prashant Bana
Numerade Educator
04:28

Problem 57

A 38.1 -mm-diameter, $0.0245-\mathrm{N}$ table tennis ball is released from the bottom of a 4 -m-deep swimming pool. Assuming that the ball has reached its terminal velocity within $1 \mathrm{m}$, estimate the time required to rise the remaining distance to the surface.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 58

A hot-air balloon roughly spherical in shape has a volume of $70,000 \mathrm{ft}^{3}$ and a weight of 500 Ib (including passengers, basket, balloon fabric, etc.). If the outside air temperature is $80^{\circ} \mathrm{F}$ and the temperature within the balloon is $165^{\circ} \mathrm{F}$, estimate the rate at which it will rise under steady-state conditions if the atmospheric pressure is 14.7 psi.

Victor Salazar
Victor Salazar
Numerade Educator
04:59

Problem 59

It is often assumed that "sharp objects can cut through the air better than blunt ones." Based on this assumption, the drag on the object shown in Fig. $\mathrm{P} 9.59$ should be less when the wind blows from right to left than when it blows from left to right. Experiments show that the opposite is true. Explain.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 60

An object falls at a rate of $100 \mathrm{ft} / \mathrm{s}$ immediately prior to the time that the parachute attached to it opens. The final descent rate with the chute open is $10 \mathrm{ft}$ ts. Calculate and plot the speed of falling as a function of time from when the chute opens. Assume that the chute opens instantly, that the drag coefficient and air density remain constant, and that the flow is quasisteady.

Victor Salazar
Victor Salazar
Numerade Educator
01:49

Problem 61

Estimate the velocity with which you would contact the ground if you jumped from an airplane at an altitude of $5,000 \mathrm{ft}$ and (a) air resistance is negligible,
(b) air resistance is important, but you forgot your parachute, or
(c) you use a $25-f$ t-diameter parachute.

Narayan Hari
Narayan Hari
Numerade Educator
02:26

Problem 62

As is discussed in Section $9.3,$ the drag on a rough golf ball may be less than that on an equal-sized smooth ball. Does it follow that a 10 -m-diameter spherical water tank resting on a $20-\mathrm{m}$ -tall support should have a rough surface so as to reduce the moment needed at the base of the support when a wind blows? Explain.

Narayan Hari
Narayan Hari
Numerade Educator
01:07

Problem 63

A 12 -mm-diameter cable is strung between a series of poles that are $50 \mathrm{m}$ apart. Determine the horizontal force this cable puts on each pole if the wind velocity is $30 \mathrm{m} / \mathrm{s}$

Narayan Hari
Narayan Hari
Numerade Educator
04:59

Problem 64

A strong wind can blow a golf ball off the tee by pivoting it about point I as shown in Fig. P9.64. Determine the wind speed necessary to do this.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 65

A 22 in. by 34 in. speed limit sign is supported on a 3-in.-wide, 5-ft-long pole. Estimate the bending moment in the pole at ground level when a 30 -mph wind blows against the sign. (See Video $\vee 9.14 .$ ) List any assumptions used in your calculations.

Victor Salazar
Victor Salazar
Numerade Educator
03:10

Problem 66

A 20 -m/s wind blows against a 20 -m-tall, 0.12 -m-diameter flag pole. (a) Determine the anchoring moment at the base of the pole.
(b) Determine the anchoring moment if a $2-\mathrm{m}$ by $2.5-\mathrm{m}$ flag is attached to the top of the pole, See Fig. 9.32 for drag coefficient data for flags.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:59

Problem 67

During a flash flood, water rushes over a road as shown in Fig. $P 9.67$ with a speed of 12 mph. Estimate the maximum water depth, $h,$ that would allow a car to pass without being swept away. List all assumptions and show all calculations.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 68

With the rider in the racing position, how much more power is required to pedal a bicycle at $15 \mathrm{mph}$ into a 20 -mph head-win than at 15 mph through still air? Pedaling at 15 mph in still ai how much more power is required for an upright position instead of a racing position?

Victor Salazar
Victor Salazar
Numerade Educator
02:08

Problem 69

Estimate the wind velocity necessary to knock over a $20-1 b$ garbage can that is $3 \mathrm{ft}$ tall and $2 \mathrm{ft}$ in diameter. List your assumptions.

Narayan Hari
Narayan Hari
Numerade Educator
02:36

Problem 70

On a day without any wind, your car consumes $x$ gallons of gasoline when you drive at a constant speed, $U$, from point $A$ to point $B$ and back to point $A$. Assume that you repeat the journey. driving at the same speed, on another day when there is a steady wind blowing from $B$ to $A$. Would you expect your fuel consumption to be less than, equal to, or greater than $x$ gallons for this windy round-trip? Support your answer with appropriate analysis.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 71

The structure shown in Fig. $\mathrm{P} 9.71$ consists of three cylindrical support posts to which an elliptical flat plate sign is attached. Estimate the drag on the structure when a 50 -mph wind blows against it.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 72

A 25 -ton $(50,000-1 b)$ truck coasts down a steep $7 \%$ mountain grade without brakes, as shown in Fig. P9.72. The truck's ultimate steady-state speed, $V$, is determined by a balance between weight, rolling resistance, and aerodynamic drag. Determine $V$ if the rolling resistance for a truck on concrete is $1.2 \%$ of the weight and the drag coefficient based on frontal area is 0.76

Victor Salazar
Victor Salazar
Numerade Educator
03:30

Problem 73

Phil's Pizza Parlor decides to place a thin, rectangular, plastic sign on top of its delivery van as shown in Fig. P9.73. The sign measures 2 ft by 5 ft. (a) Estimate the extra power required to drive the van in standard still air at 35 mph if the sign faces forward rather than sideways.
(b) The supports for the sign consist of two steel pipes 10 in. long and of $1 \frac{5}{16}$ in. outside diameter. Estimate the power required to overcome air drag on these two supports with the sign facing forward.

Chai Santi
Chai Santi
Numerade Educator
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Problem 74

As shown in Video $\mathrm{V} 9.19$ and Fig. $\mathrm{P} 9.74$, the aerodynamic drag on a truck can be reduced by the use of appropriate air deflectors. A reduction in drag coefficient from $C_{D}=0.96$ to $C_{D}=0.70$ corresponds to a reduction of how many horsepower needed at a highway speed of 65 mph?

Victor Salazar
Victor Salazar
Numerade Educator
04:35

Problem 75

A full-sized automobile has a frontal area of $24 \mathrm{ft}^{2},$ and a compact car has a frontal area of $13 \mathrm{ft}^{2}$. Both have a drag coefficient of 0.5 based on the frontal area. Find the horsepower required to move each automobile along a level road in still air at 55 mph. Assume that the power required to deform the tires continuously at this speed (called rolling resistance) is equal to the power to overcome the air resistance. Estimate the gas mileage of both automobiles if the energy supplied to the drive wheels is $\frac{1}{4}$ that available in the fuel. A gallon of fuel has $1.0 \times 10^{8} \mathrm{ft} \cdot 1 \mathrm{b}$ available energy.

Mario Munoz
Mario Munoz
Numerade Educator
04:15

Problem 76

Estimate the energy required for an average person (see Fig. 9.32 to run a mile in 4 minutes in still standard air. Compare your estimate if you instead modeled the person as a cylinder $6 \mathrm{ft}$ tall and $2 \mathrm{ft}$ in diameter.

Narayan Hari
Narayan Hari
Numerade Educator
01:40

Problem 77

As shown in Video $\mathrm{V} 9.11$ and Fig. $\mathrm{P} 9.77,$ a vertical wind tunnel can be used for skydiving practice. Estimate the vertical wind speed needed if a 160 -lb person is to be able to "float" motionless when the person
(a) curls up as in a crouching position or
(b) lies flat. See Fig. 9.30 for appropriate drag coefficient data.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 78

Compare the rise velocity of an $\frac{1}{8}$ -in.-diameter air bubble in water to the fall velocity of an $\frac{1}{8}$ -in.- -diameter water drop in air. Assume each to behave as a solid sphere.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 79

A 50 -lb box shaped like a 1 -ft cube falls from the cargo hold of an airplane at an altitude of $30,000 \mathrm{ft}$. If the drag coefficient of the falling box is $1.2,$ determine the time it takes for the box to tit the ocean. Assume that it falls at the terminal velocity corresponding to its current altitude and use a standard atmosphere (see Table $C .1$ ).

Victor Salazar
Victor Salazar
Numerade Educator
02:38

Problem 80

A 500 -N cube of specific gravity $S G=1.8$ falls through water at a constant speed $U$. Determine $U$ if the cube falls (a) as oriented in Fig. $\mathrm{P} 9.80 a,$ (b) as oriented in Fig. $\mathrm{P} 9.80 \mathrm{b}$

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 81

The helium-filled balloon shown in Fig P9.81 is to be used as a wind-speed indicator. The specific weight of the helium is $y=0.011 \mathrm{lb} / \mathrm{ft}^{3},$ the weight of the balloon material is $0.20 \mathrm{lb}$, and the weight of the anchoring cable is negligible. Plot a graph of $\theta$ as a function of $U$ for $1 \leq U \leq 50$ mph. Would this be an effective device over the range of $U$ indicated? Explain.

Victor Salazar
Victor Salazar
Numerade Educator
05:46

Problem 82

A 0.30 -m-diameter cork ball $(S G=0.21)$ is tied to an object on the bottom of a river as is shown in Fig. P9.82. Estimate the speed of the river current. Neglect the weight of the cable and the $\operatorname{drag}$ on it.

Narayan Hari
Narayan Hari
Numerade Educator
05:24

Problem 83

A shortwave radio antenna is constructed from circular tubing, as is illustrated in Fig. $\mathrm{P} 9.83 .$ Estimate the wind force on the antenna in a $100-\mathrm{km} / \mathrm{hr}$ wind.

Narayan Hari
Narayan Hari
Numerade Educator
01:51

Problem 84

Estimate the wind force on your hand when you hold it out of your car window while driving 55 mph. Repeat your calculations if you were to hold your hand out of the window of an airplane flying 550 mph.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 85

Estimate the energy that a runner expends to overcome aerodynamic drag while running a complete marathon race. This expenditure of energy is equivalent to climbing a hill of what height? List all assumptions and show all calculations.

Victor Salazar
Victor Salazar
Numerade Educator
02:53

Problem 86

A 2 -mm-diameter meteor of specific gravity 2.9 has a speed of $6 \mathrm{km} / \mathrm{s}$ at an altitude of $50,000 \mathrm{m}$ where the air density is $1.03 \times$ $10^{-3} \mathrm{kg} / \mathrm{m}^{3} .$ If the drag coefficient at this large Mach number condition is $1.5,$ determine the deceleration of the meteor

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 87

Air flows past two equal sized spheres (one rough, one smooth) that are attached to the arm of a balance as is indicated in Fig. $\mathrm{P} 9.87 .$ With $U=0$ the beam is balanced. What is the minimum air velocity for which the balance arm will rotate clockwise?

Victor Salazar
Victor Salazar
Numerade Educator
01:55

Problem 88

A 2 -in.- -diameter sphere weighing 0.14 lb is suspended by the jet of air shown in Fig. $\mathrm{P} 9.88$ and Video $\mathrm{V} 3.2 .$ The drag coefficient for the sphere is $0.5 .$ Determine the reading on the pressure gage if friction and gravity effects can be neglected for the flow between the pressure gage and the nozzle exit.

Narayan Hari
Narayan Hari
Numerade Educator
03:36

Problem 89

A smooth orange ball weighs $\frac{1}{64}$ Ib (at sea level) and has a diameter of 1.5 in. The discharge of a vacuum cleaner is directed upward and supports the ball $\frac{1}{2}$ in. above the hose outlet similar to that shown in Video $\mathrm{V} 3.2 .$ If the hose inside diameter is 4.0 in. estimate the volume flow rate through the vacuum cleaner. The air temperature is $100^{\circ} \mathrm{F}$

Narayan Hari
Narayan Hari
Numerade Educator
01:38

Problem 90

A $60 \mathrm{mph}$ wind blows against a football stadium scoreboard that is $36 \mathrm{ft}$ tall, $80 \mathrm{ft}$ wide, and $8 \mathrm{ft}$ thick (parallel to the wind). Estimate the wind force on the scoreboard. See Fig. 9.30 for drag coefficient data.

Narayan Hari
Narayan Hari
Numerade Educator
03:31

Problem 91

A marine location marker is a smoke-producing device usually dropped from an airplane and used to mark a reference point in the ocean. One is being tested in a wind tunnel to determine the drag force when it is carried by an airplane at a velocity of 200 mph. The marker is a cylinder with flat ends, has a diameter of 12 in., and is 48 in. long. Calculate the drag force on the marker in $10^{\circ} \mathrm{F}$ air if the air flow is parallel to the cylinder's axis.

Supratim Pal
Supratim Pal
Numerade Educator
02:44

Problem 92

The United Nations Building in New York is approximately $87.5 \mathrm{m}$ wide and $154 \mathrm{m}$ tall. (a) Determine the drag on this building if the drag coefficient is 1.3 and the wind speed is a uniform $20 \mathrm{m} / \mathrm{s}$
(b) Repeat your calculations if the velocity profile against the building is a typical profile for an urban area (see Problem 9.21 ) and the wind speed halfway up the building is $20 \mathrm{m} / \mathrm{s}$.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 93

An airplane flies at $150 \mathrm{km} / \mathrm{hr}$. (a) The airplane is towing a banner that is $b=0.8 \mathrm{m}$ tall and $\ell=25 \mathrm{m}$ long. If the drag coefficient based on area $b \ell$ is $C_{D}=0.06,$ estimate the power required to tow the banner.
(b) For comparison, determine the power required if the airplane was instead able to tow a rigid flat plate of the same size.
(c) Explain why one had a larger power requirement (and larger drag) than the other.
(d) Finally, determine the power required if the airplane was towing a smooth spherical balloon with a diameter of $2 \mathrm{m}$

Victor Salazar
Victor Salazar
Numerade Educator
01:29

Problem 94

The paint stirrer shown in Fig. $\mathrm{P} 9.94$ consists of two circular disks attached to the end of a thin rod that rotates at 80 rpm. The specific gravity of the paint is $S G=1.1$ and its viscosity is $\mu=2 \times$ $10^{-2} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2} .$ Estimate the power required to drive the mixer if the induced motion of the liquid is neglected.

Anand Jangid
Anand Jangid
Numerade Educator
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Problem 95

If the wind becomes strong enough, it is "impossible" to paddle a canoe into the wind. Estimate the wind speed at which this will happen. List all assumptions and show all calculations.

Victor Salazar
Victor Salazar
Numerade Educator
01:42

Problem 96

By appropriate streamlining, the drag coetficient for an airplane is reduced by $12 \%$ while the frontal area remains the same. For the same power output, by what percentage is the flight speed increased?

Narayan Hari
Narayan Hari
Numerade Educator
02:57

Problem 97

As indicated in Fig. $\mathrm{P} 9.97$, the orientation of leaves on a tree is a function of the wind speed, with the tree becoming "more streamlined" as the wind increases. The resulting drag coefficient for the tree (based on the frontal area of the tree, $H W$ ) as a function of Reynolds number (based on the leaf length, $L$ ) is approximated as shown. Consider a tree with leaves of length $L=0.3 \mathrm{ft}$. What wind speed will produce a drag on the tree that is 6 times greater than the drag on the tree in a $15 \mathrm{ft} / \mathrm{s}$ wind?

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 98

(See The Wide World of Fluids article "Dimpled Baseball Bats," Section $9.3 .3 .$.) How fast must a 3.5 -in.-diameter, dimpled baseball bat move through the air in order to take advantage of drag reduction produced by the dimples on the bat? Although there are differences, assume the bat (a cylinder) acts the same as a golf ball in terms of how the dimples affect the transition from a laminar to a turbulent boundary layer.

Narayan Hari
Narayan Hari
Numerade Educator
02:51

Problem 99

(See The Wide World of Fluids article "At $12,600 \mathrm{mpg}$ It Doesn't cost Much to 'Fill 'er Up," section 9.3.3.) (a) Determine the power it takes to overcome aerodynamic drag on a small $\left(6 \mathrm{ft}^{2}\right.$ cross section), streamlined $\left(C_{D}=0.12\right)$ vehicle traveling $15 \mathrm{mph}$
(b) Compare the power calculated in part (a) with that for a large
$\left(36 \mathrm{ft}^{2} \text { cross-sectional area }\right),$ nonstreamlined $\left(C_{D}=0.48\right) \mathrm{SUV}$ traveling 65 mph on the interstate.

Narayan Hari
Narayan Hari
Numerade Educator
01:06

Problem 100

A rectangular wing with an aspect ratio of 6 is to generate 000 lb of lift when it flies at a speed of 200 ft/s. Determine the length of the wing if its lift coefficient is 1.0

Narayan Hari
Narayan Hari
Numerade Educator
01:44

Problem 101

A $1.2-1 b$ kite with an area of $6 \mathrm{ft}^{2}$ flies in a $20-\mathrm{ft} / \mathrm{s}$ wind such that the weightless string makes an angle of $55^{\circ}$ relative to the horizontal. If the pull on the string is 1.5 lb, determine the lift and drag coefficients based on the kite area.

Narayan Hari
Narayan Hari
Numerade Educator
01:05

Problem 102

A Piper Cub airplane has a gross weight of $1750 \mathrm{lb}$, a cruising speed of $115 \mathrm{mph}$, and a wing area of $179 \mathrm{ft}^{2}$. Determine the lift coefficient of this airplane for these conditions.

Narayan Hari
Narayan Hari
Numerade Educator
01:39

Problem 103

A light aircraft with a wing area of $200 \mathrm{ft}^{2}$ and a weight of
2000 Ib has a lift coefficient of 0.40 and a drag coefficient of 0.05 Determine the power required to maintain level flight.

Narayan Hari
Narayan Hari
Numerade Educator
01:52

Problem 104

An airplane weighs $320,000 \mathrm{lb}$, has a wing area of $2800 \mathrm{ft}^{2}$ and has a wing length of $140 \mathrm{ft}$. The atmospheric pressure and temperature are 14.67 psia and $60^{\circ} \mathrm{F}$, respectively. The airplane is moving along a runway at $200 \mathrm{mph}$, and the wing has the lift and drag coefficients given in Fig. $9.39 .$ Find the lift and drag on the airplane wings for an angle of attack of $10^{\circ}$

Narayan Hari
Narayan Hari
Numerade Educator
03:11

Problem 105

As shown in $\mathrm{V} 9.25$ and Fig. $\mathrm{P} 9.105,$ a spoiler (i.e., an upside-down airfoil) is mounted above the rear wheels of a race car to produce negative lift (i.e., downforce), thereby improving tractive force. The spoiler's airfoil is angled $10^{\circ}$ with the race track and has lift and drag coefficient characteristics of Fig. $9.39 .$ The race car is traveling at $200 \mathrm{mph}$ in air at $90^{\circ} \mathrm{F}$ and 14.72 psia.
(a) If the coefficient of friction between the wheels and pavement is $0.6,$ by how much would use of the spoiler increase the maximum tractive force that would be generated between the wheels and the track? Assume the airspeed past the spoiler equals the car speed and that the airfoil acts directly over the drive wheels. (b) How much drag is added to the car with the use of the spoiler's airfoil?

Narayan Hari
Narayan Hari
Numerade Educator
02:02

Problem 106

The wings of old airplanes are often strengthened by the use of wires that provided cross-bracing as shown in Fig. P9.106. If the drag coefficient for the wings was 0.020 (based on the platform area), determine the ratio of the drag from the wire bracing to that from the wings.

Narayan Hari
Narayan Hari
Numerade Educator
01:59

Problem 107

A wing generates a lift $\mathscr{L}$ when moving through sea-level air with a velocity $U$. How fast must the wing move through the air at an altitude of $10,000 \mathrm{m}$ with the same lift coefficient if it is to generate the same lift?

Narayan Hari
Narayan Hari
Numerade Educator
03:11

Problem 108

A design group has two possible wing designs $(A \text { and } B$ ) for an airplane wing. The planform area of either wing is $130 \mathrm{m}^{2}$ and each must provide a lift of $1,550,000 \mathrm{N}$

Narayan Hari
Narayan Hari
Numerade Educator
04:33

Problem 109

Air blows over the flat-bottomed, two-dimensional object shown in Fig. P9.109. The shape of the object, $y=y(x),$ and the fluid speed along the surface, $u=u(x),$ are given in the table. Determine the lift coefficient for this object.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 110

When air flows past the airfoil shown in Fig. $\mathrm{P} 9.110$, the velocity just outside the boundary layer, $u,$ is as indicated. Estimate the lift coefficient for these conditions.

Victor Salazar
Victor Salazar
Numerade Educator
02:01

Problem 111

A Boeing 747 aircraft weighing 580,000 lb when loaded with fuel and 100 passengers takes off with an airspeed of 140 mph. With the same configuration (i.e., angle of attack, flap settings, etc.), what is its takeoff speed if it is loaded with 372 passengers? Assume each passenger with luggage weighs $200 \mathrm{lb}$.

Narayan Hari
Narayan Hari
Numerade Educator
01:55

Problem 112

Show that for unpowered flight (for which the lift, drag, and weight forces are in equilibrium) the glide slope angle, $\theta$, is given by $\tan \theta=C_{D} / C_{L}$

Narayan Hari
Narayan Hari
Numerade Educator
01:02

Problem 113

A sail plane with a lift-to-drag ratio of 25 flies with a speed of 50 mph. It maintains or increases its altitude by flying in thermals, columns of vertically rising air produced by buoyancy effects of nonuniformly heated air. What vertical airspeed is needed if the sail plane is to maintain a constant altitude?

Narayan Hari
Narayan Hari
Numerade Educator
02:33

Problem 114

If the lift coefficient for a Boeing 777 aircraft is 15 times greater than its drag coefficient, can it glide from an altitude of 30,000 ft to an airport $80 \mathrm{mi}$ away if it loses power from its engines? Explain. (See Problem 9.112 .)

Narayan Hari
Narayan Hari
Numerade Educator
03:01

Problem 115

Over the years there has been a dramatic increase in the flight speed $(U)$, altitude $(h),$ weight $(\mathcal{W}),$ and wing loading $(\mathscr{W} / \mathrm{A}=$ weight divided by wing area) of aircraft. Use the data given in the table below to determine the lift coefficient for each of the aircraft listed.

Narayan Hari
Narayan Hari
Numerade Educator
01:09

Problem 116

If the required takeoff speed of a particular airplane is $120 \mathrm{mi} / \mathrm{hr}$ at sea level, what will be required at Denver (elevation $5000 \mathrm{ft}$ ? Use properties of the U.S. Standard Atmosphere.

Narayan Hari
Narayan Hari
Numerade Educator
01:56

Problem 117

The landing speed of a winged aircraft such as the Space Shuttle is dependent on the air density. (See Video $\sqrt{9.1} .)$ By what percent must the landing speed be increased on a day when the temperature is $110^{\circ} \mathrm{F}$ compared to a day when it is $50^{\circ} \mathrm{F}$ ? Assume that the atmospheric pressure remains constant.

Narayan Hari
Narayan Hari
Numerade Educator
02:05

Problem 118

Commercial airliners normally cruise at relatively high altitudes $(30,000 \text { to } 35,000 \mathrm{ft}) .$ Discuss how flying at this high altitude (rather than $10,000 \mathrm{ft}$, for example) can save fuel costs.

Narayan Hari
Narayan Hari
Numerade Educator
03:43

Problem 119

A pitcher can pitch a "curve ball" by putting sufficient spin on the ball when it is thrown. A ball that has absolutely no spin will follow a "straight" path. A ball that is pitched with a very small amount of spin (on the order of one revolution during its flight between the pitcher's mound and home plate) is termed a knuckle ball. A ball pitched this way tends to "jump around" and "zig-zag", back and forth. Explain this phenomenon. Note: A baseball has seams.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:22

Problem 120

For many years, hitters have claimed that some baseball pitchers have the ability to actually throw a rising fastball. Assuming that a top major leaguer pitcher can throw a 95 -mph pitch and impart an 1800 -rpm spin to the ball, is it possible for the ball to actually rise? Assume the baseball diameter is 2.9 in. and its weight is 5.25 oz.

Narayan Hari
Narayan Hari
Numerade Educator
03:36

Problem 121

A baseball leaves the pitcher's hand with horizontal velocity of $90 \mathrm{mph}$ and travels a distance of $45 \mathrm{ft}$. Neglect air drag and gravity, so the ball moves in a horizontal plane. The ball has a mass of 5 oz, a circumference of 9 in., a rotational speed of 1600 rev/min, and a baseball lift coefficient of $0.75 .$ How far does the baseball "break" in the horizontal plane in $60^{\circ} \mathrm{F}$, still air?

Narayan Hari
Narayan Hari
Numerade Educator
02:15

Problem 122

(See The Wide World of Fluids article "Learning from Nature," Section 9.4.1.) As indicated in Fig. P9.122, birds can significantly alter their body shape and increase their platform area, $A,$ by spreading their wing and tail feathers, thereby reducing their flight speed. If during landing the platform area is increased by $50 \%$ and the lift coefficient increased by $30 \%$ while all other parameters are held constant, by what percent is the flight speed reduced?

Narayan Hari
Narayan Hari
Numerade Educator
01:18

Problem 123

(See The Wide World of Fluids article "Why Winglets?", Section $9.4 .2 .)$ It is estimated that by installing appropriately designed winglets on a certain airplane the drag coefficient will be reduced by $5 \%$. For the same engine thrust, by what percent will the aircraft speed be increased by use of the winglets?

Narayan Hari
Narayan Hari
Numerade Educator