Chapter 11 Flow Over Bodies: Drag and Lift Lift Notes | EduRev

: Chapter 11 Flow Over Bodies: Drag and Lift Lift Notes | EduRev

 Page 1


Chapter 11  Flow Over Bodies: Drag and Lift 
 
Lift 
 
11-71C The contribution of viscous effects to lift is usually negligible for airfoils since the wall shear is 
parallel to the surfaces of such devices and thus nearly normal to the direction of lift. 
 
11-72C When air flows past a symmetrical airfoil at zero angle of attack, (a) the lift will be zero, but (b) the 
drag acting on the airfoil will be nonzero. 
 
11-73C When air flows past a nonsymmetrical airfoil at zero angle of attack, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-74C When air flows past a symmetrical airfoil at an angle of attack of 5°, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-75C The decrease of lift with an increase in the angle of attack is called stall. When the flow separates 
over nearly the entire upper half of the airfoil, the lift is reduced dramatically (the separation point is near 
the leading edge). Stall is caused by flow separation and the formation of a wide wake region over the top 
surface of the airfoil. The commercial aircraft are not allowed to fly at velocities near the stall velocity for 
safety reasons. Airfoils stall at high angles of attack (flow cannot negotiate the curve around the leading 
edge). If a plane stalls, it loses mush of its lift, and it can crash. 
 
11-76C Both the lift and the drag of an airfoil increase with an increase in the angle of attack, but in 
general lift increases at a much higher rate than does the drag.  
 
11-77C Flaps are used at the leading and trailing edges of the wings of large aircraft during takeoff and 
landing to alter the shape of the wings to maximize lift and to enable the aircraft to land or takeoff at low 
speeds. An aircraft can takeoff or land without flaps, but it can do so at very high velocities, which is 
undesirable during takeoff and landing. 
 
11-78C Flaps increase both the lift and the drag of the wings. But the increase in drag during takeoff and 
landing is not much of a concern because of the relatively short time periods involved. This is the penalty 
we pay willingly to takeoff and land at safe speeds. 
 
11-79C The effect of wing tip vortices is to increase drag (induced drag) and to decrease lift. This effect is 
also due to the downwash, which causes an effectively smaller angle of attack.   
 
11-80C Induced drag is the additional drag caused by the tip vortices. The tip vortices have a lot of kinetic 
energy, all of which is wasted and is ultimately dissipated as heat in the air downstream. The induced drag 
can be reduced by using long and narrow wings. 
 
11-81C When air is flowing past a spherical ball, the lift exerted on the ball will be zero if the ball is not 
spinning, and it will be nonzero if the ball is spinning about an axis normal to the free stream velocity (no 
lift will be generated if the ball is spinning about an axis parallel to the free stream velocity).  
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-42
Page 2


Chapter 11  Flow Over Bodies: Drag and Lift 
 
Lift 
 
11-71C The contribution of viscous effects to lift is usually negligible for airfoils since the wall shear is 
parallel to the surfaces of such devices and thus nearly normal to the direction of lift. 
 
11-72C When air flows past a symmetrical airfoil at zero angle of attack, (a) the lift will be zero, but (b) the 
drag acting on the airfoil will be nonzero. 
 
11-73C When air flows past a nonsymmetrical airfoil at zero angle of attack, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-74C When air flows past a symmetrical airfoil at an angle of attack of 5°, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-75C The decrease of lift with an increase in the angle of attack is called stall. When the flow separates 
over nearly the entire upper half of the airfoil, the lift is reduced dramatically (the separation point is near 
the leading edge). Stall is caused by flow separation and the formation of a wide wake region over the top 
surface of the airfoil. The commercial aircraft are not allowed to fly at velocities near the stall velocity for 
safety reasons. Airfoils stall at high angles of attack (flow cannot negotiate the curve around the leading 
edge). If a plane stalls, it loses mush of its lift, and it can crash. 
 
11-76C Both the lift and the drag of an airfoil increase with an increase in the angle of attack, but in 
general lift increases at a much higher rate than does the drag.  
 
11-77C Flaps are used at the leading and trailing edges of the wings of large aircraft during takeoff and 
landing to alter the shape of the wings to maximize lift and to enable the aircraft to land or takeoff at low 
speeds. An aircraft can takeoff or land without flaps, but it can do so at very high velocities, which is 
undesirable during takeoff and landing. 
 
11-78C Flaps increase both the lift and the drag of the wings. But the increase in drag during takeoff and 
landing is not much of a concern because of the relatively short time periods involved. This is the penalty 
we pay willingly to takeoff and land at safe speeds. 
 
11-79C The effect of wing tip vortices is to increase drag (induced drag) and to decrease lift. This effect is 
also due to the downwash, which causes an effectively smaller angle of attack.   
 
11-80C Induced drag is the additional drag caused by the tip vortices. The tip vortices have a lot of kinetic 
energy, all of which is wasted and is ultimately dissipated as heat in the air downstream. The induced drag 
can be reduced by using long and narrow wings. 
 
11-81C When air is flowing past a spherical ball, the lift exerted on the ball will be zero if the ball is not 
spinning, and it will be nonzero if the ball is spinning about an axis normal to the free stream velocity (no 
lift will be generated if the ball is spinning about an axis parallel to the free stream velocity).  
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-42
Chapter 11  Flow Over Bodies: Drag and Lift 
11-82 A tennis ball is hit with a backspin. It is to be determined if the ball will fall or rise after being hit. 
Assumptions 1 The outer surface of the ball is smooth enough for Fig. 11-53 to be applicable. 2 The ball is 
hit horizontally so that it starts its motion horizontally.    
Properties The density and kinematic viscosity of air at 1 atm and 25°C are ? = 1.184 kg/m
3
 and ? = 
1.562×10
-5
 m
2
/s. 
Analysis  The ball is hit horizontally, and thus it would normally fall under the effect of gravity without the 
spin. The backspin will generate a lift, and the ball will rise if the lift is greater than the weight of the ball. 
The lift can be determined from  
2
2
V
A C F
L L
?
= 
 11-43
where A is the frontal area of the ball, which is . The 
regular and angular velocities of the ball are 
4 /
2
D A p =
4200 rpm 
m/s 56 . 25
 km/h 3.6
m/s 1
 km/h) 92 ( =
?
?
?
?
?
?
= V 
92 km/h
rad/s 440
s 60
min 1
rev 1
rad 2
rev/min) 4200 ( = ?
?
?
?
?
?
?
?
?
?
?
?
=
p
? 
Then, 
 rad 551 . 0
m/s) 56 . 25 ( 2
m) 064 rad/s)(0. 440 (
2
= =
V
D ?
 
From Fig. 11-53, the lift coefficient corresponding to this value is C
L
 = 0.11. Then the lift acting on the ball 
is 
N 14 . 0
m/s kg 1
N 1
2
m/s) 56 . 25 )( kg/m 184 . 1 (
4
m) 064 . 0 (
) 11 . 0 (
2
2 3 2
=
?
?
?
?
?
?
?
?
·
=
p
L
F 
The weight of the ball is 
N 56 . 0
m/s kg 1
N 1
) m/s 81 . 9 )( kg 057 . 0 (
2
2
=
?
?
?
?
?
?
?
?
·
= = mg W 
which is more than the lift. Therefore, the ball will drop under the combined effect of gravity and lift due 
to spinning after hitting, with a net force of 0.56 - 0.14 = 0.42 N.   
Discussion The Reynolds number for this problem is  
 
5
2 5
10 05 . 1
/s m 10 562 . 1
m) .064 5.56m/s)(0 2 (
Re × =
×
= =
-
?
VD
L
 
which is close enough to 6×10
4
 for which Fig. 11-53 is prepared. Therefore, the result should be close 
enough to the actual answer. 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
Page 3


Chapter 11  Flow Over Bodies: Drag and Lift 
 
Lift 
 
11-71C The contribution of viscous effects to lift is usually negligible for airfoils since the wall shear is 
parallel to the surfaces of such devices and thus nearly normal to the direction of lift. 
 
11-72C When air flows past a symmetrical airfoil at zero angle of attack, (a) the lift will be zero, but (b) the 
drag acting on the airfoil will be nonzero. 
 
11-73C When air flows past a nonsymmetrical airfoil at zero angle of attack, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-74C When air flows past a symmetrical airfoil at an angle of attack of 5°, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-75C The decrease of lift with an increase in the angle of attack is called stall. When the flow separates 
over nearly the entire upper half of the airfoil, the lift is reduced dramatically (the separation point is near 
the leading edge). Stall is caused by flow separation and the formation of a wide wake region over the top 
surface of the airfoil. The commercial aircraft are not allowed to fly at velocities near the stall velocity for 
safety reasons. Airfoils stall at high angles of attack (flow cannot negotiate the curve around the leading 
edge). If a plane stalls, it loses mush of its lift, and it can crash. 
 
11-76C Both the lift and the drag of an airfoil increase with an increase in the angle of attack, but in 
general lift increases at a much higher rate than does the drag.  
 
11-77C Flaps are used at the leading and trailing edges of the wings of large aircraft during takeoff and 
landing to alter the shape of the wings to maximize lift and to enable the aircraft to land or takeoff at low 
speeds. An aircraft can takeoff or land without flaps, but it can do so at very high velocities, which is 
undesirable during takeoff and landing. 
 
11-78C Flaps increase both the lift and the drag of the wings. But the increase in drag during takeoff and 
landing is not much of a concern because of the relatively short time periods involved. This is the penalty 
we pay willingly to takeoff and land at safe speeds. 
 
11-79C The effect of wing tip vortices is to increase drag (induced drag) and to decrease lift. This effect is 
also due to the downwash, which causes an effectively smaller angle of attack.   
 
11-80C Induced drag is the additional drag caused by the tip vortices. The tip vortices have a lot of kinetic 
energy, all of which is wasted and is ultimately dissipated as heat in the air downstream. The induced drag 
can be reduced by using long and narrow wings. 
 
11-81C When air is flowing past a spherical ball, the lift exerted on the ball will be zero if the ball is not 
spinning, and it will be nonzero if the ball is spinning about an axis normal to the free stream velocity (no 
lift will be generated if the ball is spinning about an axis parallel to the free stream velocity).  
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-42
Chapter 11  Flow Over Bodies: Drag and Lift 
11-82 A tennis ball is hit with a backspin. It is to be determined if the ball will fall or rise after being hit. 
Assumptions 1 The outer surface of the ball is smooth enough for Fig. 11-53 to be applicable. 2 The ball is 
hit horizontally so that it starts its motion horizontally.    
Properties The density and kinematic viscosity of air at 1 atm and 25°C are ? = 1.184 kg/m
3
 and ? = 
1.562×10
-5
 m
2
/s. 
Analysis  The ball is hit horizontally, and thus it would normally fall under the effect of gravity without the 
spin. The backspin will generate a lift, and the ball will rise if the lift is greater than the weight of the ball. 
The lift can be determined from  
2
2
V
A C F
L L
?
= 
 11-43
where A is the frontal area of the ball, which is . The 
regular and angular velocities of the ball are 
4 /
2
D A p =
4200 rpm 
m/s 56 . 25
 km/h 3.6
m/s 1
 km/h) 92 ( =
?
?
?
?
?
?
= V 
92 km/h
rad/s 440
s 60
min 1
rev 1
rad 2
rev/min) 4200 ( = ?
?
?
?
?
?
?
?
?
?
?
?
=
p
? 
Then, 
 rad 551 . 0
m/s) 56 . 25 ( 2
m) 064 rad/s)(0. 440 (
2
= =
V
D ?
 
From Fig. 11-53, the lift coefficient corresponding to this value is C
L
 = 0.11. Then the lift acting on the ball 
is 
N 14 . 0
m/s kg 1
N 1
2
m/s) 56 . 25 )( kg/m 184 . 1 (
4
m) 064 . 0 (
) 11 . 0 (
2
2 3 2
=
?
?
?
?
?
?
?
?
·
=
p
L
F 
The weight of the ball is 
N 56 . 0
m/s kg 1
N 1
) m/s 81 . 9 )( kg 057 . 0 (
2
2
=
?
?
?
?
?
?
?
?
·
= = mg W 
which is more than the lift. Therefore, the ball will drop under the combined effect of gravity and lift due 
to spinning after hitting, with a net force of 0.56 - 0.14 = 0.42 N.   
Discussion The Reynolds number for this problem is  
 
5
2 5
10 05 . 1
/s m 10 562 . 1
m) .064 5.56m/s)(0 2 (
Re × =
×
= =
-
?
VD
L
 
which is close enough to 6×10
4
 for which Fig. 11-53 is prepared. Therefore, the result should be close 
enough to the actual answer. 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
Chapter 11  Flow Over Bodies: Drag and Lift 
11-83 The takeoff speed of an aircraft when it is fully loaded is given. The required takeoff speed when the 
weight of the aircraft is increased by 20% as a result of overloading is to be determined.  v 
Assumptions 1 The atmospheric conditions (and thus the properties of air) remain the same. 2 The settings 
of the plane during takeoff are maintained the same so that the lift coefficient of the plane remains the 
same.   
Analysis  An aircraft will takeoff when lift equals the total weight. Therefore,  
A C
W
V A V C W F W
L
L L
?
?
2
                      
2
2
1
= ? = ? = 
Takeoff 
V = 190 km/h 
We note that the takeoff velocity is proportional to the square root 
of the weight of the aircraft. When the density, lift coefficient, and 
area remain constant, the ratio of the velocities of the overloaded 
and fully loaded aircraft becomes 
1
2
1 2
1
2
1
2
1
2
                   
/ 2
/ 2
W
W
V V
W
W
A C W
A C W
V
V
L
L
= ? = =
?
?
 
Substituting, the takeoff velocity of the overloaded aircraft is determined to be 
km/h 208 = = = 1.2 km/h) 190 (
2 . 1
1
1
1 2
W
W
V V 
Discussion A similar analysis can be performed for the effect of the variations in density, lift coefficient, 
and planform area on the takeoff velocity. 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-44
Page 4


Chapter 11  Flow Over Bodies: Drag and Lift 
 
Lift 
 
11-71C The contribution of viscous effects to lift is usually negligible for airfoils since the wall shear is 
parallel to the surfaces of such devices and thus nearly normal to the direction of lift. 
 
11-72C When air flows past a symmetrical airfoil at zero angle of attack, (a) the lift will be zero, but (b) the 
drag acting on the airfoil will be nonzero. 
 
11-73C When air flows past a nonsymmetrical airfoil at zero angle of attack, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-74C When air flows past a symmetrical airfoil at an angle of attack of 5°, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-75C The decrease of lift with an increase in the angle of attack is called stall. When the flow separates 
over nearly the entire upper half of the airfoil, the lift is reduced dramatically (the separation point is near 
the leading edge). Stall is caused by flow separation and the formation of a wide wake region over the top 
surface of the airfoil. The commercial aircraft are not allowed to fly at velocities near the stall velocity for 
safety reasons. Airfoils stall at high angles of attack (flow cannot negotiate the curve around the leading 
edge). If a plane stalls, it loses mush of its lift, and it can crash. 
 
11-76C Both the lift and the drag of an airfoil increase with an increase in the angle of attack, but in 
general lift increases at a much higher rate than does the drag.  
 
11-77C Flaps are used at the leading and trailing edges of the wings of large aircraft during takeoff and 
landing to alter the shape of the wings to maximize lift and to enable the aircraft to land or takeoff at low 
speeds. An aircraft can takeoff or land without flaps, but it can do so at very high velocities, which is 
undesirable during takeoff and landing. 
 
11-78C Flaps increase both the lift and the drag of the wings. But the increase in drag during takeoff and 
landing is not much of a concern because of the relatively short time periods involved. This is the penalty 
we pay willingly to takeoff and land at safe speeds. 
 
11-79C The effect of wing tip vortices is to increase drag (induced drag) and to decrease lift. This effect is 
also due to the downwash, which causes an effectively smaller angle of attack.   
 
11-80C Induced drag is the additional drag caused by the tip vortices. The tip vortices have a lot of kinetic 
energy, all of which is wasted and is ultimately dissipated as heat in the air downstream. The induced drag 
can be reduced by using long and narrow wings. 
 
11-81C When air is flowing past a spherical ball, the lift exerted on the ball will be zero if the ball is not 
spinning, and it will be nonzero if the ball is spinning about an axis normal to the free stream velocity (no 
lift will be generated if the ball is spinning about an axis parallel to the free stream velocity).  
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-42
Chapter 11  Flow Over Bodies: Drag and Lift 
11-82 A tennis ball is hit with a backspin. It is to be determined if the ball will fall or rise after being hit. 
Assumptions 1 The outer surface of the ball is smooth enough for Fig. 11-53 to be applicable. 2 The ball is 
hit horizontally so that it starts its motion horizontally.    
Properties The density and kinematic viscosity of air at 1 atm and 25°C are ? = 1.184 kg/m
3
 and ? = 
1.562×10
-5
 m
2
/s. 
Analysis  The ball is hit horizontally, and thus it would normally fall under the effect of gravity without the 
spin. The backspin will generate a lift, and the ball will rise if the lift is greater than the weight of the ball. 
The lift can be determined from  
2
2
V
A C F
L L
?
= 
 11-43
where A is the frontal area of the ball, which is . The 
regular and angular velocities of the ball are 
4 /
2
D A p =
4200 rpm 
m/s 56 . 25
 km/h 3.6
m/s 1
 km/h) 92 ( =
?
?
?
?
?
?
= V 
92 km/h
rad/s 440
s 60
min 1
rev 1
rad 2
rev/min) 4200 ( = ?
?
?
?
?
?
?
?
?
?
?
?
=
p
? 
Then, 
 rad 551 . 0
m/s) 56 . 25 ( 2
m) 064 rad/s)(0. 440 (
2
= =
V
D ?
 
From Fig. 11-53, the lift coefficient corresponding to this value is C
L
 = 0.11. Then the lift acting on the ball 
is 
N 14 . 0
m/s kg 1
N 1
2
m/s) 56 . 25 )( kg/m 184 . 1 (
4
m) 064 . 0 (
) 11 . 0 (
2
2 3 2
=
?
?
?
?
?
?
?
?
·
=
p
L
F 
The weight of the ball is 
N 56 . 0
m/s kg 1
N 1
) m/s 81 . 9 )( kg 057 . 0 (
2
2
=
?
?
?
?
?
?
?
?
·
= = mg W 
which is more than the lift. Therefore, the ball will drop under the combined effect of gravity and lift due 
to spinning after hitting, with a net force of 0.56 - 0.14 = 0.42 N.   
Discussion The Reynolds number for this problem is  
 
5
2 5
10 05 . 1
/s m 10 562 . 1
m) .064 5.56m/s)(0 2 (
Re × =
×
= =
-
?
VD
L
 
which is close enough to 6×10
4
 for which Fig. 11-53 is prepared. Therefore, the result should be close 
enough to the actual answer. 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
Chapter 11  Flow Over Bodies: Drag and Lift 
11-83 The takeoff speed of an aircraft when it is fully loaded is given. The required takeoff speed when the 
weight of the aircraft is increased by 20% as a result of overloading is to be determined.  v 
Assumptions 1 The atmospheric conditions (and thus the properties of air) remain the same. 2 The settings 
of the plane during takeoff are maintained the same so that the lift coefficient of the plane remains the 
same.   
Analysis  An aircraft will takeoff when lift equals the total weight. Therefore,  
A C
W
V A V C W F W
L
L L
?
?
2
                      
2
2
1
= ? = ? = 
Takeoff 
V = 190 km/h 
We note that the takeoff velocity is proportional to the square root 
of the weight of the aircraft. When the density, lift coefficient, and 
area remain constant, the ratio of the velocities of the overloaded 
and fully loaded aircraft becomes 
1
2
1 2
1
2
1
2
1
2
                   
/ 2
/ 2
W
W
V V
W
W
A C W
A C W
V
V
L
L
= ? = =
?
?
 
Substituting, the takeoff velocity of the overloaded aircraft is determined to be 
km/h 208 = = = 1.2 km/h) 190 (
2 . 1
1
1
1 2
W
W
V V 
Discussion A similar analysis can be performed for the effect of the variations in density, lift coefficient, 
and planform area on the takeoff velocity. 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-44
Chapter 11  Flow Over Bodies: Drag and Lift 
11-84 The takeoff speed and takeoff time of an aircraft at sea level are given. The required takeoff speed, 
takeoff time, and the additional runway length required at a higher elevation are to be determined.  
Assumptions 1 Standard atmospheric conditions exist. 2 The settings of the plane during takeoff are 
maintained the same so that the lift coefficient of the plane and the planform area remain constant. 3 The 
acceleration of the aircraft during takeoff remains constant.  
Properties The density of standard air is ?
1
 = 1.225 kg/m
3
 at sea level, and ?
2
 = 1.048 kg/m
3
 at 1600 m 
altitude.  
Analysis  (a) An aircraft will takeoff when lift equals the total weight. Therefore,  
A C
W
V A V C W F W
L
L L
?
?
2
                      
2
2
1
= ? = ? = 
We note that the takeoff speed is inversely proportional to the square root of air density. When the weight, 
lift coefficient, and area remain constant, the ratio of the speeds of the aircraft at high altitude and at sea 
level becomes 
km/h 238 = = = ? = =
048 . 1
225 . 1
km/h) 220 (                   
/ 2
/ 2
2
1
1 2
2
1
1
2
1
2
?
?
?
?
?
?
V V
A C W
A C W
V
V
L
L
 
Therefore, the takeoff velocity of the aircraft at higher altitude is 238 km/h. 
 
(b) The acceleration of the aircraft at sea level is    
2
m/s 074 . 4
 km/h 3.6
m/s 1
s 15
0 -  km/h 220
=
?
?
?
?
?
?
=
?
?
=
t
V
a 
Takeoff 
V = 220 km/h 
which is assumed to be constant both at sea level and the higher 
altitude. Then the takeoff time at the higher altitude becomes  
s 16.2 = ?
?
?
?
?
?
=
?
= ? ?
?
?
=
km/h 3.6
m/s 1
m/s 4.074
0 - km/h   238
            
2
a
V
t
t
V
a 
(c) The additional runway length is determined by calculating the distance 
traveled during takeoff for both cases, and taking their difference:    
m 458 s) 15 )( m/s 074 . 4 (
2 2
2
1 2
1
2
1
1
= = = at L 
m 535 s) 2 . 16 )( m/s 074 . 4 (
2 2
2
1 2
2
2
1
2
= = = at L 
m 77 = - = - = ? 458 535
1 2
L L L 
Discussion Note that altitude has a significant effect on the length of the runways, and it should be a major 
consideration on the design of airports. It is interesting that a 1.2 second increase in takeoff time increases 
the required runway length by about 100 m. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-45
Page 5


Chapter 11  Flow Over Bodies: Drag and Lift 
 
Lift 
 
11-71C The contribution of viscous effects to lift is usually negligible for airfoils since the wall shear is 
parallel to the surfaces of such devices and thus nearly normal to the direction of lift. 
 
11-72C When air flows past a symmetrical airfoil at zero angle of attack, (a) the lift will be zero, but (b) the 
drag acting on the airfoil will be nonzero. 
 
11-73C When air flows past a nonsymmetrical airfoil at zero angle of attack, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-74C When air flows past a symmetrical airfoil at an angle of attack of 5°, both the (a) lift and (b) drag 
acting on the airfoil will be nonzero. 
 
11-75C The decrease of lift with an increase in the angle of attack is called stall. When the flow separates 
over nearly the entire upper half of the airfoil, the lift is reduced dramatically (the separation point is near 
the leading edge). Stall is caused by flow separation and the formation of a wide wake region over the top 
surface of the airfoil. The commercial aircraft are not allowed to fly at velocities near the stall velocity for 
safety reasons. Airfoils stall at high angles of attack (flow cannot negotiate the curve around the leading 
edge). If a plane stalls, it loses mush of its lift, and it can crash. 
 
11-76C Both the lift and the drag of an airfoil increase with an increase in the angle of attack, but in 
general lift increases at a much higher rate than does the drag.  
 
11-77C Flaps are used at the leading and trailing edges of the wings of large aircraft during takeoff and 
landing to alter the shape of the wings to maximize lift and to enable the aircraft to land or takeoff at low 
speeds. An aircraft can takeoff or land without flaps, but it can do so at very high velocities, which is 
undesirable during takeoff and landing. 
 
11-78C Flaps increase both the lift and the drag of the wings. But the increase in drag during takeoff and 
landing is not much of a concern because of the relatively short time periods involved. This is the penalty 
we pay willingly to takeoff and land at safe speeds. 
 
11-79C The effect of wing tip vortices is to increase drag (induced drag) and to decrease lift. This effect is 
also due to the downwash, which causes an effectively smaller angle of attack.   
 
11-80C Induced drag is the additional drag caused by the tip vortices. The tip vortices have a lot of kinetic 
energy, all of which is wasted and is ultimately dissipated as heat in the air downstream. The induced drag 
can be reduced by using long and narrow wings. 
 
11-81C When air is flowing past a spherical ball, the lift exerted on the ball will be zero if the ball is not 
spinning, and it will be nonzero if the ball is spinning about an axis normal to the free stream velocity (no 
lift will be generated if the ball is spinning about an axis parallel to the free stream velocity).  
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-42
Chapter 11  Flow Over Bodies: Drag and Lift 
11-82 A tennis ball is hit with a backspin. It is to be determined if the ball will fall or rise after being hit. 
Assumptions 1 The outer surface of the ball is smooth enough for Fig. 11-53 to be applicable. 2 The ball is 
hit horizontally so that it starts its motion horizontally.    
Properties The density and kinematic viscosity of air at 1 atm and 25°C are ? = 1.184 kg/m
3
 and ? = 
1.562×10
-5
 m
2
/s. 
Analysis  The ball is hit horizontally, and thus it would normally fall under the effect of gravity without the 
spin. The backspin will generate a lift, and the ball will rise if the lift is greater than the weight of the ball. 
The lift can be determined from  
2
2
V
A C F
L L
?
= 
 11-43
where A is the frontal area of the ball, which is . The 
regular and angular velocities of the ball are 
4 /
2
D A p =
4200 rpm 
m/s 56 . 25
 km/h 3.6
m/s 1
 km/h) 92 ( =
?
?
?
?
?
?
= V 
92 km/h
rad/s 440
s 60
min 1
rev 1
rad 2
rev/min) 4200 ( = ?
?
?
?
?
?
?
?
?
?
?
?
=
p
? 
Then, 
 rad 551 . 0
m/s) 56 . 25 ( 2
m) 064 rad/s)(0. 440 (
2
= =
V
D ?
 
From Fig. 11-53, the lift coefficient corresponding to this value is C
L
 = 0.11. Then the lift acting on the ball 
is 
N 14 . 0
m/s kg 1
N 1
2
m/s) 56 . 25 )( kg/m 184 . 1 (
4
m) 064 . 0 (
) 11 . 0 (
2
2 3 2
=
?
?
?
?
?
?
?
?
·
=
p
L
F 
The weight of the ball is 
N 56 . 0
m/s kg 1
N 1
) m/s 81 . 9 )( kg 057 . 0 (
2
2
=
?
?
?
?
?
?
?
?
·
= = mg W 
which is more than the lift. Therefore, the ball will drop under the combined effect of gravity and lift due 
to spinning after hitting, with a net force of 0.56 - 0.14 = 0.42 N.   
Discussion The Reynolds number for this problem is  
 
5
2 5
10 05 . 1
/s m 10 562 . 1
m) .064 5.56m/s)(0 2 (
Re × =
×
= =
-
?
VD
L
 
which is close enough to 6×10
4
 for which Fig. 11-53 is prepared. Therefore, the result should be close 
enough to the actual answer. 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
Chapter 11  Flow Over Bodies: Drag and Lift 
11-83 The takeoff speed of an aircraft when it is fully loaded is given. The required takeoff speed when the 
weight of the aircraft is increased by 20% as a result of overloading is to be determined.  v 
Assumptions 1 The atmospheric conditions (and thus the properties of air) remain the same. 2 The settings 
of the plane during takeoff are maintained the same so that the lift coefficient of the plane remains the 
same.   
Analysis  An aircraft will takeoff when lift equals the total weight. Therefore,  
A C
W
V A V C W F W
L
L L
?
?
2
                      
2
2
1
= ? = ? = 
Takeoff 
V = 190 km/h 
We note that the takeoff velocity is proportional to the square root 
of the weight of the aircraft. When the density, lift coefficient, and 
area remain constant, the ratio of the velocities of the overloaded 
and fully loaded aircraft becomes 
1
2
1 2
1
2
1
2
1
2
                   
/ 2
/ 2
W
W
V V
W
W
A C W
A C W
V
V
L
L
= ? = =
?
?
 
Substituting, the takeoff velocity of the overloaded aircraft is determined to be 
km/h 208 = = = 1.2 km/h) 190 (
2 . 1
1
1
1 2
W
W
V V 
Discussion A similar analysis can be performed for the effect of the variations in density, lift coefficient, 
and planform area on the takeoff velocity. 
 
 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-44
Chapter 11  Flow Over Bodies: Drag and Lift 
11-84 The takeoff speed and takeoff time of an aircraft at sea level are given. The required takeoff speed, 
takeoff time, and the additional runway length required at a higher elevation are to be determined.  
Assumptions 1 Standard atmospheric conditions exist. 2 The settings of the plane during takeoff are 
maintained the same so that the lift coefficient of the plane and the planform area remain constant. 3 The 
acceleration of the aircraft during takeoff remains constant.  
Properties The density of standard air is ?
1
 = 1.225 kg/m
3
 at sea level, and ?
2
 = 1.048 kg/m
3
 at 1600 m 
altitude.  
Analysis  (a) An aircraft will takeoff when lift equals the total weight. Therefore,  
A C
W
V A V C W F W
L
L L
?
?
2
                      
2
2
1
= ? = ? = 
We note that the takeoff speed is inversely proportional to the square root of air density. When the weight, 
lift coefficient, and area remain constant, the ratio of the speeds of the aircraft at high altitude and at sea 
level becomes 
km/h 238 = = = ? = =
048 . 1
225 . 1
km/h) 220 (                   
/ 2
/ 2
2
1
1 2
2
1
1
2
1
2
?
?
?
?
?
?
V V
A C W
A C W
V
V
L
L
 
Therefore, the takeoff velocity of the aircraft at higher altitude is 238 km/h. 
 
(b) The acceleration of the aircraft at sea level is    
2
m/s 074 . 4
 km/h 3.6
m/s 1
s 15
0 -  km/h 220
=
?
?
?
?
?
?
=
?
?
=
t
V
a 
Takeoff 
V = 220 km/h 
which is assumed to be constant both at sea level and the higher 
altitude. Then the takeoff time at the higher altitude becomes  
s 16.2 = ?
?
?
?
?
?
=
?
= ? ?
?
?
=
km/h 3.6
m/s 1
m/s 4.074
0 - km/h   238
            
2
a
V
t
t
V
a 
(c) The additional runway length is determined by calculating the distance 
traveled during takeoff for both cases, and taking their difference:    
m 458 s) 15 )( m/s 074 . 4 (
2 2
2
1 2
1
2
1
1
= = = at L 
m 535 s) 2 . 16 )( m/s 074 . 4 (
2 2
2
1 2
2
2
1
2
= = = at L 
m 77 = - = - = ? 458 535
1 2
L L L 
Discussion Note that altitude has a significant effect on the length of the runways, and it should be a major 
consideration on the design of airports. It is interesting that a 1.2 second increase in takeoff time increases 
the required runway length by about 100 m. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-45
Chapter 11  Flow Over Bodies: Drag and Lift 
11-85E The rate of fuel consumption of an aircraft while flying at a low altitude is given. The rate of fuel 
consumption at a higher altitude is to be determined for the same flight velocity.  
Assumptions 1 Standard atmospheric conditions exist. 2 The settings of the plane during takeoff are 
maintained the same so that the drag coefficient of the plane and the planform area remain constant. 3 The 
velocity of the aircraft and the propulsive efficiency remain constant. 4 The fuel is used primarily to 
provide propulsive power to overcome drag, and thus the energy consumed by auxiliary equipment (lights, 
etc) is negligible. 
Properties The density of standard air is ?
1
 = 0.05648 lbm/ft
3
 at 10,000 ft, and ?
2
 = 0.02866 lbm/ft
3
 at 
30,000 ft altitude. 
Analysis When an aircraft cruises steadily (zero acceleration) at a constant altitude, the net force acting on 
the aircraft is zero, and thus the thrust provided by the engines must be equal to the drag force. Also, power 
is force times velocity (distance per unit time), and thus the propulsive power required to overcome drag is 
equal to the thrust times the cruising velocity. Therefore, 
Cruising 
m
fuel
 = 5 gal/min
2 2
Thrust
3 2
propulsive
V
A C V
V
A C V F V W
D D D
? ?
= = = × =
&
 
The propulsive power is also equal to the product of the rate of fuel energy 
supplied (which is the rate of fuel consumption times the heating value of 
the fuel,  ) and the propulsive efficiency. Then,  HV
fuel
m &
HV
2
          HV
fuel prop
3
fuel prop prop
m
V
A C m W
D
& &
&
?
?
? = ? = 
We note that the rate of fuel consumption is proportional to the density of air. When the drag coefficient, 
the wing area, the velocity, and the propulsive efficiency remain constant, the ratio of the rates of fuel 
consumptions of the aircraft at high and low altitudes becomes 
gal/min 2.54 = = = ? = =
0.05648
0.02866
gal/min) 5 (        
HV 2 /
HV 2 /
1
2
1 fuel, 2 fuel,
1
2
prop
3
1
prop
3
2
1 fuel,
2 fuel,
?
?
?
?
? ?
? ?
m m
V A C
V A C
m
m
D
D
& &
&
&
 
Discussion Note the fuel consumption drops by half when the aircraft flies at 30,000 ft instead of 10,000 ft 
altitude. Therefore, large passenger planes routinely fly at high altitudes (usually between 30,000 and 
40,000 ft) to save fuel. This is especially the case for long flights. 
 
 
PROPRIETARY MATERIAL. © 2006 The McGraw-Hill Companies, Inc.  Limited distribution 
permitted only to teachers and educators for course preparation.  If you are a student using this Manual, you 
are using it without permission.   
11-46
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