Mechanical Engineering Chapter 3 Problem Problem Shows Pulley Systemssolution Extrapolation The Previous Problem Containing One Two

Problem 3.59 Problem 3.58 shows pulley systems

containing one, two, and three pulleys. The number of

pulleys in the type of system shown could obviously be

Solution: By extrapolation of the previous problem

(a) TDW

Problem 3.60 A 14,000-kg airplane is in steady ﬂight

in the vertical plane. The ﬂight path angle is D10°,

the angle of attack is ˛D4°, and the thrust force exerted

by the engine is TD60 kN. What are the magnitudes

of the lift and drag forces acting on the airplane? (See

Example 3.4).

Solution: Let us draw a more detailed free body diagram to see

the angles involved more clearly. Then we will write the equations of

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Problem 3.61 An airplane is in steady ﬂight, the angle

of attack ˛D0, the thrust-to-drag ratio T/D D2, and

the lift-to-drag ratio L/D D4. What is the ﬂight path

angle ? (See Example 3.4).

Problem 3.62 An airplane glides in steady ﬂight (TD

0), and its lift-to-drag ratio is L/D D4.

(a) What is the ﬂight path angle ?

(b) If the airplane glides from an altitude of 1000 m

to zero altitude, what horizontal distance does it

travel?

(See Example 3.4.)

Solution: See Example 3.4. The angle between the thrust vector

and the positive xaxis is ˛:

y

Path

126

Problem 3.63 In Active Example 3.5, suppose that the

attachment point Bis moved to the point (5,0,0) m. What

are the tensions in cables AB,AC, and AD?(⫺3, 0, 3) m

(⫺2, 0, ⫺2) m

D

(4, 0, 2)

m

B

y

C

x

Problem 3.65 Suppose that you want to apply a

1000-lb force Fat point Ain a direction such that the

resulting tensions in cables AB,AC, and AD are equal.

Determine the components of F.

y

z

(12, 4, 2) ft

(0, 4, 6) ft

(6, 0, 0) ftB

C

A

F

(0, 6, 0) ftD

x

Solution: We ﬁrst write the position vectors

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Problem 3.66 The 10-lb metal disk Ais supported by

the smooth inclined surface and the strings AB and AC.

The disk is located at coordinates (5,1,4) ft. What are

the tensions in the strings? (0, 6, 0) ft

(8, 4, 0) ft

C

y

B

2 ft

x

Solution: The position vectors are

Problem 3.67 The bulldozer exerts a force FD2i(kip)

at A. What are the tensions in cables AB,AC, and AD?

y

C

A

3 ft

2 ft

8 ft

6 ft

B

Solution: Isolate the cable juncture. Express the tensions in terms

of unit vectors. Solve the equilibrium equations. The coordinates of

points A,B,C,Dare:

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Problem 3.68 Prior to its launch, a balloon carrying a

set of experiments to high altitude is held in place by

groups of student volunteers holding the tethers at B,C,

and D. The mass of the balloon, experiments package,

and the gas it contains is 90 kg, and the buoyancy force

on the balloon is 1000 N. The supervising professor

conservatively estimates that each student can exert at

least a 40-N tension on the tether for the necessary length

of time. Based on this estimate, what minimum numbers

of students are needed at B,C, and D?

y

zB

x

C (10,0, –12) m

(16, 0, 16) m

D

(–16, 0, 4) m

(0, 8, 0) m

A

Solution:

FyD1000 ⊲90⊳⊲9.81⊳TD0

1000 N

Problem 3.69 The 20-kg mass is suspended by cables

attached to three vertical 2-m posts. Point Ais at

(0, 1.2, 0) m. Determine the tensions in cables AB,AC,

and AD.B

A

C

D

y

z

x

2 m

0.3 m

1 m

Solution: Points A,B,C, and Dare located at

yC

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Problem 3.70 The weight of the horizontal wall sec-

tion is WD20,000 lb. Determine the tensions in the

cables AB,AC, and AD.

Solution: Set the coordinate origin at Awith axes as shown. The

eAC D0.566i0.424j0.707k,

and eAD D0iC0.625j0.781k.

C

B

6 ft

4 ft 8 ft

W

D

B

C

TC

14 ft

6 ft

10 ft

Problem 3.71 The car in Fig. a and the pallet

supporting it weigh 3000 lb. They are supported by

four cables AB,AC,AD, and AE. The locations of the

attachment points on the pallet are shown in Fig. b.

the tensions in the cables.

C

y

B

(a)

x

8 ft 6 ft

z

Solution: Isolate the knot at A. Let TAB,TAC ,TAD and TAE be

the forces exerted by the tensions in the cables. The force exerted by

the vertical cable is (3000 lb)j.Weﬁrst ﬁnd the position vectors and

then express all of the forces as vectors.

rAB D⊲5i10jC5k⊳ft

rAD D⊲8i10j4k⊳ft

rAE D⊲6i10jC5k⊳ft

TAB DTAB

rAB

jrABjDTAB⊲0.408i0.816jC0.408k⊳

TAC DTAC

rAC

jrACjDTAC ⊲0.473i0.788j0.394k⊳

rAD

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Problem 3.72 The 680-kg load suspended from the

helicopter is in equilibrium. The aerodynamic drag force

on the load is horizontal. The yaxis is vertical, and cable

OA lies in the x–yplane. Determine the magnitude of the

drag force and the tension in cable OA.

yA

10°

x

O

B

CD

Solution:

y

Problem 3.73 In Problem 3.72, the coordinates of the

three cable attachment points B,C, and Dare (3.3,

4.5, 0) m, (1.1, 5.3, 1) m, and (1.6, 5.4, 1) m,

respectively. What are the tensions in cables OB,OC,

and OD?

eOD D0.280i0.944j0.175k.

Using these unit vectors, we obtain the equilibrium equations

y

TOA

10°

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Problem 3.74 If the mass of the bar AB is negligible

compared to the mass of the suspended object E, the

bar exerts a force on the “ball” at Bthat points from A

toward B. The mass of the object Eis 200 kg. The y-axis

points upward. Determine the tensions in the cables BC

and CD.

Strategy: Draw a free-body diagram of the ball at B.

(The weight of the ball is negligible.)

x

y

z

C

B

D

A

(0, 5, 5) m

(0, 4, ⫺3) m

(4, 3, 1) m

E

Solution:

Problem 3.75* The 3400-lb car is at rest on the plane

surface. The unit vector enD0.456iC0.570jC0.684k

is perpendicular to the surface. Determine the magni-

tudes of the total normal force Nand the total friction

force fexerted on the surface by the car’s wheels.

y

en

138

Problem 3.76 The system shown anchors a stanchion

of a cable-suspended roof. If the tension in cable AB is

900 kN, what are the tensions in cables EF and EG?

y

z

x

F

(0, 1.4, ⫺1.2) m

(2.2, 0, 1) m

(3.4, 1, 0) m

G

E

C

Solution: Using the coordinates for the points we ﬁnd

eBC D0.140i0.700jC0.700k

Problem 3.77* The cables of the system will each

safely support a tension of 1500 kN. Based on this

criterion, what is the largest safe value of the tension

in cable AB?

y

F

(0, 1.4, ⫺1.2) m

(2, 1, 0) m

(2.2, 0, ⫺1) m

(3.4, 1, 0) m

G

E

Solution: From Problem 3.76 we know that if the tension in AB

results from Problem 3.76

Problem 3.78 The 200-kg slider at Ais held in place

on the smooth vertical bar by the cable AB.

(a) Determine the tension in the cable.

(b) Determine the force exerted on the slider by

the bar.

z

y

x

A

B

2 m

5 m

2 m

Solution: The coordinates of the points A,Bare A⊲2,2,0⊳,

B⊲0,5,2⊳. The vector positions

The weight is

T

below). In this general situation, it will be necessary to eliminate the

slider bar normal force by some procedure equivalent to that used

above. End Note.

140

c

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Problem 3.79 In Example 3.6, suppose that the cable

AC is replaced by a longer one so that the distance from

point Bto the slider Cincreases from 6 ft to 8 ft. Deter-

mine the tension in the cable.

y

A

B

4 ft

Solution: The vector from Bto Cis now