Mechanical Engineering Chapter 3 Problem The Mass Suspended From Three Cables Cable Equipped With Turnbuckle Thatits

cables. Cable AC is equipped with a turnbuckle so that

its tension can be adjusted and a strain gauge that allows

its tension to be measured. If the tension in cable AC is

40 N, what are the tensions in cables AB and AD?

BC

A

D

0.4 m0.4 m 0.48 m

0.64 m

Solution:

TAB

TAC

5

Problem 3.34 The structural joint is in equilibrium. If

FAD1000 lb and FDD5000 lb, what are FBand FC?

FB

FC

FAFD

80⬚

65⬚

35⬚

Solution: The equilibrium equations are

104

Problem 3.35 The collar Aslides on the smooth

vertical bar. The masses mAD20 kg and mBD10 kg.

When hD0.1 m, the spring is unstretched. When the

system is in equilibrium, hD0.3 m. Determine the

spring constant k.

B

A

h

k

0.25 m

Solution: The triangles formed by the rope segments and the hori-

zontal line level with Acan be used to determine the lengths Luand

Ls. The equations are

163.5 N and TD255.4 N. Now we go to the free body diagram for B,

Lu

Ls

0.1 m

0.3 m

0.25 m

mBg

B

K

δ

Problem 3.36* Suppose that you want to design a

cable system to suspend an object of weight Wfrom

the ceiling. The two wires must be identical, and the

dimension bis ﬁxed. The ratio of the tension Tin each

wire to its cross-sectional area Amust equal a speciﬁed

value T/A D. The “cost” of your design is the total

volume of material in the two wires, VD2Apb2Ch2.

Determine the value of hthat minimizes the cost.

W

b

h

b

Solution: From the equation

TT

θθ

106

Problem 3.37 The system of cables suspends a

1000-lb bank of lights above a movie set. Determine

the tensions in cables AB,CD, and CE.

D

C

E

20 ft

B

18 ft

Solution: Isolate juncture A, and solve the equilibrium equations.

Repeat for the cable juncture C.

0.9659 D732.05 lb

Isolate juncture C. The angle between the positive xaxis and the cable

CA is ⊲180°˛⊳. The tension is

TCA DjTCAj⊲icos⊲180°C˛⊳ Cjsin⊲180°C˛⊳⊳,

W

y

x

D

Problem 3.38 Consider the 1000-lb bank of lights in

Problem 3.37. A technician changes the position of the

lights by removing the cable CE. What is the tension in

cable AB after the change?

Solution: The original conﬁguration in Problem 3.35 is used to

solve for the dimensions and the angles. Isolate the juncture A, and

The new angles are given by the cosine law

Reduce and solve:

Isolate the juncture A. The angle between the cable AD and the positive

xaxis is ˛. The tension is:

The equilibrium conditions are

20 ft 18 ft

B

y

D

Solve: jTABjDcos ˛

0.989 D621.03 lb,

108

Problem 3.39 While working on another exhibit, a

curator at the Smithsonian Institution pulls the suspended

Voyager aircraft to one side by attaching three horizontal

cables as shown. The mass of the aircraft is 1250 kg.

Determine the tensions in the cable segments AB,BC,

and CD.

B

C

D

50°

30°

Solution: Isolate each cable juncture, beginning with Aand solve

Isolate juncture B. The angles are ˛D50°,ˇD70°, and the tension

cable BC is TBC DjTBCj⊲icos ˛Cjsin ˛⊳. The angle between the

cable BA and the positive xaxis is ⊲180 Cˇ⊳; the tension is

y

x

C

Problem 3.40 A truck dealer wants to suspend a 4000-

kg truck as shown for advertising. The distance bD

15 m, and the sum of the lengths of the cables AB and

BC is 42 m. Points A and C are at the same height. What

are the tensions in the cables?

C

A

b

40 m

B

Solution: Determine the dimensions and angles of the cables. Iso-

late the cable juncture B, and solve the equilibrium conditions. The

dimensions of the triangles formed by the cables:

15 m 25 m

b

L

110

Problem 3.41 The distance hD12 in, and the tension

in cable AD is 200 lb. What are the tensions in cables

AB and AC?

12 in.

8 in.

8 in. h

D

B

A

C

Solution: Isolated the cable juncture. From the sketch, the angles

are found from

y

B

α

12 in

Problem 3.42 You are designing a cable system to

support a suspended object of weight W. Because your

design requires points Aand Bto be placed as shown,

you have no control over the angle ˛, but you can choose

the angle ˇby placing point Cwherever you wish. Show

that to minimize the tensions in cables AB and BC, you

must choose ˇD˛if the angle ˛½45°.

Strategy: Draw a diagram of the sum of the forces

exerted by the three cables at A.W

C

B

A

β

α

Solution: Draw the free body diagram of the knot at point A. Then

draw the force triangle involving the three forces. Remember that ˛is

In this case, we solved the problem without writing the equations of

equilibrium. For reference, these equations are:

FxDTAB cos ˛CTAC cos ˇD0

y

TAB

TAC

Possible locations

for C lie on line

B

α

C?C?

Problem 3.43* The length of the cable ABC is 1.4 m.

The 2-kN force is applied to a small pulley. The system

is stationary. What is the tension in the cable?

C

B

A1 m

0.75 m

15⬚2 kN

0.75 m ,tan ˇDh

0.25 m

)hD0.458 m,˛D31.39°,ˇD61.35°

Now draw a FBD and solve for the tension. We can use either of the

equilibrium equations

Fx:Tcos ˛CTcos ˇC⊲2kN⊳sin 15°D0

TD1.38 kN

β

α

T

112

c

currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

Problem 3.44 The masses m1D12 kg and m2D6kg

are suspended by the cable system shown. The cable BC

is horizontal. Determine the angle ˛and the tensions in

the cables AB,BC, and CD.

C

A

α

B

D

70⬚

m2

m1

117.7 N

TAB

TBC

α

B

Solution: We have 4 unknowns and 4 equations

Solving we ﬁnd

˛D79.7°,T

AB D119.7N,T

BC D21.4N,T

CD D62.6N

58.86 N

Problem 3.45 The weights W1D50 lb and W2are

suspended by the cable system shown. Determine the

weight W2and the tensions in the cables AB,BC,

and CD.

AD

BC

W1

W2

30 in 30 in 30 in

20 in 16 in

Solution: We have 4 unknowns and 4 equilibrium equations to use

TAB

Problem 3.46 In the system shown in Problem 3.45,

assume that W2DW1/2. If you don’t want the tension

anywhere in the supporting cable to exceed 200 lb, what

is the largest acceptable value of W1?

Solution:

FCy :2

p229 TBC C8

TAB

114

Problem 3.47 The hydraulic cylinder is subjected to

three forces. An 8-kN force is exerted on the cylinder

at Bthat is parallel to the cylinder and points from B

toward C. The link AC exerts a force at Cthat is parallel

to the line from Ato C. The link CD exerts a force at

Cthat is parallel to the line from Cto D.

(a) Draw the free-body diagram of the cylinder. (The

cylinder’s weight is negligible).

(b) Determine the magnitudes of the forces exerted by

the links AC and CD.

1 m

0.6 m Scoop

AB

D

C

0.15 m

0.6 m

1 m

Hydraulic

cylinder

Solution: From the ﬁgure, if Cis at the origin, then points A,B,

and Dare located at

A⊲0.15,0.6⊳

B⊲0.75,0.6⊳

y

D

FCD

Problem 3.48 The 50-lb cylinder rests on two smooth

surfaces.

(a) Draw the free-body diagram of the cylinder.

(b) If ˛D30°, what are the magnitudes of the forces

exerted on the cylinder by the left and right

surfaces?

α

45°

FyD⊲jNRjcos ˇCjNLjcos ˛jWj⊳jD0.

Problem 3.49 For the 50-lb cylinder in Problem 3.48,

obtain an equation for the force exerted on the cylinder

by the left surface in terms of the angle ˛in two ways:

(a) using a coordinate system with the yaxis vertical,

(b) using a coordinate system with the yaxis parallel to

the right surface.

Solution: The solution for Part (a) is given in Problem 3.48 (see

free body diagram).

y

αβ

116

Problem 3.50 The two springs are identical, with

unstretched length 0.4 m. When the 50-kg mass is

suspended at B, the length of each spring increases to

0.6 m. What is the spring constant k?

C

B

A

k

0.6 m

Solution:

kD1416 N/m

F

490.5 N

Problem 3.51 The cable AB is 0.5 m in length. The

unstretched length of the spring is 0.4 m. When the

50-kg mass is suspended at B, the length of the spring

increases to 0.45 m. What is the spring constant k?

C

B

A

k

0.7 m

Solution: The Geometry

0.72D0.52C0.4522⊲0.5⊳⊲0.45⊳cos ˇ

0.45 m Dsin 

0.5mDsin ˇ

0.7m

ˇD94.8°,D39.8°D45.4°

Now do the statics

FDk⊲0.45 m 0.4m⊳

Fy:TAB sin CFsin 490.5ND0

Solving: kD7560 N/m

θφ

β

0.7 m

490.5 N

F

TAB

Problem 3.52 The small sphere of mass mis attached

to a string of length Land rests on the smooth surface

of a ﬁxed sphere of radius R. The center of the sphere

is directly below the point where the string is attached.

Obtain an equation for the tension in the string in terms

of m,L,h, and R.

R

hL

m

Solution: From the geometry we have

Problem 3.53 The inclined surface is smooth. Deter-

mine the force Tthat must be exerted on the cable to

hold the 100-kg crate in equilibrium and compare your

answer to the answer of Problem 3.11.

T

60⬚

Solution:

60°

N

3T

118

c

currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

Problem 3.54 In Example 3.3, suppose that the mass

of the suspended object is mAand the masses of the

pulleys are mBD0.3mA,mCD0.2mA, and mDD0.2mA.

Show that the force Tnecessary for the system to be in

equilibrium is 0.275mAg.

T

A

B

DD

CC

Solution: From the free-body diagram of pulley C

TD

Problem 3.55 The mass of each pulley of the system

is mand the mass of the suspended object Ais mA.

Determine the force Tnecessary for the system to be in

equilibrium.

Solution: Draw free body diagrams of each pulley and the object

120

Problem 3.56 The suspended mass m1D50 kg. Neg-

lecting the masses of the pulleys, determine the value

of the mass m2necessary for the system to be in

equilibrium.

A

B

C

m1

m2

Solution:

TT1

T

Problem 3.57 The boy is lifting himself using the

block and tackle shown. If the weight of the block and

tackle is negligible, and the combined weight of the boy

and the beam he is sitting on is 120 lb, what force

does he have to exert on the rope to raise himself at

a constant rate? (Neglect the deviation of the ropes from

the vertical.)

Solution: A free-body diagram can be obtained by cutting the four

122

Problem 3.58 Pulley systems containing one, two, and

three pulleys are shown. Neglecting the weights of the

pulleys, determine the force Trequired to support the

weight Win each case.

T T T

W

(a) One pulley

(b) Two pulleys

(c) Three pulleys

Solution:

(b) For two pulleys

TT

W