Mechanical Engineering Chapter 9 Problem The Direction Cosines The Cranes Cable Are Cos Cos Cos The

Problem 9.61 The direction cosines of the crane’s

cable are cos xD0.558,cos yD0.766,cos zD

0.260. The yaxis is vertical. The stationary caisson to

which the cable is attached weights 2000 lb and rests

on horizontal ground. If the coefﬁcient of static friction

between the caisson and the ground is sD0.4, what

tension in the cable necessary to cause the caisson to

slip?

y

x

z

0.766jC0.260k⊳.IfNDNjis the normal force exerted on the caisson

(1) The magnitude of the horizontal force exerted by the cable

(2) slip impends when fDsN.

(3) Solving equation (1) for Nand solving equation (2) for fand

Problem 9.62* The 10-lb metal disk Ais at the center

of the inclined surface. The tension in the string AB is

5 lb. What minimum coefﬁcient of static friction between

the disk and the surface is necessary to keep the disk

from slipping?

y

B(0, 6, 0) ft

ND3.16

7.31 D0.432.

8 ft

2 ft

y

z

β

e

726

c

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Problem 9.63* The 5-kg box is at rest on the sloping

surface. The yaxis points upward. The unit vector

0.557iC0.743jC0.371kis perpendicular to the sloping

surface. What is the magnitude of the friction force

exerted on the box by the surface?

x

y

Problem 9.64* In Problem 9.63, what is the minimum

coefﬁcient of static friction necessary for the box to

remain at rest on the sloping surface?

Problem 9.65 In Active Example 9.4, the coefﬁcients

of friction between the wedge and the log are sD0.22

and kD0.20. What is the largest value of the wedge

angle ˛for which the wedge would remain in place in

the log when the force Fis removed?

a

Problem 9.66 The wedge shown is being used to split

the log. The wedge weighs 20 lb and the angle ˛equals

30°. The coefﬁcient of kinetic friction between the faces

of the wedge and the log is 0.28. If the normal force

exerted by each face of the wedge must equal 150 lb to

split the log, what vertical force Fis necessary to drive

the wedge into the log at a constant rate? (See Active

Example 9.4.)

F

Fa

Solution: The free-body diagram is shown.

F

Problem 9.67 The coefﬁcient of static friction

between the faces of the wedge and the log in Problem

9.66 is 0.30. Will the wedge remain in place in the

log when the vertical force Fis removed? (See Active

Example 9.4.)

a

Solution: Assume that FD0 and the wedge is on the verge of

728

Problem 9.68 The weights of the blocks are WAD

100 lb and WBD25 lb. Between all of the contacting

surfaces, sD0.32 and kD0.30. What force Fis

necessary to move Bto the left at a constant rate?

F

A

f1D0.3N1,f

2D0.3N2,f

3D0.3N3

25 lb

B

F

f3

Problem 9.69 The masses of the blocks are mAD

30 kg and mBD70 kg. Between all of the contacting

surfaces, sD0.1. What is the largest force Fthat can

be applied without causing the blocks to slip?

A

F

30⬚

730

Problem 9.70 Each block weighs 200 lb. Between all

of the contacting surfaces, sD0.1. What is the largest

force Fthat can be applied without causing block Bto

slip upward?

F

AC

B

Problem 9.71 Small wedges called shims can be used

to hold an object in place. The coefﬁcient of kinetic fric-

tion between the contacting surfaces is 0.4. What force F

is needed to push the shim downward until the horizontal

force exerted on the object Ais 200 N?

A5°

5°

Shims

F

NLD200 N (3)

Fx:NLNRcos 5°CfRsin 5°D0(4)

Fy:FCfLCfRcos 5°CNRsin 5°D0(5)

Unknowns: fL,N

L,F

R,N

R,F

(5 eqns. in 5 unknowns)

Solving,

FD181 N

fLfR

NL

NR

x

Problem 9.72 The coefﬁcient of static friction

between the contacting surfaces in Problem 9.71 is 0.44.

If the shims are in place and exert a 200-N horizontal

force on the object A, what upward force must be exerted

on the left shim to loosen it?

732

Problem 9.73 The crate Aweighs 600 lb. Between all

contacting surfaces, sD0.32 and kD0.30. Neglect

the weights of the wedges. What force Fis required to

move Ato the right at a constant rate? F

QD204.4lb,

PD81.34 lb,

Problem 9.74 Suppose that between all contacting

surfaces in Problem 9.73, sD0.32 and kD0.30.

Neglect the weights of the 5°wedges. If a force FD

800 N is required to move Ato the right at a constant

rate, what is the mass of A?

FxDPcos 5°0.3Psin 5°0.3QD0,

FyDQPsin 5°0.3Pcos 5°9.81 m D0.

Solving them, we obtain

0.3 N

5°

0.3 P

NP

P

Problem 9.75 The box Ahas a mass of 80 kg, and the

wedge Bhas a mass of 40 kg. Between all contacting

surfaces, sD0.15 and kD0.12. What force Fis

required to raise Aat a constant rate?

A

734

Problem 9.76 Suppose that in Problem 9.75, Aweighs

800 lb and Bweighs 400 lb. The coefﬁcients of fric-

tion between all of the contacting surfaces are sD0.15

and kD0.12. Will Bremain in place if the force Fis

removed?

Q

µ

SN

µ

SP

N

Problem 9.77 Between Aand B,sD0.20, and

between Band C,sD0.18. Between Cand the wall,

sD0.30. The weights WBD20 lb and WCD80 lb.

What force Fis required to start Cmoving upward?

C

Solution: The active contact surfaces are between the wall and C,

between the wedge Band C, and between the wedge Band A.For the

weight C : Denote the normal force exerted by the wall by Q, and the

normal force between Band Cby N. Denote the several coefﬁcients

of static friction by subscripts. The equilibrium conditions are:

Problem 9.78 The masses of A,B, and Care 8 kg,

12 kg, and 80 kg, respectively. Between all contacting

surfaces, sD0.4. What force Fis required to start C

moving upward?

F

A

C

D0.

FxDNcos ˛sNsin ˛sPsScos ˇSsin ˇD0.

ND1293.5N,

736

Problem 9.79 In Active Example 9.5, suppose that the

pitch of the thread is changed from pD0.2intopD

0.24 in. What is the slope of the thread? What is the

magnitude of the couple that must be applied to the collar

Cto cause it to turn at a constant rate and move the

suspended object upward?

20 in

B

C

A

Problem 9.80 Suppose that in Problem 9.79, the pitch

of the threaded shaft is pD2 mm and the mean radius

0.22, and kD0.20. The weight WD500 N. Neglect

the weight of the threaded shaft. What couple must be

applied to the threaded shaft to lower the weight at a

constant rate?

W

C

Solution: The angle of kinetic friction is

˛Dtan1p

2r Dtan12

2⊲20⊳D0.9118°.

MD0.02⊲500⊳tan⊲11.31 0.9118⊳D1.835 N-m.

Problem 9.81 The position of the horizontal beam can

be adjusted by turning the machine screw A. Neglect the

weight of the beam. The pitch of the screw is pD1 mm,

and the mean radius of the thread is rD4 mm. The

400 N

MD0.004⊲300⊳tan⊲11.31°2.28°⊳D0.19 N-m

Problem 9.82 The pitch of the threaded shaft of the C

clamp is pD0.05 in and the mean radius of the thread

is rD0.15 in. The coefﬁcients of friction between the

threaded shaft and the mating collar are sD0.18 and

kD0.16.

(a) What maximum couple must be applied to the shaft

to exert a 30-lb force on the clamped object?

(b) If a 30-lb force is exerted on the clamped object,

what couple must be applied to the shaft to begin

loosening the clamp?

738

Problem 9.83 The mass of block Ais 60 kg. Neglect

the weight of the 5°wedge. The coefﬁcient of kinetic

friction between the contacting surfaces of the block A,

the wedge, the table, and the wall is kD0.4. The pitch

of the threaded shaft is 5 mm, the mean radius of the

thread is 15 mm, and the coefﬁcient of kinetic friction

between the thread and the mating groove is 0.2. What

couple must be exerted on the threaded shaft to raise the

block Aat a constant rate?

Solution: Denote the wedge angle by ˇD5°and the normal force

on the top by Nand on the lower surface by P. The free body diagrams

of the wedge and block are as shown. The equilibrium equations for

wedge:

FxDFkPNsin 5°kNcos 5°D0,

FyDPNcos 5°CkNsin 5°D0.

µ

kN

µ

kN

µ

kP

N

P

F

Problem 9.84 The vise exerts 80-lb forces on A. The

threaded shafts are subjected only to axial loads by the

jaws of the vise. The pitch of their threads is pD1/8 in,

the mean radius of the threads is rD1 in, and the coefﬁ–

4 in

A

FxD80 CBCD0,

from which CD80 lb ⊲C⊳. The pitch angle is

˛Dtan11

16D1.14°.

The static friction angle is sDtan1⊲0.2⊳D11.31°. The moments

required to loosen the vise are

MBD1

12 ⊲160⊳tan⊲11.31°1.14°⊳D2.39 ft lb,

and MCDrC tan⊲s˛⊳ D1.2 ft-lb.

Problem 9.85 Suppose that you want to tighten the

vise in Problem 9.84 by turning one of the shafts. Deter-

mine the couple you must apply (a) to shaft B; (b) to

shaft C.

740

Problem 9.86 The threaded shaft has a ball and socket

support at B. The 400-lb load Acan be raised or lowered

by rotating the threaded shaft, causing the threaded collar

at Cto move relative to the shaft. Neglect the weights

of the members. The pitch of the shaft is pD1

4in, the

mean radius of the thread is rD1 in, and the coefﬁ–

cient of static friction between the thread and the mating

groove is 0.24. If the system is stationary in the posi-

tion shown, what couple is necessary to start the shaft

rotating to raise the load?

C

A

B

9 in

12 in

The sum of the moments about Dis

MDDLCDFcos⊲90 ˇ⊳ 18WD0,

from which FD500 lb. The pitch angle is

˛Dtan1p

2r D2.28°.

The angle of static friction is sDtan1⊲0.24⊳D13.5°. The moment

needed to start the threaded collar in motion is

MDrF tan⊲sC˛⊳ D1

12 ⊲500⊳tan⊲13.5°C2.28°⊳

D11.77 ft-lb

Dx

LCD

18 in

Problem 9.87 In Problem 9.86, if the system is

stationary in the position shown, what couple is

necessary to start the shaft rotating to lower the load?

Problem 9.88 The car jack is operated by turning the

horizontal threaded shaft at A. The threaded shaft ﬁts

into a mating threading collar at B. As the shaft turns,

points Aand Bmove closer together or farther apart,

thereby raising or lowering the jack. The pitch of the

threaded shaft is pD0.1 in, the mean radius of the

thread is rD0.2 in, and the coefﬁcient of kinetic friction

between the threaded shaft and the mating collar at Bis

0.15. What couple must be applied at Ato rotate the

shaft at a constant rate and raise the jack when it is in

the position shown if the load LD1400 lb?

3 in

B

L

A

742

Problem 9.89 The car jack is operated by turning the

horizontal threaded shaft at A. The threaded shaft ﬁts into

a mating threading collar at B. As the shaft turns, points

Aand Bmove closer together or farther apart, thereby

raising or lowering the jack. The pitch of the threaded

shaft is pD0.1 in, the mean radius of the thread is rD

0.2 in, and the coefﬁcient of kinetic friction between

the threaded shaft and the mating collar at Bis 0.15.

What couple must be applied at Ato rotate the shaft

at a constant rate and lower the jack when it is in the

position shown if the load LD1400 lb?

3 in

B

L

A

Problem 9.90 Aturnbuckle, used to adjust the length

or tension of a bar or cable, is threaded at both ends.

0.25 in, and the coefﬁcient of static friction between the

threads and the mating grooves is 0.24. If TD200 lb,

what couple must be exerted on the turnbuckle to start

tightening it?

Solution: The slope of the threads is

Using these values, one half of the required couple is

MDrF tan⊲kC˛⊳ D⊲0.25 in⊳⊲200 lb⊳tan⊲13.5°C1.82°⊳D13.7 in-lb.

Problem 9.91 Suppose that the pitch of the threads of

the turnbuckle is pD0.05 in, their mean radius is rD

0.25 in, and the coefﬁcient of static friction between the

threads and the mating grooves is 0.24. If TD200 lb,

what couple must be exerted on the turnbuckle to start

loosening it?

T

744