Mechanical Engineering Chapter 9 Problem The Coefcient Static Friction Bet Ween The Blades The Shears And

Problem 9.34 The coefﬁcient of static friction bet-

ween the blades of the shears and the object they are

gripping is 0.36. What is the largest value of the angle

˛for which the object will not slip out? Neglect the

object’s weight.

Strategy: Draw the free-body diagram of the object

and assume that slip is impending.

a

Solution:

f

N

α

Problem 9.35 The stationary disk of 300-mm radius is

attached to a pin support at D. The disk is held in place

by the brake ABC in contact with the disk at C. The

hydraulic actuator BE exerts a horizontal 400-N force

on the brake at B. The coefﬁcients of friction between

the disk and the brake are sD0.6 and kD0.5. What

couple must be applied to the stationary disk to cause it

to slip in the counterclockwise direction?

200 mm

B

E

A

200 mm

CD

300

mm

Problem 9.36 The ﬁgure shows a preliminary concep-

tual idea for a device to exert a braking force on a rope

when the rope is pulled downward by the force T. The

coefﬁcient of kinetic friction between the rope and the

two bars is kD0.28. Determine the force Tnecessary

to pull the rope downward at a constant rate if FD10 lb

and (a) ˛D30°; (b) ˛D20°.

F

FF

T

6 in

3 in

aa

F

Solution: From the rope we see that TD2f

On the verge of slipping we have

fD0.28 N

(a) Solving if ˛D30°TD9.42 lb

T

ff

N

Ax

Ay

α

706

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 9.37 The mass of block Bis 8 kg. The coefﬁ–

cient of static friction between the surfaces of the clamp

and the block is sD0.2. When the clamp is aligned

as shown, what minimum force must the spring exert to

prevent the block from slipping out?

45°

160 mm

0.16

D236 N.

Problem 9.38 By altering its dimensions, redesign the

clamp in Problem 9.37 so that the minimum force the

spring must exert to prevent the block from slipping out

is 180 N. Draw a sketch of your new design.

Solution: This problem does not have a unique solution.

Problem 9.39 The horizontal bar is attached to a collar

that slides on the smooth vertical bar. The collar at P

slides on the smooth horizontal bar. The total mass of the

horizontal bar and the two collars is 12 kg. The system

is held in place by the pin in the circular slot. The pin

contacts only the lower surface of the slot, and the coef-

ﬁcient of static friction between the pin and the slot is

0.8. If the system is in equilibrium and yD260 mm,

what is the magnitude of the friction force exerted on

the pin by the slot?

P

y

and friction forces exerted by the curved slot.

yD260 mm D⊲300 mm⊳sin ,

so the angle D60.1°.

From the equilibrium equations,

FxDfCm2gcos CN2cos D0,

FyDNm2gsin N2sin D0,

we obtain

fD⊲m2gCN2⊳cos D⊲m2Cm1⊳cos D⊲12⊳⊲9.81⊳cos 60.1°

D58.7N.

m2g

x

N

Problem 9.40 In Problem 9.39, what is the minimum

height yat which the system can be in equilibrium?

708

Problem 9.41 The rectangular 100-lb plate is suppor-

ted by the pins Aand B. If friction can be neglected at

Aand the coefﬁcient of static friction between the pin

at Band the slot is sD0.4, what is the largest angle

˛for which the plate will not slip?

B

A

2 ft 3 in

α

The component of weight causing the plate to slide is FDWsin ˛.

This must be balanced by the friction force: 0 DWsin ˛CsB, from

which

Wsin ˛

s

DW

4⊲2 cos ˛C2.25 sin ˛⊳.

Reduce algebraically to obtain

˛Dtan1s

21.125sD14.47°

B

W

A

Problem 9.42 If you can neglect friction at Bin

Problem 9.41 and the coefﬁcient of static friction

between the pin at Aand the slot is sD0.4, what is

the largest angle ˛for which the 100-lb plate will not

slide?

MBD2.25Wsin ˛C2Wcos ˛4AND0,

Wsin ˛

s

DW

4⊲2.25 sin ˛C2 cos ˛⊳.

Check: The normal reactions at Aand Bare unequal: as the slots

are inclined from the horizontal, the parallel component of the gravity

AN

W

Problem 9.43 The airplane’s weight is WD2400 lb.

Its brakes keep the rear wheels locked, and the

coefﬁcient of static friction between the wheels and

the runway is sD0.6. The front (nose) wheel can

turn freely and so exerts only a normal force on the

runway. Determine the largest horizontal thrust force T

the plane’s propeller can generate without causing the

rear wheels to slip.

W

AB

5 ft 2 ft

T

4 ft

Solution: The free body diagram when slip of the rear wheels

impends is shown. From the equilibrium equations

we obtain

AD1120 lb,

BD1280 lb,

and TD766 lb.

y

Problem 9.44 The refrigerator weighs 220 lb. It is

supported at Aand B. The coefﬁcient of static friction

between the supports and the ﬂoor is sD0.2. If you

assume that the refrigerator does not tip over before it

slips, what force Fis necessary for impending slip?

h

AB

F

b

710

Problem 9.45 The refrigerator weighs 220 lb. It is

supported at Aand B. The coefﬁcient of static fric-

tion between the supports and the ﬂoor is sD0.2. The

distance hD60-in and the dimension bD30 in. When

the force Fis applied to push the refrigerator across the

ﬂoor, will it tip over before it slips? (See Example 9.3.)

F

Problem 9.46 To obtain a preliminary evaluation of

the stability of a turning car, imagine subjecting the

stationary car to an increasing lateral force Fat the height

of its center of mass, and determine whether the car will

slip (skid) laterally before it tips over. Show that this will

be the case if b/h > 2s. (Notice the importance of the

height of the center of mass relative to the width of the

car. This reﬂects on recent discussions of the stability of

sport utility vehicles and vans that have relatively high

centers of mass.)

F

b

_

2

b

_

2

h

Fy:NLCNRmg D0

712

Problem 9.47 The man exerts a force Pon the car at

an angle ˛D20°. The 1760-kg car has front wheel drive.

The driver spins the front wheels, and the coefﬁcient of

kinetic friction is kD0.02. Snow behind the rear tires

exerts a horizontal resisting force S. Getting the car to

move requires overcoming a resisting force SD420 N.

What force Pmust the man exert?

P

S

0.90 m

1.62 m

2.55 m

3.40 m

α

Solution:

y

Problem 9.48 In Problem 9.47, what value of the angle

˛minimizes the magnitude of the force Pthe man must

exert to overcome the resisting force SD420 N exerted

on the rear tires by the snow? What force must he exert?

NRCNFmg Psin ˛D0(2)

From Eqn (1),

NFD1

⊲S Pcos ˛⊳ (a)

kdP

d˛ cos ˛CPsin ˛C0.90 dP

d˛ cos ˛0.90 Psin ˛

3.40 dP

d˛ D0D

0.90 cos ˛3.40 sin ˛2.55

k

cos ˛

714

Problem 9.49 The coefﬁcient of static friction bet-

ween the 3000-lb car’s tires and the road is sD0.5.

Determine the steepest grade (the largest value of the

angle ˛) the car can drive up at constant speed if the car

has (a) rear-wheel drive; (b) front-wheel drive; (c) four-

wheel drive.

19 in

72 in 35 in

α

Solution: The friction force acts parallel to the incline, and the

˛Dtan1





35

107

s

19





D10.18°

(c) For four wheel drive: Use the reactions of the front and rear

72 in

α

Problem 9.50 The stationary cabinet has weight W.

Determine the force Fthat must be exerted to cause it

to move if (a) the coefﬁcient of static friction at Aand

Bis s; (b) if the coefﬁcient of static friction at Ais sA

and the coefﬁcient of static friction at Bis sB.

b

—

b

—

H

h

AB

FG

from which

716

Problem 9.51 The table weighs 50 lb and the

coefﬁcient of static friction between its legs and the

inclined surface is 0.7.

(a) If you apply a force at Aparallel to the inclined

surface to push the table up the inclined surface,

will the table tip over before it slips? If not, what

force is required to start the table moving up

the surface?

(b) If you apply a force at Bparallel to the inclined

surface to push the table down the inclined surface,

will the table tip over before it slips? If not, what

force is required to start the table moving down

the surface?

A

B

28 in

23 in

32 in

20⬚

f1D0.7N1,f

2D0.7N2

f1

N2

Problem 9.52 The coefﬁcient of static friction between

the right bar and the surface at Ais sD0.6. Neglect the

weights of the bars. If ˛D20°, what is the magnitude

of the friction force exerted at A?

A

F

αα

Solution: Note that the condition of impending slip does not

necessarily apply. The moments about the left pin support:

FF

Problem 9.53 Consider the system shown in Problem

9.52. The coefﬁcient of static friction between the right

bar and the surface at Ais sD0.6. Neglect the weight

of the bars. What is the largest angle ˛at which the truss

will remain stationary without slipping?

30.96°

718

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 9.54 The bar BC is supported by a rough

ﬂoor at C.IfFD2 kN and the bar BC does not slip

at C, what is the magnitude of the friction force exerted

on the bar at C?

F

AB

600 mm

Problem 9.55 The bar BC is supported by a rough

ﬂoor at C.IfFD2 kN what is the minimum coefﬁcient

of static friction for which bar BC will not slip at C?

F

AB

600 mm

Solution: See the solution to Problem 9.54. Equilibrium requires

that fD1

Problem 9.56 The weight of the box is WD20 lb and

the coefﬁcient of static friction between the box and the

ﬂoor is sD0.65. Neglect the weights of the bars. What

is the largest value of the force Fthat will not cause the

box to slip?

4 in

F

8 in

W

8 in

4 in

Solution: Note that BC is a two force member.

Fx:AxFBC cos 45°D0(1)

Fy:AyFCFBC sin 45°D0(2)

AX

F

720

Problem 9.57 The mass of the suspended object is

6 kg. The structure is supported at Bby the normal and

friction forces exerted on the plate by the wall. Neglect

the weights of the bars.

(a) What is the magnitude of the friction force exerted

on the plate at B?

(b) What is the minimum coefﬁcient of static friction

at Bnecessary for the structure to remain in equi-

librium?

30°

8°

Plate

A

C

B

Problem 9.58 Suppose that the lengths of the bars in

Problem 9.57 are LAB D1.2 m and LAC D1.0 m and

their masses are mAB D3.6kgandmAC D3.0 kg.

(a) What is the magnitude of the friction force exerted

on the plate at B?

(b) What is the minimum coefﬁcient of static friction

at Bnecessary for the structure to remain in equi-

librium?

C⊲1.2 cos 8°⊳⊲AymLg⊳ D0(3)

mAC D3.0kg

y

0.5 m

60°

722

Problem 9.59 The frame is supported by the normal

and friction forces exerted on the plates at Aand Gby

the ﬁxed surfaces. The coefﬁcient of static friction at A

is sD0.6. Will the frame slip at Awhen it is subjected

to the loads shown? AB C

6 kN

8 kN

EGD

1 m 1 m 1 m

1 m

Solution: The strategy is to write the equilibrium equations and

solve for the unknown. The complete structure as a free body: Denote

the normal forces as Aand G, and the friction forces as fAand fG.

The sum of forces:

(7) MCDC2DxDyC8D0,

(9) FyDDyCyD0.

(10) MDD2GED0

CyD18 kN,

CxD13 kN,

8 kN

fGDX

DX

DY

E

G

The assumed directions are shown in the Figure; a negative sign means

sA DfA

Problem 9.60 The frame is supported by the normal

and friction forces exerted on the plate at Aby the wall.

(a) What is the magnitude of the friction force exerted

on the plate at A?

(b) What is the minimum coefﬁcient of static friction

at Anecessary for the structure to remain in equi-

librium?

A

E

D

B

C

6 kN

1 m

724