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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
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
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
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
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,
FxDfCm2gcos CN2cos D0,
FyDNm2gsin 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 DWsin ˛CsB, from
which
Wsin ˛
s
DW
4⊲2 cos ˛C2.25 sin ˛⊳.
Reduce algebraically to obtain
˛Dtan1s
21.125sD14.47°
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?
MBD2.25Wsin ˛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
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 > 2s. (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:NLCNRmg 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?
NRCNFmg Psin ˛D0(2)
From Eqn (1),
NFD1
⊲S Pcos ˛⊳ (a)
kdP
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
˛Dtan1
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
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
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
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:AxFBC cos 45°D0(1)
Fy:AyFCFBC 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°⊳⊲AymLg⊳ 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
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) MCDC2DxDyC8D0,
(9) FyDDyCyD0.
(10) MDD2GED0
CyD18 kN,
CxD13 kN,
8 kN
fGDX
DX
DY
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
1 m
724
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