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Problem 9.1 In Active Example 9.1, suppose that the
coefficient of static friction between the 180-lb crate
and the ramp is sD0.3. What is the magnitude of
the smallest horizontal force the rope must exert on the
crate to prevent it from sliding down the ramp?
Problem 9.2 A person places a 2-lb book on a table
that is tilted at 5°relative to the horizontal. She finds that
if she exerts a very small force on the book as shown,
the book remains in equilibrium, but if she removes the
force, the book slides down the table. What force would
she need to exert on the book (in the direction parallel
to the table) to cause it to slide up the table?
15⬚
Problem 9.3 A student pushes a 200-lb box of books
across the floor. The coefficient of kinetic friction
between the carpet and the box is kD0.15.
(a) If he exerts the force Fat angle ˛D25°, what is
the magnitude of the force he must exert to slide
the box across the floor?
(b) If he bends his knees more and exerts the force
Fat angle ˛D10°, what is the magnitude of the
force he must exert to slide the box?
a
F
N
Problem 9.4 The 2975-lb car is parked on a sloped
street. The brakes are applied to both its front and rear
wheels.
(a) If the coefficient of static friction between the car’s
tires and the road is sD0.8, what is the steepest
slope (in degrees relative to the horizontal) on
which the car could remain in equilibrium?
(b) If the street were icy and the coefficient of static
friction between the car’s tires and the road was
sD0.2, what is the steepest slope on which the
car could remain in equilibrium?
686
Problem 9.5 The truck’s winch exerts a horizontal
force on the 200-kg crate in an effort to pull it down
the ramp. The coefficient of static friction between the
crate and the ramp is sD0.6.
(a) If the winch exerts a 200-N horizontal force on the
crate, what is the magnitude of the friction force
exerted on the crate by the ramp?
(b) What is the magnitude of the horizontal force the
winch must exert on the crate to cause it to start
moving down the ramp?
20⬚
Problem 9.6 The device shown is designed to position
pieces of luggage on a ramp. It exerts a force parallel
to the ramp. The suitcase weighs 40 lb. The coefficients
of friction between the suitcase and ramp are sD0.20
and kD0.18.
(a) Will the suitcase remain stationary on the ramp
when the device exerts no force on it?
(b) What force must the device exert to push the
suitcase up the ramp at a constant speed?
20⬚
688
Problem 9.7 The coefficient of static friction between
the 50-kg crate and the ramp is sD0.35. The
unstretched length of the spring is 800 mm, and the
spring constant is kD660 N/m.
What is the minimum value of xat which the crate can
remain stationary on the ramp?
k
x
FsD⊲660 N/m⊳⊲0.8mx⊳
Fs
490.5 N
Problem 9.8 The coefficient of kinetic friction bet-
ween the 40-kg crate and the slanting floor is kD0.3.
If the angle ˛D20°, what tension must the person exert
on the rope to move the crate at constant speed?
10⬚
a
Solution:
fD0.3N
T
392.4 N
f
Problem 9.9 In Problem 9.8, for what angle ˛is the
tension necessary to move the crate at constant speed a
minimum? What is the necessary tension?
fD0.3N
)TD184.1N
)
TD176.3N
Problem 9.10 Box Aweighs 100 lb, and box Bweighs
30 lb. The coefficients of friction between box Aand the
ramp are sD0.30 and kD0.28. What is the magni-
tude of the friction force exerted on box Aby the ramp?
A
B
690
Problem 9.11 In Problem 9.10, box Aweighs 100 lb,
and the coefficients of friction between box Aand the
ramp are sD0.30 and kD0.28. For what range
of the weights of the box Bwill the system remain
stationary?
D100⊲sin 30°C0.3 cos 30°⊳D75.98 lb
For impending motion down the ramp:
FDAsin ˛BMIN sAcos ˛D0,
from which
BDA⊲sin ˛scos ˛⊳
D100⊲sin 30°0.3 cos 30°⊳D24.02 lb
N
Problem 9.12 The mass of the box on the left is 30 kg,
and the mass of the box on the right is 40 kg. The coeffi-
cient of static friction between each box and the inclined
surface is sD0.2. Determine the minimum angle ˛for
with the boxes will remain stationary.
30⬚
a
Solution: If the boxes slip when ˛is decreased, they will slip
toward the right. Assume that slip toward the right impends, the free
body diagrams are as shown.
FxDT0.2N
A⊲30⊳⊲9.81⊳sin ˛D0,(1)
0.2 NA
0.2 NB
NANB
α
x
x
y
yTT
30°
Problem 9.13 The coefficient of kinetic friction between
the 100-kg box and the inclined surface is 0.35. Deter-
mine the tension Tnecessary to pull the box up the
T
692
Problem 9.14 The box is stationary on the inclined
surface. The coefficient of static friction between the
box and the surface is s.
(a) If the mass of the box is 10 kg, ˛D20°,ˇD30°,
and sD0.24, what force Tis necessary to start
the box sliding up the surface?
T
β
gD9.81 m/s2
Fx:Tcos ˇCfCmg sin ˛D0
Problem 9.15 To explain observations of ship launch-
ings at the port of Rochefort in 1779, Coulomb analyzed
the system shown in Problem 9.14 to determine the
minimum force Tnecessary to hold the box stationary
on the inclined surface. Show that the result is
694
Problem 9.16 Two sheets of plywood Aand Blie on
the bed of a truck. They have the same weight W, and
the coefficient of static friction between the two sheets
of wood and between sheet Band the truck bed is s.
(a) If you apply a horizontal force to sheet Aand apply
no force to sheet B, can you slide sheet Aoff the
truck without causing sheet Bto move? What force
is necessary to cause sheet Ato start moving?
(b) If you prevent sheet Afrom moving by applying
a horizontal force on it, what horizontal force on
sheet Bis necessary to start it moving?
B
A
FDf
AB DsW.
(b) The force on Bis the friction between Aand Band the friction
between Band the truck bed. Thus the force required to start B
in motion is
FBDf
AB Cf
BT D3sW.
fBT
FB
2W
Problem 9.17 The weights of the two boxes are W1D
100 lb and W2D50 lb. The coefficients of kinetic fric-
tion between the left box and the inclined surface are
sD0.12 and kD0.10. Determine the tension the
man must exert on the rope to pull the boxes upward
at a constant rate.
30⬚
30⬚
W2
W1
Problem 9.18 In Problem 9.17, for what range of
tensions exerted on the rope by the man will the boxes
remain stationary?
fD0.12 N
Next solve for the smallest force Tmin. We need to turn the friction
89.6lb<T<110.4lb
Problem 9.19 Each box weighs 10 lb. The coefficient
of static friction between box Aand box Bis 0.24, and
the coefficient of static friction between box Band the
inclined surface is 0.3. What is the largest angle ˛for
which box Bwill not slip?
A
B
696
Problem 9.20 The masses of the boxes are mAD
15 kg and mBD60 kg. The coefficient of static friction
between boxes Aand Band between box Band the
inclined surface is 0.12. What is the largest force Ffor
which the boxes will not slip? FA
B
f1D0.12N1,f
2D0.12N2
Solving we find FD267 N
20°
N1
f1
B
588.6 N
Problem 9.21 In Problem 9.20, what is the smallest
force Ffor which the boxes will not slip?
Problem 9.22 In Example 9.2, what clockwise couple
Mwould need to be applied to the disk to cause it to
rotate at a constant rate in the clockwise direction?
C
M
weighs 20 lb. The homogeneous disk weighs 30 lb. The
coefficient of kinetic friction between the disk and the
sloping surface is kD0.24. What is the magnitude of
the couple that would need to be applied to the disk
to cause it to rotate at a constant rate in the clockwise
direction?
AB
1 ft
MD11.2 ft-lb.
698
c
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Problem 9.24 The homogeneous horizontal bar AB
weighs 20 lb. The homogeneous disk weighs 30 lb. The
coefficient of kinetic friction between the disk and the
sloping surface is kD0.24. What is the magnitude of
the couple that would need to be applied to the disk to
cause it to rotate at a constant rate in the counterclock-
wise direction?
AB
5 ft
1 ft
Problem 9.25 The mass of the bar is 4 kg. The coef-
ficient of static friction between the bar and the floor is
0.3. Neglect friction between the bar and the wall.
(a) If ˛D20°, what is the magnitude of the friction
force exerted on the bar by the floor?
(b) What is the maximum angle ˛for which the bar
will not slip?
1 m
a
Solving: fBD7.14 N,f
max D11.77 N
Since fB<f
max,f
BD7.14 N
1.0 m
Problem 9.26 In Problem 9.25, suppose that the coef-
ficient of static friction between the bar and the floor
and between the 4-kg bar and the wall is 0.3. What is
the maximum angle ˛for which the bar will not slip?
fBD0.3NB,f
AD0.3NB
Fy:NBCfA39.24 N D0
MB:⊲39.24 N⊳⊲0.5m⊳sin ˛NA⊲1.0m⊳cos ˛
fA⊲1.0m⊳sin ˛D0
NA
1.0 m
700
Problem 9.27 The ladder and the person weigh 30 lb
and 180 lb, respectively. The center of mass of the 12-ft
ladder is at its midpoint. The angle ˛D30°. Assume that
the wall exerts a negligible friction force on the ladder.
(a) If xD4 ft, what is the magnitude of the friction
force exerted on the ladder by the floor?
(b) What minimum coefficient of static friction
between the ladder and the floor is necessary for
the person to be able to climb to the top of the
ladder without slipping?
x
a
Problem 9.28 In Problem 9.27, the ladder and the per-
son weigh 30 lb and 180 lb, respectively. The center of
mass of the 12-ft ladder is at its midpoint. The coefficient
of static friction between the ladder and the floor is sD
0.5. What is the largest value of the angle ˛for which
the person could climb to the top of the ladder without
it slipping?
Problem 9.29 In Problem 9.27, the ladder and the per-
son weigh 30 lb and 180 lb, respectively. The center of
mass of the 12-ft ladder is at its midpoint. The coefficient
of static friction between the ladder and the floor is 0.5
and the coefficient of friction between the ladder and the
wall is 0.3. What is the largest value of the angle ˛for
which the person could climb to the top of the ladder
without it slipping? Compare your answer to the answer
to Problem 9.28.
Solution:
xD⊲12 ft⊳sin ˛
Fx:fBNAD0
Fy:NB210 lb CfAD0
MB:NA⊲12 ft cos ˛⊳ 30 lb ⊲6 ft sin ˛⊳
180 lb xCfAxD0
fBD0.5NB,f
AD0.3NA
fA
NA
180 lb
α
702
Problem 9.30 The disk weighs 50 lb and the bar
weighs 25 lb. The coefficients of friction between the
disk and the inclined surface are sD0.6 and kD0.5.
(a) What is the largest couple Mthat can be applied
to the stationary disk without causing it to
start rotating?
(b) What couple Mis necessary to rotate the disk at a
constant rate?
M
5 in
20 in
fD0.6N
)MD162.4lbin
(b) Steady rotation. Replace the last equation with
fD0.5N )MD135.3lbin
30°
M
f
N
B2
Ax
25 lb
Problem 9.31 The radius of the 40-kg homogeneous
cylinder is R = 0.15 m. The slanted wall is smooth
and the angle ˛D30°. The coefficient of static friction
between the cylinder and the floor is sD0.2. What is
the largest couple Mthat can be applied to the cylinder
without causing it to slip?
M
R
a
Problem 9.32 The homogeneous cylinder has weight
W. The coefficient of static friction between the cylinder
and both surfaces is s. What is the largest couple M
that can be applied to the cylinder without causing it to
slip? (Assume that the cylinder slips before rolling up
M
R
Problem 9.33 The homogeneous cylinder has weight
W. The coefficient of static friction between the cylinder
and both surfaces is s. What is the minimum value of
sfor which the couple Mwill cause the cylinder to
roll up the inclined surface without slipping?
M
R
a
704
