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Problem 9.122* The “Morse taper” is used to support
the workpiece on a machinist’s lathe. The taper is driven
into the spindle and is held in place by friction. If the
spindle exerts a uniform pressure pD15 psi on the taper
and sD0.2, what couple must be exerted about the
axis of the taper to loosen it?
Taper
2 in
Spindle
1.25 in
9 in
Solution: The outer radius of the taper is roD1 in, and the inner
radius is riD0.6.5 in. The angle of the taper is
check. The normal force on the taper is NDpA D689.8 lb, and the
axial force on the taper is FDNcos ˛D28.72 lb. The taper is equiv-
Problem 9.123 In Active Example 9.9, suppose that
the left fixed cylinder is replaced by a pully. Assume
that the tensions in the rope on each side of the pulley
are approximately equal. What is the smallest force the
woman needs to exert on the rope to support the
stationary box?
Problem 9.124 Suppose that you want to lift a 50-lb
crate off the ground by using a rope looped over a tree
limb as shown. The coefficient of static friction between
the rope and the limb is 0.2, and the rope is wound 135°
around the limb. What force must you exert to begin
lifting the crate?
Problem 9.125 Winches are used on sailboats to help
support the forces exerted by the sails on the ropes
(sheets) holding them in position. The winch shown is a
post that will rotate in the clockwise direction (seen from
above), but will not rotate in the counterclockwise direc-
tion. The sail exerts a tension TSD800 N on the sheet,
which is wrapped two complete turns around the winch.
The coefficient of static friction between the sheet and
the winch is sD0.2. What tension TCmust the crew
member exert on the sheet to prevent it from slipping
on the winch?
TS
TC
TsD800 N sD0.2
ˇD4
Solving,
TcD64.8N
Problem 9.126 The coefficient of kinetic friction
between the sheet and the winch in Problem 9.125 is
kD0.16. If the crew member wants to let the sheet slip
at a constant rate, releasing the sail, what initial tension
TCmust he exert on the sheet as it begins slipping?
766
Problem 9.127 The box Aweights 20 lb. The rope
is wrapped one and one-fourth turns around the fixed
wooden post. The coefficients of friction between the
rope and post are sD0.15 and kD0.12.
(a) What minimum force does the man need to exert
to support the stationary box?
(b) What force would the man have to exert to raise
the box at a constant rate?
Solution:
Problem 9.128 The weight of the block Ais W. The
disk is supported by a smooth bearing. The coefficient
of kinetic friction between the disk and the belt is k.
What couple Mis necessary to turn the disk at a constant
rate?
rM
Problem 9.129 The couple required to turn the wheel
of the exercise bicycle is adjusted by changing the
weight W. The coefficient of kinetic friction between
the wheel and the belt is k. Assume the wheel turns
clockwise.
(a) Show that the couple Mrequired to turn the wheel
is MDWR⊲1e3.4k⊳.
(b) If WD40 lb and kD0.2, what force will the
scale S indicate when the bicycle is in use?
15°
R
S
Problem 9.130 The box Bweighs 50 lb. The coeffi-
cient of friction between the cable and the fixed round
supports are sD0.4 and kD0.3.
(a) What is the minimum force Frequired to support
the box?
(b) What force Fis required to move the box upward
at a constant rate? BF
Problem 9.131 The coefficient of static friction
between the 50-lb box and the inclined surface is 0.10.
The coefficient of static friction between the rope and the
fixed cylinder is 0.05. Determine the force the woman
must exert on the rope to cause the box to start moving
up the inclined surface.
20⬚30⬚
45⬚
A
768
Problem 9.132 In Problem 9.131, what is the mini-
mum force the woman must exert on the rope to hold
the box in equilibrium on the inclined surface?
Solution: See 9.131 - Change the friction force
F%:T2cos 25°Cf50 lb sin 20°D0
F-:T2sin 25°CN50 lb cos 20°D0
fD0.1N
T2DTe0.05⊲7/12⊳
Solving TD13.10 lb
50 lb
N
f
T2
Problem 9.133 Blocks Band Ceach have a mass of
20 kg. The coefficient of static friction at the contacting
surfaces is 0.2. Block Ais suspended by a rope that
passes over a fixed cylinder and is attached to block B.
The coefficient of static friction between the rope and
the cylinder is 0.3. What is the largest mass block Acan
have without causing block Bto slip the left?
A
C
B
20⬚
Problem 9.134 If the force Fin Example 9.10 is
increased to 400 N, what are the largest values of the
couples MAand MBfor which the belt will not slip?
MA
b
RA
F
MB
F
T2
T2
MB
a
a
MA
MA
RB
Ax
Solution: From Example 9.10, bD500 mm, sD0.8, RaD
200 mm, RbD100 mm. The angle of contact for pulley Ais ˇaD
C2˛. The angle of contact for pulley Bis ˇbD2˛, where
˛Dsin1RaRb
bDsin10.1
0.5D0.2014 radians.
The belt contact is less for pulley B, so it is most likely to slip first.
The couples are in opposition so that the tension in the upper belt is
greater than the tension in the lower belt: For belt B:
Tupper DTloweresˇbD8.945Tlower.
770
Problem 9.135 The spring exerts a 320-N force on the
left pulley. The coefficient of static friction between the
flat belt and the pulleys is sD0.5. The right pulley
cannot rotate. What is the largest couple Mthat can be
exerted on the left pulley without causing the belt to
slip?
100 mm
M
260 mm
40 mm
Solution: The angle of the belt relative to the horizontal is
Since the angle of contact is less on the right pulley, it should slip
Problem 9.136 The weight of the box is WD30 lb,
and the force Fis perpendicular to the inclined surface.
The coefficient of static friction between the box and the
inclined surface is sD0.2.
(a) If FD30 lb, what is the magnitude of the friction
force exerted on the stationary box?
(b) If FD10 lb, show that the box cannot remain at
rest on the inclined surface.
F
20°
W
Problem 9.137 In Problem 9.136, what is the smallest
force Fnecessary to hold the box stationary on the
inclined surface?
Problem 9.138 Blocks Aand Bare connected by a
horizontal bar. The coefficient of static friction between
the inclined surface and the 400-lb block Ais 0.3. The
coefficient of static friction between the surface and the
300-lb block Bis 0.5. What is the smallest force Fthat
will prevent the blocks from slipping down the surface?
F
A
B
45°
Solution: The (horizontal) connecting bar exerts a component of
772
Problem 9.139 What force Fis necessary to cause the
blocks in Problem 9.138 to start sliding up the plane?
flink DB
⊲1sB⊳⊲sin ˛CsBcos ˛⊳ D636.4lb.
Isolate A. The resultant force on Ais
Fflink DA⊲sin ˛CsAcos ˛⊳ sAflink D176.78 lb.
The resultant force required to cause the blocks to start to move up
the plane is FD636.4C176.8D813.2lb
F
A
NA
SANA
µ
Problem 9.140 The masses of crates Aand Bare 25 kg
and 30 kg, respectively. The coefficient of static friction
between the contacting surfaces is sD0.34. What is
the largest value of ˛for which the crates will remain
in equilibrium? A
B
α
Problem 9.141 The side of a soil embankment has a
45°slope (Fig. a). If the coefficient of static friction of
soil on soil is sD0.6, will the embankment be stable
or will it collapse? If it will collapse, what is the smallest
slope that can be stable?
Strategy: Draw a free-body diagram by isolating part
of the embankment as shown in Fig. b.
(a)
45°
(b)
θ
774
Problem 9.142 The mass of the van is 2250 kg, and
the coefficient of static friction between its tires and the
road is 0.6. If its front wheels are locked and its rear
wheels can turn freely, what is the largest value of ˛for
which it can remain in equilibrium?
1 m
Solution: Choose a coordinate system with the xaxis parallel
to the incline. The weight of the van is WDmg D22072.5 N. The
moment about the point of contact of the rear wheels is
MRD⊲31.2⊳W cos ˛C1Wsin ˛3ND0,
3.
The sum of the forces parallel to the inclined surface is
s1.8
3cos ˛Cs
31sin ˛D0,
from which
˛Dtan11.8s
3sDtan1⊲0.45⊳D24.2°
1 m
1.2 m
1.8 m
N
Problem 9.143 In Problem 9.142, what is the largest
value of ˛for which the van can remain in equilibrium
if it points up the slope?
Problem 9.144 The shelf is designed so that it can be
placed at any height on the vertical beam. The shelf is
supported by friction between the two horizontal cylin-
ders and the vertical beam. The combined weight of the
shelf and camera is W. If the coefficient of static friction
between the vertical beam and the horizontal cylinders
is s, what is the minimum distance bnecessary for the
shelf to stay in place?
bt
h
W
776
