Problem 9.145 The 20-lb homogenous object is
supported at Aand B. The distance hD4 in, friction can
be neglected at B, and the coefficient of static friction
at Ais 0.4. Determine the largest force Fthat can be
exerted without causing the object to slip. F
6 in
2 in
h
AB
4 in
Solution: Choose a coordinate system with origin at Aand the x
The sum of the forces parallel to xis
Problem 9.146 In Problem 9.145, suppose that the
coefficient of static friction at Bis 0.36. What is the
largest value of hfor which the object will slip before
it tips over?
htip D⊲4x⊳W
F.
The sum of the forces parallel to the yaxis is
FyDACBWD0,
Problem 9.147 The 180-lb climber is supported in the
“chimney” by the normal and friction forces exerted on
his shoes and back. The static coefficients of friction
between his shoes and the wall and between his back
and the wall are 0.8 and 0.6, respectively. What is the
minimum normal force his shoes must exert?
D0.
Reduce to two simultaneous equations in two unknowns:
where
det Da11a22 a12 a21 D1.4529
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Problem 9.148 The sides of the 200-lb door fit loosely
into grooves in the walls. Cables at Aand Braise the
door at a constant rate. The coefficient of kinetic friction
between the door and the grooves is kD0.3. What
force must the cable Aexert to continue raising the door
at a constant rate if the cable at Bbreaks?
3 ft3 ft
AB
3 ft
Problem 9.149 The coefficients of static friction
between the tires of the 1000-kg tractor and the ground
and between the 450-kg crate and the ground are 0.8 and
0.3, respectively. Starting from rest, what torque must
the tractor’s engine exert on the rear wheels to cause the
crate to move? (The front wheels can turn freely.)
0.8 m
0.4 m
1.4 m 0.8 m
exerted by the ground on the tires,
Problem 9.150 In Problem 9.149, what is the most
massive crate the tractor can cause to move from rest
if its engine can exert sufficient torque? What torque is
necessary?
WcD0.8stWt
gD1134.75 D1134.8kg.
780
Problem 9.151 The mass of the vehicle is 900 kg, it
has rear-wheel drive, and the coefficient of static friction
between its tires and the surface is 0.65. The coefficient
of static friction between the crate and the surface is 0.4.
If the vehicle attempts to pull the crate up the incline,
what is the largest value of the mass of the crate for
which it will slip up the incline before the vehicle’s tires
slip?
0.8 m
20°
2.5 m
1.5 m
1.2 m
20°
Solution: The normal force between the crate and the incline
is NcDWccos , where D20°. The drawbar force parallel to the
incline is FdDWcsin scNc. For brevity write Dsin C
sc cos . The horizontal component of the drawbar force at the
Problem 9.152 Each 1-m bar has a mass of 4 kg. The
coefficient of static friction between the bar and the
surface at Bis 0.2. If the system is in equilibrium, what
is the magnitude of the friction force exerted on the bar
at B?
A
45°
M⊲leftend⊳D⊲1⊳cos 45°Ay⊲1⊳cos 45°Ax⊲0.5⊳cos 45°mg
D0,
Right bar:
FxDAxCfcos 30°Nsin 30°D0,
FyDAymg Cfsin 30°CNcos 30°D0,
M⊲rightend⊳D⊲1⊳cos 45°AxC⊲1⊳cos 45°AyC⊲0.5⊳cos 45°mg
D0,
Solving, we obtain ND43.8 N and fD2.63 N.
Problem 9.153 In Problem 9.152, what is the mini-
mum coefficient of static friction between the bar and
the surface at Bnecessary for the system to be in equi-
librium?
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Problem 9.154 The collars Aand Beach have a mass
of 2 kg. If friction between collar Band the bar can be
neglected, what minimum coefficient of static friction
between collar Aand the bar is necessary for the collars
to remain in equilibrium in the position shown?
B
Problem 9.155 In Problem 9.154, if the coefficient of
static friction has the same value sbetween collars
Aand Band the bars, what minimum value of s,is
necessary for the collars to remain in equilibrium in the
position shown? (Assume that slip impends at Aand B.)
Isolate Collar B: The sum of forces:
FxDNBcos CsNBcos Tcos ˛D0.
FyDNBsin sNBsin Tsin ˛D0.
These are four equations in four unknowns. Solve: TD45.4N,NAD
42.6N,NBD55 N, and sD0.0963 . This is the minimum coeffi–
cient of friction required to maintain equilibrium.
Problem 9.156 The clamp presses two pieces of wood
together. The pitch of the threads is pD2 mm, the mean
radius of the thread is rD8 mm, and the coefficient of
kinetic friction between the thread and the mating groove
is 0.24. What couple must be exerted on the thread shaft
to press the pieces of wood together with a force of
200 N?
50 mm
50 mm
125 mm
A
C
BE
125 mm 125 mm
we find that BE D500 N. The compressive load in BE is 500 N.
D1.13 N-m.
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Problem 9.157 In Problem 9.156, the coefficient of
static friction between the thread and the mating groove
is 0.28. After the threaded shaft is rotated sufficiently to
press the pieces of wood together with a force of 200 N,
what couple must be exerted on the shaft to loosen it?
BE D500 N (compression)
CxD500 N
CyD200 N
We don’t have to solve for additional forces because we used the fact
that member BE was a two force member.
From Problem 9.170, PD2mm,rD8 mm. We have sD0.28. We
want to loosen the clamp (Turn the clamp such that the motion is in
the direction of the axial force.
To do this,
MDrF tan⊲s˛⊳
where
tan sDsD0.28
sD15.64°
200 N 0.05 m
0.05 m
A
y
BE
B
x
C
CY
CX
0.125 m 0.125 m
Problem 9.158 The axles of the tram are supported
by journal bearings. The radius of the wheels is 75 mm,
If the weight of the tram and its load is evenly divided
between the axles, what force Pis necessary to push the
tram at a constant speed?
P
Solution: Assume that there are two bearings per axle. The weight
of the tram is WDmg D1569.6 N. This load is divided between four
4D392.4N.
The angle of kinetic friction is kDtan1⊲k⊳D7.97°. The moment
RD0.8161
0.075 D10.88 N.
Problem 9.159 The two pulleys have a radius of 6 in
and are mounted on shafts of 1-in radius supported by
journal bearings. Neglect the weights of the pulleys and
shafts. The coefficient of kinetic friction between the
shafts and the bearings is kD0.2. If a force TD200 lb
is required to raise the man at a constant rate, what is
his weight?
Solution: Denote the tension in the horizontal portion of the cable
by H. The angle of kinetic friction is kDtan1⊲0.2⊳D11.31°
Consider the right pulley: The force on the right pulley is
190.97 D191 lb.
Consider the left pulley: The force on the pulley is Fleft DpW2CH2.
The applied moment is Mapplied D⊲H W⊳R, from which ⊲H
199.9 lb, 182.25 lb. By an analogous argument to that used in
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Problem 9.160 If the man in Problem 9.159 weighs
160 lb, what force Tis necessary to lower him at a
constant rate?
Problem 9.161 If the two cylinders are held fixed, what
is the range of Wfor which the two weights will remain
stationary?
W100 lb
s = 0.34
k = 0.32
µ
µ
s = 0.30
k = 0.28
µ
µ
2.
Begin on the left with the known weight. Suppose that the known
Problem 9.162 In Problem 9.161, if the system is
initially stationary and the left cylinder is slowly rotated,
determine the largest weight Wthat can be (a) raised;
109.88 D110 lb, where the kinetic coefficient of friction is used, since
the rope slips at a steady rate. Thus (a) WD110 lb .
788