Problem 11.1 Determine the reactions at A.
Strategy: Subject the beam to three virtual motions:
(1) a horizontal displacement υx; (2) a vertical displace-
ment υy; and (3) a rotation υ about A.
300 N
2 m
A
800 N-m
2 m
y
x
Solution:
Problem 11.2
(a) Determine the virtual work done by the 2-kN force
and the 2.4 kN-m couple when the beam is rotated
through a counterclockwise angle υ about point A.
(b) Use the result of (a) to determine the reaction at B.
30°
400 mm400 mm
B
A
2.4 kN-m
2 kN
800 mm
Solution:
(a) The virtual work done by the 2-kN force and the 2.4 kN-m couple
is
so
Ax
Ay
2 kN 2.4 kN-m
0.4 m
B
0.4 m0.8 m
30°
Problem 11.3 Determine the tension in the cable.
Solution: When the beam rotates through a counterclockwise
angle υ, the virtual work is
Problem 11.4 The L-shaped bar is in equilibrium.
Determine F.
60 N
600 mm
F
Solution: Perform a virtual rotation about the pinned support:
Problem 11.5 The dimension LD4 ft and w0D
300 lb/ft. Determine the reactions at Aand B.
Strategy: To determine the virtual work done by the
distributed load, represent it by an equivalent force.
A
B
868
Problem 11.6 Determine the reactions at Aand B.
3 ft 3 ft 2 ft
AB
x
y
100 lb/ft
300 lb/ft
600 lb
To find Bwe do a virtual rotation about point A.
υU D⊲600 lb⊳⊲3ftυ⊳ ⊲900 lb⊳⊲1.5ftυ⊳ ⊲200 lb⊳⊲3.67 ft υ⊳
⊲200 lb⊳⊲4ftυ⊳ CB⊲5ftυ⊳
D[⊲600 lb⊳⊲3⊳⊲900 lb⊳⊲1.5⊳⊲200 lb⊳⊲3.67⊳
⊲200 lb⊳⊲4⊳CB⊲5⊳]ftυ D0
)BD937 lb
To find Aywe do a virtual rotation about point B.
υU D⊲600 lb⊳⊲8ftυ⊳ Ay⊲5ftυ⊳ C⊲900 lb⊳⊲3.5ftυ⊳
C⊲200 lb⊳⊲1.33 ft υ⊳ C⊲200 lb⊳⊲1ftυ⊳
D[⊲600 lb⊳⊲8⊳Ay⊲5⊳C⊲900 lb⊳⊲3.5⊳C⊲200 lb⊳⊲1.33⊳
C⊲200 lb⊳⊲1⊳]ftυ D0
)AyD237 lb
In summary AxD0,A
yD237 lb,BD937 lb
B
Ay
Ax
Problem 11.7 The mechanism is in equilibrium.
Determine the force Rin terms of F.
F
B
F
D
60°
Problem 11.8 Determine the reaction at the roller
support.
1.5 m
1 m
1 m
1.5 m
A
B
CD
E
F
1.5 m
200 N
Solution: Assume that Band Eremain fixed, and give bar ABC a
F
200 N
Problem 11.9 Determine the couple Mnecessary for
the mechanism to be in equilibrium.
600 N/m
0.3 m
0.5 m
0.9 m
M
Solution: Notice that ⊲0.3⊳υ D⊲0.4⊳υˇ.
0.3 m
870
Problem 11.10 The system is in equilibrium. The total
mass of the suspended load and assembly Ais 120 kg.
(a) By using equilibrium, determine the force F.
(b) Using the result of (a) and the principle of virtual
work, determine the distance the suspended load
rises if the cable is pulled upward 300 mm at B.B
F
Solution: The weight of assembly and load is WD1177.2N.
(b) Do a virtual translation of the assembly Ain the vertical direction.
The work done is υU DWυy CFυx D0, from which the ratio
Problem 11.11 Determine the force Pnecessary for
the mechanism to be in equilibrium.
P
200 mm
400 mm
400 mm 400 mm
Solution: Denote axial forces in the left horizontal member by R1
and the right horizontal member by R2. Do a virtual rotation of the
from which
2.
Do a virtual rotation of the middle vertical member about the pin
support:
Problem 11.12* Show that υx is related to υ˛ by
υx D⊲L1tan ˇ⊳υ˛.
L1
L2
b
Solution: The distance xcan be written
xDL1cos ˛CL2cos ˇ
υx DL1sin ˛υ˛L2sin ˇυˇ
872
Problem 11.13 The horizontal surface is smooth.
Determine the horizontal force Fnecessary for the
system to be in equilibrium. (See Active Example 11.1.)
9 in
400 in-lb 50⬚
6 in
Problem 11.14* Show that υx is related to υ˛ by
υx DL1xsin ˛
xL1cos ˛υ˛.
Strategy: Write the law of cosines in terms of ˛and
L1L2
α
Solution: Denote the horizontal distance from the pin support to
Problem 11.15 The linkage is in equilibrium. What is
the force F? (See Active Example 11.1.)
F
Problem 11.16 The linkage is in equilibrium. What is
the force F? (See Active Example 11.1.)
3 ft
6 ft
400 lb
4 ft 8 ft
F
4 ft
Solution: Denote the horizontal reaction at the roller support by
From the dimensions given,
874
Problem 11.17 Bar AC is connected to bar BD by a
pin that fits in the smooth vertical slot. The masses of
the bars are negligible. If MAD30 N-m, what couple
MBis necessary for the system to be in equilibrium?
0.7 m
0.4 m
D
C
Solution:
υx Ddx,
yD1
0.4υx. (1)
The virtual work is
so MADMBD30 N-m.
MA
δβ
x
MB
Problem 11.18 The angle ˛D20°, and the force
exerted on the stationary piston by pressure is 4 kN
toward the left. What couple Mis necessary to keep
the system in equilibrium?
M
240
mm
130
mm
α
Solution: Perform a virtual rotation of the crank: υU DMυ˛ C
Fυx D0, from which
Problem 11.19 The structure is subjected to a 400-N
load and is held in place by a horizontal cable. Determine
the tension in the cable.
2 m
400 N
60° 60°
2 m
Solution: Perform a virtual rotation about the pin supports. The
load can only move vertically, since the structure is a parallelogram.
and
Problem 11.20 If the load on the car jack is LD
6.5 kN, what is the tension in the threaded shaft between
Aand B?
65 mm
65 mm
120
mm
B
L
A
Solution: We have the constraint
876
Problem 11.21 Determine the reactions at Aand B.
(Use the equilibrium equations to determine the hori-
zontal components of the reactions, and use the proce-
dure described in Example 11.12 to determine the vertical
components.)
A
300 lb
60°
Solution: Denote the angle ˛D60°and the distance AC by L.
The sum of the moments about Ais
υU D300υx Ay
2CBy
2D0,
from which
The vertical reaction is
Problem 11.22 This device raises a load Wby ex-
tending the hydraulic actuator DE. The bars AD and
BC are each 2 m long, and the distances bD1.4 m and
hD0.8m.IfWD4 kN, what force must the actuator
exert to hold the load in equilibrium?
B
D
A
C
W
h
b
E
Solution: Perform a virtual vertical displacement of the load.
Denote the distance CD by x. The virtual work is υU DWυh C
Dυx D0, from which
878
Problem 11.23 Determine the force Pnecessary for
the mechanism to be in equilibrium.
P
F
F
600 mm
Solution: The height of pt. Cis
Problem 11.24 The collar Aslides on the smooth
vertical bar. The masses are mAD20 kg and mBD
10 kg.
(a) If the collar Ais given an upward virtual
displacement υy, what is the resulting downward
displacement of the mass B?
(b) Use virtual work to determine the tension in
the spring.
B
A
0.2 m
0.25 m
Solution: The motion is constrained by the constant length Lof
the string.
0DyAυyA
yA2Cd2CυyB)υyBD yAυyA
yA2Cd2
(a) If the motion of a is υy DυyAthen we have
υyBD 0.2m
⊲0.2m⊳2C⊲0.25 m⊳2⊲υy⊳ D0.625υy
Where positive means down
(b) υU D⊲20 kg⊳⊲9.81 m/s2⊳υy
C[⊲10 kg⊳⊲9.81 m/s2⊳Fspring]⊲0.625υy⊳
D[196.2NC⊲98.1N⊳⊲0.625⊳Fspring⊲0.625⊳]υy D0
Fspring D216 N
A
B
d = 0.25 m
880
c
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Problem 11.25 The potential energy of a conservative
system is given by VD2x3C3x212x.
(a) For what values of xis the system in equilibrium?
(b) Determine whether the equilibrium positions you
found in (a) are stable or unstable.
Solution:
(b) The stable and unstable positions are determined by the sign of
Problem 11.26 The potential energy of a conservative
system is given by VD2q321q2C72q.
(a) For what values of qis the system in equilibrium?
(b) Determine whether the equilibrium positions you
found in (a) are stable or unstable.
Solution:
Problem 11.27 The mass mD2 kg and the spring
constant kD100 N/m. The spring is unstretched when
xD0.
(a) Determine the value of xfor which the mass is in
equilibrium.
(b) In the equilibrium position stable or unstable?
(See Example 11.3.)
k
x
m
Solution:
Problem 11.28 The nonlinear spring exerts a force
kx Cεx3on the mass, where kand εare constants.
Determine the potential energy Vassociated with
the force exerted on the mass by the spring. (See
Example 11.3.)
x
Solution: The potential energy of the spring is
882
Problem 11.29 The 1-kg mass is suspended from
the nonlinear spring described in Problem 11.28. The
constants kD10 and εD1, where xis in meters.
(a) Show that the mass is in equilibrium when xD
1.12 m and when xD2.45 m.
(b) Determine whether the equilibrium positions are
stable or unstable.
(See Example 11.3.) x
Problem 11.30 The two straight segments of the bar
are each of weight Wand length L. Determine whether
the equilibrium position shown is stable if (a) 0 <˛
0<
90°; (b) 90°<˛
0<180°.
α
0
α
0
Solution: From a heuristic argument, if the bars hang straight
down (˛0D0) they are equivalent to one bar suspended at one end,
Choose a coordinate system with origin at the pin support with the x
The coordinates of the center of weight of the composite is
884
Problem 11.31 The homogeneous composite object
consists of a hemisphere and a cylinder. It is at rest on
the plane surface. Show that this equilibrium position is
stable only if L<R/
p2. (See Example 11.4.)
R
L
Solution: An angular disturbance will cause the composite system
to rock about the radial center of the hemisphere. The change in
Retain only the numerator, since the denominator must be positive
always: The condition for stability is
Problem 11.32 The homogeneous composite object
consists of a half-cylinder and a triangular prism. It is
at rest on the plane surface. Show that this equilibrium
position is stable only if h<p2R. (See Example 11.4.)
h
R
886