PROBLEM 10.88
Collar A can slide freely on the semicircular rod shown. Knowing that the constant
of the spring is k and that the unstretched length of the spring is equal to the radius
r, determine the value of
corresponding to equilibrium when
50 lb, 9 in.,Wr
and k 15 lb/in.
SOLUTION
Stretch of spring
2
2( cos )
(2cos 1)
1cos 2
2
sABr r r
sr
VksWr
PROBLEM 10.89
Two bars AB and BC of negligible weight are attached to a
single spring of constant k that is unstretched when the bars
are horizontal. Determine the range of values of the
magnitude P of two equal and opposite forces
P
and
P
for
which the equilibrium of the system is stable in the position
shown.
SOLUTION
PROBLEM 10.89 (Continued)
Note: For 1
2,
P
kl we have 2
20
dV
d
and we must determine which is the first derivative to be 0.
Differentiating Eq. (1):
PROBLEM 10.90
A vertical bar AD is attached to two springs of constant k and is in
equilibrium in the position shown. Determine the range of values of the
magnitude P of two equal and opposite vertical forces
P
and
P
for which
the equilibrium position is stable if (a)
,AB CD
(b)
2.AB CD
SOLUTION
For both (a) and (b): Since
P
and
P
are vertical, they form a couple of moment
sin
P
MPl
PROBLEM 10.90 (Continued)
For Stability:
2
2
2
1
0, 0
2
dV Pl ka
d
or (for Parts a and b)
2
2
ka
Pl
PROBLEM 10.91
Rod AB is attached to a hinge at A and to two springs, each of
constant k. If
25 in., 12 in.,hd
and
80 lb,W
determine the
range of values of k for which the equilibrium of the rod is stable
in the position shown. Each spring can act in either tension or
compression.
SOLUTION
We have
sin cos
CB
xd yh
Potential Energy:
2
22
1
22
sin cos
CB
VkxWy
kd Wh
PROBLEM 10.91 (Continued)
Since 4
4
21
2,4 0,
dV
d
kd Wh Wh Wh
we conclude that the equilibrium is unstable for 21
2
kd Wh and
the sign in Equation (2) is correct.
With 80 lb, 25 in., and 12 in.Wh d
PROBLEM 10.92
Rod AB is attached to a hinge at A and to two springs, each of
constant k. If 45 in., 6 lb/in.,
hk and 60 lb,W determine the
smallest distance d for which the equilibrium of the rod is stable in
the position shown. Each spring can act in either tension or
compression.
SOLUTION
Using Equation (2) of Problem 10.91 with
45 in., 6 lb/in., and 60 lbhk W
PROBLEM 10.93
Two bars are attached to a single spring of constant k
that is unstretched when the bars are vertical.
Determine the range of values of P for which the
equilibrium of the system is stable in the position
shown.
SOLUTION
2
sin sin
33
LL
s
For small values of
and
2
2
21
cos cos
33 2
LL
VP ks
PROBLEM 10.94
Two bars are attached to a single spring of constant k
that is unstretched when the bars are vertical.
Determine the range of values of P for which the
equilibrium of the system is stable in the position
shown.
SOLUTION
2sin sin
33
LL
a
For small values of
and
2
sin
3
L
s
PROBLEM 10.95
The horizontal bar BEH is connected to three vertical bars. The collar at
E can slide freely on bar DF. Determine the range of values of Q for
which the equilibrium of the system is stable in the position shown when
a 24 in., b 20 in., and P 150 lb.
SOLUTION
First note
sin sinAa b
For small values of
and :
ab
PROBLEM 10.95 (Continued)
Stability:
22
22
0: 2 0
dV a PQ
db
2
2
2b
P
Q
a
(1)
PROBLEM 10.96
The horizontal bar BEH is connected to three vertical bars. The collar
at E can slide freely on bar DF. Determine the range of values of P for
which the equilibrium of the system is stable in the position shown when
a 150 mm, b 200 mm, and Q 45 N.
SOLUTION
Using Equation (2) of Problem 10.95 with
45 N, 150 mm, and 200 mmQa b
PROBLEM 10.97*
Bars AB and BC, each of length l and of negligible weight, are attached to two
springs, each of constant k. The springs are undeformed, and the system is in
equilibrium when
1
2
0. Determine the range of values of P for which
the equilibrium position is stable.
SOLUTION
We have
12
12
22
sin
sin sin
cos cos
11
22
B
C
C
CBC
xl
xl l
yl l
VPy kx kx
PROBLEM 10.97* (Continued)
For 12
12
0: 0 (condition satisfied)
VV
2
222
22
12 12
0
VVV
Substituting 22 2
24 22 3 24
() ( 2)( )0
320
kl Pl kl Pl kl
kl Pl Pkl kl
PROBLEM 10.98*
Solve Problem 10.97 knowing that l 800 mm and k 2.5 kN/m.
PROBLEM 10.97* Bars AB and BC, each of length l and of negligible
weight, are attached to two springs, each of constant k. The springs are
undeformed, and the system is in equilibrium when
1
2 0. Determine
the range of values of P for which the equilibrium position is stable.
SOLUTION
From the analysis of Problem 10.97 with