PROBLEM 13.31
A 5-kg collar A is at rest on top of, but not attached to, a spring
with stiffness k1 400 N/m; when a constant 150-N force is
applied to the cable. Knowing A has a speed of 1 m/s when the
upper spring is compressed 75 mm, determine the spring
stiffness k2. Ignore friction and the mass of the pulley.
SOLUTION
Use the method of work and energy applied to the collar A.
T1 U1
T1
Since collar is initially at rest,
T2
2
0.
In position 2, where the upper spring is compressed 75 mm and v2
T2
1 2
mv2
2
1.00 m/s, the kinetic energy is
1
(5 kg)(1.00 m/s)2
2
2.5 J
As the collar is raised from level A to level B, the work of the weight force is
(U1
where m 5 kg, g
Thus, (U1
2 )g
9.81 m/s 2 and h
(5)(9.81)(0.450)
450 mm
2 )g
mgh
0.450 m
22.0725 J
In position 1, the force exerted by the lower spring is equal to the weight of collar A.
F1
mg
(5 kg)(9.81 m/s)
49.05 N
As the collar moves up a distance x1, the spring force is
F
F1 k1 x2
until the collar separates from the spring at
xf
F1
k1
49.05 N
400 N/m
0.122625 m 122.625 mm
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PROBLEM 13.31 (Continued)
Work of the force exerted by the lower spring:
(U1
xf
2 )1
0
( F1
F1 x f
k1 x)dx
1 2
kx f
2
k1 x 2f
1 2
k1 x f
2
1
(400 N/m)(0.122625)2
2
In position 2, the upper spring is compressed by y
75 mm
1
k1 x 2f
2
3.0074 J
0.075 m. The work of the force exerted
by this spring is
(U1
2 )2
1
k2 y 2
2
1
k2 (0.075)2
2
0.0028125 k 2
Finally, we must calculate the work of the 150 N force applied to the cable. In position 1, the length AB is
(450) 2
(l AB )1
In position 2, the length AB is (l AB )2
(400) 2
602.08 mm
400 mm.
The displacement d of the 150 N force is
d
(l AB )1 (l AB )2
202.08 mm 0.20208 m
The work of the 150 N force P is
(U1
Total work:
2 )P
U1
Pd
(150 N)(0.20208 m) 30.312 J
22.0725 3.0074 0.0028125k2
2
30.312
11.247 0.0028125k2
Principle of work and energy:
T1 U1
2
T2
0 11.247 0.0028125k2
k2
3110 N/m
2.5
k2
3110 N/m
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PROBLEM 13.32
A piston of mass m and cross-sectional area A is equilibrium
under the pressure p at the center of a cylinder closed at both
ends. Assuming that the piston is moved to the left a distance
a/2 and released, and knowing that the pressure on each side of
the piston varies inversely with the volume, determine the
velocity of the piston as it again reaches the center of the
cylinder. Neglect friction between the piston and the cylinder
and express your answer in terms of m, a, p, and A.
SOLUTION
Pressures vary inversely as the volume
pL
P
pR
P
Initially at
At
,
pL
Aa
A(2a x)
pR
v
0
T1
0
x
,
Aa
Ax
a
2
x
1 2
mv
2
a, T2
a
pa
x
pa
(2a x)
( pL
pR ) Adx
U1
2
U1
2
paA[ln x ln (2a
U1
2
paA ln a ln a ln
U1
2
paA ln a 2
T1 U1
2
T2
v2
a/2
0
ln
3a 2
4
paA ln
4
3
2 paA ln
m
4
3
a
a/2
paA
1
x
1
dx
2a x
x)]aa/2
a
2
0.5754
ln
paA ln
3a
2
4
3
1 2
mv
2
paA
m
v
0.759
paA
m
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