PROBLEM 19.15 (Continued)
PROBLEM 19.16
A small bob is attached to a cord of length 1.2 m and is released from rest when
5.
A
θ
=°
Knowing that d = 0.6 m, determine (a) the time required for the bob to
return to Point A, (b) the amplitude .
C
θ
SOLUTION
As the pendulum moves between Points A and B, the length of the pendulum is 1.2 m.
AB
ll==
2
1
9.81 m/s 2.8592 rad/s
1.2 m
nn
AB
g
l
ωω
== = =
PROBLEM 19.16 (Continued)
A
C
S
O
O
LUTION
PROBLE
M
A 25-kg bloc
k
downward fr
o
of the result
i
amplitude of
t
M
19.17
k
is supporte
d
o
m its equilib
r
i
ng motion,
(
t
he motion is
a
a
i
i
n
n
d
by the sprin
g
r
ium position
(
b) the maxi
m
30 mm.
g
arrangement
and released,
m
um velocit
y
shown. If the
determine (a
)
y
and acceler
a
block is mov
)
the period a
n
a
tion of the
b
ed vertically
n
d frequency
b
lock if the
P
R
A 1
Kn
o
in
h
b
lo
c
SO
Eq
u
R
OBLEM 1
9
1-lb block, at
t
o
wing that th
e
h
alf, the rema
i
c
k which is at
t
LUTION
u
ivalent sprin
g
9
.18
t
ached to the
l
e
constant k o
f
i
ning two sp
r
t
ached to the
c
g
constant.
l
ower end of
a
f
a spring is in
v
r
ings each ha
v
c
enter of the s
a
spring who
s
v
ersely propo
r
v
e a spring c
o
ame spring if
s
e upper end i
s
r
tional to its l
e
o
nstant of 20
the upper an
d
s
fixed, vibra
t
e
ngth (e.g., if
lb/in.), deter
m
d
lower ends o
t
es with a per
i
you cut a 10
l
m
ine the peri
o
f the spring a
r
i
od of 7.2 s.
l
b/in. spring
o
d of a 7-lb
r
e fixed.
o
PROBLEM 19.19
Block A of mass m is supported by the spring arrangement as shown. Knowing that the mass of
the pulley is negligible and that the block is moved vertically downward from its equilibrium
position and released, determine the frequency of the motion.
SOLUTION
We first determine the constant keq of a single spring equivalent to the spring and pulley system supporting the
block by finding the total displacement A
δ
of the end of the cable under a given static load P. Owing to the
force 2P in the upper spring the pulley moves down a distance
P
P
2
P
n
m
PROBLEM 19.20
A 13.6-kg block is supported by the spring arrangement shown. If the block is
moved from its equilibrium position 44 mm vertically downward and released,
determine (a) the period and frequency of the resulting motion, (b) the maximum
velocity and acceleration of the block.
SOLUTION
0.3614 s
n
n
τ
n
PROBLEM 19.20 (Continued)
PROBLEM 19.21
A 50-kg block is supported by the spring arrangement shown. The block is moved
vertically downward from its equilibrium position and released. Knowing that the
amplitude of the resulting motion is 60 mm, determine (a) the period and frequency
of the motion, (b) the maximum velocity and maximum acceleration of the block.
SOLUTION
δ
(a) First, calculate the spring constant
(b) Now
(
)
sin
mn
xx t
ω
φ
=
+
And, since 00.060 mx=
(
)
(
)
0.060 m sin 30.984 rad/sxt
φ
⎡
⎤
=+
⎣
⎦
(
)
(
)( )
0.060 m 30.984 rad/s cos 30.984 rad/sxt
φ
⎡⎤
=+
⎣⎦
()()()
2
0.060 m 30.984 rad/s sin 30.984 rad/sxt
φ
⎡⎤
=− +
⎣⎦
Hence max 1.859 m/sv=
2
max 57.6 m/sa=
PROBLEM 19.22
A 50-kg block is supported by the spring arrangement shown. The block is moved
vertically downward from its equilibrium position and released. Knowing that the
amplitude of the resulting motion is 60 mm, determine (a) the period and frequency of the
motion, (b) the maximum velocity and maximum acceleration of the block.
PROBLEM 19.23
Two springs of constants 1
k and 2
kare connected in series to a block A that
vibrates in simple harmonic motion with a period of 5 s. When the same two
springs are connected in parallel to the same block, the block vibrates with a
period of 2 s. Determine the ratio 12
/kk
of the two spring constants.
Solution
Equivalent Springs
kk
u
u
c
s
d
d
PR
O
The
p
b
loc
k
(a) t
h
and
B
m
O
BLEM 19
p
eriod of vib
r
k
A is remo
v
h
e mass of bl
o
B
have been r
e
6 kg
m
=+
.24
r
ation of the s
y
v
ed, the perio
o
ck C, (b) the
e
moved.
0.8 s
τ
=
y
stem shown
d is observe
d
period of vib
r
is observed t
o
d
to be 0.7 s
r
ation when b
o
o
be 0.8 s. If
. Determine
o
th blocks A
PROBLEM 19.25
The 100-lb platform A is attached to springs B and D, each of which has a
constant 120 lb/ft.k
=
Knowing that the frequency of vibration of the
platform is to remain unchanged when an 80-lb block is placed on it and a
third spring C is added between springs B and D, determine the required
constant of spring C.
PROBLEM 19.26
The period of vibration for a barrel floating in salt water is found to be
0.58 s when the barrel is empty and 1.8 s when it is filled with 55 gallons of
crude oil. Knowing that the density of the oil is 900 kg/m3, determine (a) the
mass of the empty barrel, (b) the density of the salt water,
ρ
sw. [Hint: the
force of the water on the bottom of the barrel can be modeled as a spring
with constant k =
ρ
swgA.]
SOLUTION
22
2
(0.572 m) 0.2570 m
D
ππ
PROBLEM 19.26 (Continued)
(b) Density of the salt water.
sw
3
kgA
ρ
=
sw 1011 kg/m
PROBLEM 19.27
From mechanics of materials it is known that for a simply supported
beam of uniform cross section a static load P applied at the center will
cause a deflection 348 ,
A
P
LEI
δ
= where L is the length of the beam,
E is the modulus of elasticity, and I is the moment of inertia of the
cross-sectional area of the beam. Knowing that L = 15 ft,
E = 6
30 10 psi,× and 34
210 ft,I−
=× determine (a) the equivalent
spring constant of the beam, (b) the frequency of vibration of a 1500-lb
block attached to the center of the beam. Neglect the mass of the beam
and assume that the load remains in contact with the beam.
SOLUTION
()
(
)
(
)
62 22 34
48 30 10 lb/in. 144 in. /ft 2 10 ft
48
PEI
−
××
222
n
πππ
PROBLEM 19.28
From mechanics of materials it is known that when a static load P is
applied at the end B of a uniform metal rod fixed at end A, the length of
the rod will increase by an amount /,
P
LAE
δ
=
where L is the length of
the undeformed rod. A is its cross-sectional area, and E is the modulus of
elasticity of the metal. Knowing that L = 450 mm and E = 200 GPa and
that the diameter of the rod is 8 mm, and neglecting the mass of the rod,
determine (a) the equivalent spring constant of the rod, (b) the frequency
of the vertical vibrations of a block of mass m = 8 kg attached to end B
of the same rod.
SOLUTION
P
R
De
lo
a
Ne
S
O
R
OBLEM 1
noting by
st
δ
t
a
d is
glect the mas
s
O
LUTION
9.29
t
he static defl
e
s
of the beam,
e
ction of a be
and assume t
h
k
m
am under a g
i
1
2
f
π
=
h
at the load r
e
st
W
W
k
W
m
g
δ
=
=
i
ven load, sho
st
g
δ
e
mains in con
t
w that the fre
q
t
act with the
b
q
uency of vib
b
eam.
ω
ration of the
π
PROBLEM 19.30
A 40-mm deflection of the second floor of a building is measured directly under a newly installed 3500-kg
piece of rotating machinery, which has a slightly unbalanced rotor. Assuming that the deflection of the floor is
proportional to the load it supports, determine (a) the equivalent spring constant of the floor system, (b) the
speed in rpm of the rotating machinery that should be avoided if it is not to coincide with the natural
frequency of the floor-machinery system.
SOLUTION
S
O
O
LUTION
P
I
d
i
n
t
e
P
ROBLEM
f 700 m
m
h
=
d
etermine the
n
finite. Negle
e
nsion or co
m
x
19.31
m
and
50
0
d
=
mass m for w
ct the mass o
f
m
pression.
0
mm
and ea
c
hich the peri
o
f
the rod and
a
c
h spring has
o
d of small os
c
a
ssume that e
a
a constant k
c
illations is (
a
a
ch spring can
600 N/m,=
a
) 0.50 s, (b)
act in either
2