PROBLEM 19.31 (Continued)
PROBLEM 19.32
The force-deflection equation for a nonlinear spring fixed at one end is 1/2
1.5
F
x= where F is the force,
expressed in newtons, applied at the other end, and x is the deflection expressed in meters. (a) Determine the
deflection x0 if a 4-oz block is suspended from the spring and is at rest. (b) Assuming that the slope of the
force-deflection curve at the point corresponding to this loading can be used as an equivalent spring constant,
determine the frequency of vibration of the block if it is given a very small downward displacement from its
equilibrium position and released.
SOLUTION
x
PROBLEM 19.33*
Expanding the integrand in Equation (19.19) of Section 19.4 into a series of even powers of sin
φ
and
integrating, show that the period of a simple pendulum of length l may be approximated by the formula
2
1
4
21sin
2
m
l
g
θ
τπ
⎛⎞
=+
⎜⎟
⎝⎠
where m
θ
is the amplitude of the oscillations.
θ
PROBLEM 19.34*
Using the formula given in Problem 19.33, determine the amplitude m
θ
for which the period of a simple
pendulum is 1
2percent longer than the period of the same pendulum for small oscillations.
SOLUTION
l
PROBLEM 19.35*
Using the data of Table 19.1, determine the period of a simple pendulum of length 750 mml= (a) for small
oscillations, (b) for oscillations of amplitude 60 ,
m
θ
=
° (c) for oscillations of amplitude 90 .
m
θ
=°
SOLUTION
l
PROBLEM 19.36*
Using the data of Table 19.1, determine the length in inches of a simple pendulum which oscillates with a
period of 2 s and an amplitude of 90°.
SOLUTION
PROBLEM 19.37
The uniform rod shown has mass 6 kg and is attached to a spring
of constant 700 N/m.k
=
If end B of the rod is depressed 10 mm
and released, determine (a) the period of vibration, (b) the
maximum velocity of end B.
SOLUTION
=
2
(700 N/m)(0.5) 0.38
(460.53) 0
θθ
θθ
−=
+=
PROBLEM 19.37 (Continued)
m
SO
LUTION
PROBL
E
A belt is
p
to two sp
r
1.5 in. d
o
observed
t
to preven
t
flywheel,
(
E
M 19.38
p
laced around
r
ings, each of
o
wn and rel
e
t
o be 0.5 s. K
n
t
slipping, d
e
(
b) the centro
i
the rim of a
5
constant
k
=
e
ased, the pe
r
n
owing that t
h
e
termine (a) t
i
dal radius of
g
5
00-lb flywhe
85 lb/in. If e
n
r
iod of vibr
a
h
e initial tensi
o
he maximum
g
yration of th
e
el and attach
e
n
d C of the b
e
a
tion of the
f
o
n in the belt
i
angular vel
o
e
flywheel.
e
d as shown
e
lt is pulled
f
lywheel is
i
s sufficient
o
city of the
n
l
u
l
o
=
=
=
PROBLEM 19.38 (Continued)
S
O
O
LUTION
PROB
A 6-kg
u
and is a
t
is attac
h
Knowin
g
position
(b) the
m
LEM 19.3
9
u
niform cyli
n
t
tached by a
p
h
ed to two s
p
g
that the ba
r
and released,
m
agnitude of t
h
9
n
der can roll
w
p
in at point C
p
rings, each
r
is moved 1
2
determine (a
)
h
e maximum
v
w
ithout slidin
g
to the 4-kg h
o
of constant
k
2
mm to the
)
the period o
f
v
elocity of b
a
g
on a horizo
n
o
rizontal bar
A
k
= 5 kN/m
right of the
f
vibration of
a
r AB.
n
tal surface
A
B. The bar
as shown.
equilibrium
the system,
2
2
6
PROBLEM 19.40
A 6-kg uniform cylinder is assumed to roll without sliding on a horizontal
surface and is attached by a pin at point C to the 4-kg horizontal bar AB.
The bar is attached to two springs, each of constant k = 3.5 kN/m as
shown. Knowing that the coefficient of static friction between the
cylinder and the surface is 0.5, determine the maximum amplitude of the
motion of point C which is compatible with the assumption of rolling.
SOLUTION
From Problem 19.39
SO
a
a
a
n
LUTION
PR
O
A 15-
l
b
elt i
s
rod at
down
maxi
m
O
BLEM 19.
4
l
b slender ro
d
s
attached to
t
rest in the p
o
and release
d
m
um velocity
o
4
1
d
AB is rivete
d
t
he rim of th
e
o
sition shown
.
d
, determine
o
f end A.
d
to a 12-lb u
e
disk and to
a
.
If end A of
t
(a) the peri
o
niform disk
a
a
spring whi
c
t
he rod is mo
v
o
d of vibrati
o
a
s shown. A
c
h holds the
v
ed 0.75 in.
o
n, (b) the
u
o
PROBLEM 19.41 (Continued)
2
2
2
1
12
1(0.46584)(3.0)
12
0.34938 lb s ft
AB
ImL
=
=
=⋅⋅
mnm
m
SO
LUTION
P
R
A
3
b
e
l
cy
l
m
o
of
cy
l
R
OBLEM
1
3
0-lb unifor
m
l
t is attached
l
inder at rest
i
o
ved 2 in. do
w
vibration, (b
l
inder.
1
9.42
m
cylinder can
to the rim o
f
i
n the positio
n
w
n the incline
) the maxim
u
roll without s
l
f
the cylinde
r
n
shown. If th
e
and released,
u
m accelerat
i
l
iding on a 15
°
r
, and a sprin
g
e
center of th
e
determine (a
)
i
on of the c
e
°
–
incline. A
g
holds the
e
cylinder is
)
the period
e
nter of the
h
PROBLEM 19.42 (Continued)
1
8(8)(3012 lb/ft)
k
−
×
PROBL
E
A square pl
a
that each s
p
frequency
o
displaceme
n
about G an
d
E
M 19.43
a
te of mass m
p
ring can ac
t
o
f the resultin
g
n
t and releas
e
d
released.
is held by eig
h
t
in either te
g
vibration (
a
e
d, (b) if the
p
h
t springs, ea
c
nsion or co
m
a
) if the plate
p
late is
r
otat
e
c
h of constant
k
m
pression, de
t
is given a s
m
e
d through a
s
k
. Knowing
t
ermine the
m
all vertical
s
mall angle
B
t
d
a
n
:
t
B
m
o
d
e
t
d
a
n
h
l
a
e
a
e
l
k
a
r
e
o
k
r
c
u
b
n
n
f
u
m
n
f
g
i
i
m
i
w
PROBLEM 19.43 (Continued)
SO
LUTION
PRO
B
Two s
m
and wei
the angl
B
LEM 19.4
4
m
all weights
w
ght W. Deno
t
e
β
for whic
h
4
w
are attached
t
ing by
0
τ
the
h
the period o
f
at A and B t
o
period of s
m
f
small oscilla
t
o
the rim of a
m
all oscillatio
n
t
ions is
0
2.
τ
uniform dis
k
n
s when
0
β
=
k
of radius r
0
,
determine
u
2
2
2
β
τ
=
=
=
2
2
β
π
π
β
u
c
PRO
B
Two 4
0
10
0
r=
B
LEM 19.4
5
0
-g weights a
r
0
mm. Deter
m
5
r
e attached at
A
m
ine the frequ
e
A
and B to th
e
e
ncy of small
o
e
rim of a 1.5
–
o
scillations
w
–
kg uniform d
i
w
hen
60 .
β
=°
i
sk of radius