Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
CHAPTER 19
PROBLEM 19.1
A particle moves in simple harmonic motion. Knowing that the maximum velocity is 200 mm/s and the
maximum acceleration is 4 m/s2, determine the amplitude and frequency of the motion.
SOLUTION
PROBLEM 19.2
A particle moves in simple harmonic motion. Knowing that the amplitude is 15 in. and the maximum
acceleration is 15 ft/s2, determine the maximum velocity of the particle and the frequency of its motion.
PROBLEM 19.3
Determine the amplitude and maximum acceleration of a particle which moves in simple harmonic motion
with a maximum velocity of 4 ft/s and a frequency of 6 Hz.
SOLUTION
π
PROBLEM 19.4
A 32-kg block is attached to a spring and can move without friction in a slot
as shown. The block is in its equilibrium position when it is struck by a
hammer, which imparts to the block an initial velocity of 250 mm/s.
Determine (a) the period and frequency of the resulting motion, (b) the
amplitude of the motion and the maximum acceleration of the block.
SOLUTION
mmn
m
PROBLEM 19.5
A 12-kg block is supported by the spring shown. If the block is moved vertically downward
from its equilibrium position and released, determine (a) the period and frequency of the
resulting motion, (b) the maximum velocity and acceleration of the block if the amplitude
of its motion is 50 mm.
SOLUTION
PROBLEM 19.6
An instrument package A is bolted to a shaker table as shown. The
table moves vertically in simple harmonic motion at the same
frequency as the variable-speed motor which drives it. The package is
to be tested at a peak acceleration of 150 ft/s2. Knowing that the
amplitude of the shaker table is 2.3 in., determine (a) the required
speed of the motor in rpm, (b) the maximum velocity of the table.
SOLUTION
In simple harmonic motion,
2
max max
22
22
2.3
150 ft/s ft
12
(782.6 rad/s)
n
n
n
ax
ω
ω
ω
=
⎛⎞
=⎜⎟
⎝⎠
=
PROBLEM 19.7
A simple pendulum consisting of a bob attached to a cord oscillates in a vertical plane
with a period of 1.3 s. Assuming simple harmonic motion and knowing that the
maximum velocity of the bob is 0.4 m/s, determine (a) the amplitude of the motion in
degrees, (b) the maximum tangential acceleration of the bob.
SOLUTION
m
(b) Maximum tangential acceleration t
al
θ
=
The maximum tangential acceleration occurs when
θ
is maximum.
2
1.933 m/s
tm
=
2
() 1.933m/s
tm
a=
PROBLEM 19.8
A simple pendulum consisting of a bob attached to a cord of length 800 mml=
oscillates in a vertical plane. Assuming simple harmonic motion and knowing that the
bob is released from rest when 6,
θ
=
° determine (a) the frequency of oscillation, (b) the
maximum velocity of the bob.
SOLUTION
2
(9.81 m/s )
g
m
m
PROBLEM 19.9
A 10-lb block A rests on a 40-lb plate B which is attached to an unstretched
spring of constant k = 60 lb/ft. Plate B is slowly moved 2.4 in. to the left
and released from rest. Assuming that block A does not slip on the plate,
determine (a) the amplitude and frequency of the resulting motion, (b) the
corresponding smallest allowable value of the coefficient of static friction.
SOLUTION
Simple Harmonic Motion:
2
32.2 ft/s
s
g
PROBLEM 19.10
A 5-kg fragile glass vase is surrounded by packing material in a cardboard box of negligible weight. The
packing material has negligible damping and a force-deflection relationship as shown. Knowing that the box
is dropped from a height of 1 m and the impact with the ground is perfectly plastic, determine (a) the
amplitude of vibration for the vase, (b) the maximum acceleration the vase experiences in g’s.
SOLUTION
and 4.4294 m/s
nm
PROBLEM 19.10 (Continued)
4.4294 m/s 0.099045 m
x==
m
PROBLEM 19.11
A 3-lb block is supported as shown by a spring of constant 2 lb/in.k
=
which can act in
tension or compression. The block is in its equilibrium position when it is struck from below
by a hammer which imparts to the block an upward velocity of 90 in./s. Determine (a) the
time required for the block to move 3 in. upward, (b) the corresponding velocity and
acceleration of the block.
SOLUTION
PROBLEM 19.12
In Problem 19.11, determine the position, velocity, and acceleration of the block 0.90 s after it has been struck
by the hammer.
SOLUTION
PROBLEM 19.13
The bob of a simple pendulum of length 40 in.l
=
is released from rest when 5.
θ
=+ °
Assuming simple harmonic motion, determine 1.6 s after release (a) the angle ,
θ
(b) the magnitudes of the velocity and acceleration of the bob.
SOLUTION
PROBLEM 19.13 (Continued)
Angular acceleration:
22
2
5
sin( ) (3.1082 rad/s) cos (3.1082 rad/s)(1.6 s)
180 2
mn n
t
π
π
θθω ωφ
⎡⎤
=− + =− +
⎢⎥
⎣⎦
S
O
O
LUTION
PROB
L
A 150-k
g
the curr
e
and the
s
constant
maximu
m
will occ
u
0.03 s af
t
L
EM 19.14
g
electromag
n
e
nt is turned
o
s
upporting cr
200 kN/m,
d
m
velocity of
t
u
r in the cabl
e
t
er the curren
t
n
et is at rest a
n
o
ff and the st
e
ane have a t
o
d
etermine (a)
t
he resulting
m
e
during the
m
t
is turned off.
n
d is holding
1
e
el is droppe
d
o
tal stiffness
the frequenc
y
m
otion, (b) th
e
m
otion, (c) t
h
1
00 kg of scra
p
d
. Knowing t
h
equivalent to
y
, the amplit
u
e
m
inimum t
e
h
e velocity o
f
p
steel when
h
at the cable
a spring of
u
de, and the
e
nsion which
f
the magnet
PROBLEM 19.14 (Continued)
(b) Minimum value of tension occurs when .
x
x=−
min 0
12
12
()
m
TTkx
mg m g
mmg
=−
=−
=−
PROBLEM 19.15
A 5-kg collar C is released from rest in the position shown and slides without friction on a
vertical rod until it hits a spring of constant k = 720 N/m which it compresses. The velocity of
the collar is reduced to zero and the collar reverses the direction of its motion and returns to its
initial position. The cycle is then repeated. Determine (a) the period of the motion of the
collar, (b) the velocity of the collar 0.4 s after it was released. (Note. This is a periodic motion,
but not simple harmonic motion.)
SOLUTION
