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Problem 21.1 In Active Example 21.1, suppose that
the pulley has radius R=100 mm and its moment
of inertia is I=0.005 kg-m2. The mass m=2 kg,
and the spring constant is k=200 N/m. If the mass
is displaced downward from its equilibrium position
and released, what are the period and frequency of the
resulting vibration?
R
km
m+I
R2
(2kg)+(0.005 kg-m2)
(0.1m)2
Thus
Problem 21.2 In Active Example 21.1, suppose that
the pulley has radius R=4 in and its moment of inertia
is I=0.06 slug-ft2. The suspended object weighs 5 lb,
and the spring constant is k=10 lb/ft. The system is
initially at rest in its equilibrium position. At t=0, the
suspended object is given a downward velocity of 1 ft/s.
Determine the position of the suspended object relative
to its equilibrium position as a function of time.
R
km
Problem 21.3 The mass m=4 kg. The spring is un-
stretched when x=0. The period of vibration of the
mass is measured and determined to be 0.5 s. The mass
is displaced to the position x=0.1 m and released from
x
Problem 21.4 The mass m=4 kg. The spring is un-
stretched when x=0. The frequency of vibration of the
mass is measured and determined to be 6 Hz. The mass
is displaced to the position x=0.1 m and given a veloc-
x
37.7 rad/s =0.133 m.
728
Problem 21.5 The mass m=4 kg and the spring con-
stant is k=64 N/m. For vibration of the spring-mass
oscillator relative to its equilibrium position, determine
x=0. At t=0,x =0 and the mass has a velocity of
2 m/s down the inclined surface. What is the value of x
at t=0.8s?
md2x
dt2+kx =mg sin 20◦⇒d2x
dt2+k
mx=gsin 20◦
Putting in the numbers we have
Problem 21.7 Suppose that in a mechanical design
course you are asked to design a pendulum clock, and
you begin with the pendulum. The mass of the disk is
2 kg. Determine the length Lof the bar so that the period
Problem 21.8 The mass of the disk is 2 kg and the
mass of the slender bar is 0.4 kg. Determine the length
Lof the bar so that the period of small oscillations of
the pendulum is 1 s.
Strategy: Draw a graph of the value of the period for
a range of lengths Lto estimate the value of Lcorre-
sponding to a period of 1 s.
50 mm
L
Problem 21.9 The spring constant is k=785 N/m.
The spring is unstretched when x=0. Neglect the mass
of the pulley, that is, assume that the tension in the rope
is the same on both sides of the pulley. The system is
released from rest with x=0. Determine xas a function
730
Problem 21.10 The spring constant is k=785 N/m.
x=0. Determine xas a function of time.
Solution: Let T1be the tension in the rope on the left side, and
Problem 21.11 A “bungee jumper” who weighs 160
lb leaps from a bridge above a river. The bungee cord
has an unstretched length of 60 ft, and it stretches an
additional 40 ft before the jumper rebounds. Model the
cord as a linear spring. When his motion has nearly
stopped, what are the period and frequency of his vertical
oscillations? (You can’t model the cord as a linear spring
during the early part of his motion. Why not?)
f=3.13 s
Problem 21.12 The spring constant is k=800 N/m,
Solution: The equations of motion are
(b) The solution to the differential equation is
Problem 21.13 The spring constant is k=800 N/m,
and the spring is unstretched when x=0. The mass of
each object is 30 kg. The inclined surface is smooth. The
Solution: Let T1be the tension in the rope on the left of the
pulley, and T2be the tension in the rope on the right of the pulley.
The equations of motion are
3.59 rad/s =0.279 m.
732
Problem 21.14 The 20-lb disk rolls on the horizontal
surface. Its radius is R=6 in. Determine the spring con-
stant kso that the frequency of vibration of the system
relative to its equilibrium position is f=1 Hz.
kR
Problem 21.15 The 20-lb disk rolls on the horizontal
surface. Its radius is R=6 in. The spring constant is
k=15 lb/ft. At t=0, the spring is unstretched and the
disk has a clockwise angular velocity of 2 rad/s. What
is the amplitude of the resulting vibrations of the center
of the disk?
Problem 21.16 The 2-lb bar is pinned to the 5-lb disk.
The disk rolls on the circular surface. What is the fre-
quency of small vibrations of the system relative to its
vertical equilibrium position?
15 in
4 in
Solution: Use energy methods.
Problem 21.17 The mass of the suspended object A
is 4 kg. The mass of the pulley is 2 kg and its moment
of inertia is 0.018 N-m2. For vibration of the system
relative to its equilibrium position, determine (a) the fre-
quency in Hz and (b) the period.
k
Solution: Use energy methods
734
Problem 21.18 The mass of the suspended object Ais
x=0. At t=0, the system is released from rest with
x=0. What is the velocity of the object Aat t=1s?
Problem 21.19 The thin rectangular plate is attached
to the rectangular frame by pins. The frame rotates
with constant angular velocity ω0=6 rad/s.The angle β
between the zaxis of the body-fixed coordinate system
z
b
v0
Problem 21.20 Consider the system described in
Problem 21.19. At t=0, the angle β=0.01 rad and
dβ/dt =0. Determine βas a function of time.
Problem 21.21 A slender bar of mass mand length l
is pinned to a fixed support as shown. A torsional
spring of constant kattached to the bar at the support
is unstretched when the bar is vertical. Show that the
equation governing small vibrations of the bar from its
dt2+ω2θ=0,where ω2=(k −1
2mgl)
1
3ml2.
θ
2Idθ
dt 2
The potential energy is the sum of the energy in the spring and the
gravitational energy associated with the change in height of the center
of mass of the bar,
For small amplitude vibrations sin θ→θ, and the canonical form (see
Eq. (21.4)) of the equation of motion is
k−mgL
736
Problem 21.22 The initial conditions of the slender
bar in Problem 21.21 are
t=0θ=0
dθ
dt =˙
θ0.
(a) If k>1
2mgl, show that θis given as a function of
time by
θ=˙
θ0
ωsin ωt, where ω2=(k −1
2mgl)
1
3ml2.
(b) If k<1
2mgl, show that θis given as a function of
time by
θ=˙
θ0
2h(eht −e−ht ), where h2=(1
2mgl −k)
1
3ml2.
Problem 21.23 Engineers use the device shown to
measure an astronaut’s moment of inertia. The horizontal
board is pinned at Oand supported by the linear spring
with constant k=12 kN/m. When the astronaut is not
present, the frequency of small vibrations of the board
about Ois measured and determined to be 6.0 Hz.
When the astronaut is lying on the board as shown, the
frequency of small vibrations of the board about Ois
2.8 Hz. What is the astronaut’s moment of inertia about
the zaxis?
y'
x'
x
y
1.90 m
O
Problem 21.24 In Problem 21.23, the astronaut’s cen-
ter of mass is at x=1.01 m, y=0.16 m, and his mass
is 81.6 kg. What is his moment of inertia about the z′
axis through his center of mass?
=24.2 kg-m2.
738
c
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Problem 21.25* Afloating sonobuoy (sound-measur-
ing device) is in equilibrium in the vertical position
shown. (Its center of mass is low enough that it is stable
in this position.) The device is a 10-kg cylinder 1 m
in length and 125 mm in diameter. The water density
is 1025 kg/m3, and the buoyancy force supporting the
buoy equals the weight of the water that would occupy
the volume of the part of the cylinder below the surface.
If you push the sonobuoy slightly deeper and release it,
what is the frequency of the resulting vertical vibrations?
Problem 21.26 The disk rotates in the horizontal plane
with constant angular velocity =12 rad/s. The mass
m=2 kg slides in a smooth slot in the disk and is
attached to a spring with constant k=860 N/m. The
radial position of the mass when the spring is unstretched
is r=0.2m.
(a) Determine the “equilibrium” position of the mass,
the value of rat which it will remain stationary
relative to the center of the disk.
(b) What is the frequency of vibration of the mass re-
lative to its equilibrium position?
k
m
r
V
Problem 21.27 The disk rotates in the horizontal plane
with constant angular velocity =12 rad/s. The mass
m=2 kg slides in a smooth slot in the disk and is
0.4 m and dr/dt =0. Determine the position ras a
function of time.
Solution: Using polar coordinates, Newton’s second law in the r
direction is
¨r+(16.9 rad/s)2r=86 m/s2.
740
Problem 21.28 A homogeneous 100-lb disk with
radius R=1 ft is attached to two identical cylindrical
steel bars of length L=1 ft. The relation between the
moment Mexerted on the disk by one of the bars and
the angle of rotation, θ, of the disk is
disk is to be 10 Hz.
R
θ
∂θ =2GJ
L.
Problem 21.29 The moments of inertia of gears Aand
Bare IA=0.025 kg-m2and IB=0.100 kg-m2. Gear A
is connected to a torsional spring with constant k=
10 N-m/rad. What is the frequency of small angular
vibrations of the gears?
Problem 21.30 At t=0, the torsional spring in Prob-
lem 21.29 is unstretched and gear Bhas a counterclock-
wise angular velocity of 2 rad/s. Determine the coun-
terclockwise angular position of gear Brelative to its
equilibrium position as a function of time.
M=11.62 rad/s.
Assume a solution of the form θA=Asin ωt +Bcos ωt. Apply the
initial conditions,
742
Problem 21.31 Each 2-kg slender bar is 1 m in length.
What are the period and frequency of small vibrations
of the system?
Problem 21.32* The masses of the slender bar and
the homogeneous disk are mand md, respectively. The
spring is unstretched when θ=0. Assume that the disk
rolls on the horizontal surface.
(a) Show that the motion of the system is governed by
the equation
1
3+3md
2mcos2θd2θ
dt2−3md
2msin θcos θdθ
dt 2
−g
2lsin θ+k
m(1−cos θ)sin θ=0.
(b) If the system is in equilibrium at the angle θ=θe
and ˜
θ=θ−θeshow that the equation governing
small vibrations relative to the equilibrium posi-
tion is
1
3+3md
2mcos2θed2˜
θ
dt2+k
m(cos θe−cos2θe
+sin2θe)−g
2lcos θe˜
θ=0.
k
l
R
θ
Solution: (See Example 21.2.) The system is conservative.
from the center of mass to the center of rotation is
r=L−L
sin2θ+L−L
cos2θ=L
θ
L
k
−g
2Lsin θ+k
m(1−cos θ)sin θ=0,
744
Problem 21.33* The masses of the bar and disk
in Problem 21.32 are m=2 kg and md=4 kg,
respectively. The dimensions l=1 m and R=0.28 m,
and the spring constant is k=70 N/m.
(a) Determine the angle θeat which the system is in
equilibrium.
(b) The system is at rest in the equilibrium position,
and the disk is given a clockwise angular velocity
of 0.1 rad/s. Determine θas a function of time.
Solution:
746
