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331
Oscillatory Motion
CHAPTER OUTLINE
12.1
Motion of a Particle
Attached to a Spring
12.2
Mathematical
Representation of Simple
Harmonic Motion
12.3
Energy Considerations
in Simple Harmonic
Motion
12.4
The Simple Pendulum
12.5
The Physical Pendulum
12.6
Damped Oscillations
12.7
Forced Oscillations
12.8
Context Connection—
Resonance in Structures
ANSWERS TO QUESTIONS
Q12.1
Neither are examples of simple harmonic motion, although they are
both periodic motion. In neither case is the acceleration proportional
to the position. Neither motion is so smooth as SHM. The ball’s
acceleration is very large when it is in contact with the floor, and the
student’s when the dismissal bell rings.
Q12.2
The two will be equal if and only if the position of the particle at time
zero is its equilibrium position, which we choose as the origin of
coordinates.
Q12.3
You can take
φ
π
=
, or equally well,
=−
. At
t
=
0, the particle is at its turning point on the
negative side of equilibrium, at
x
A
.
Q12.4
No. It is necessary to know both the position and velocity at time zero.
Q12.5
(a)
In simple harmonic motion, onehalf of the time, the velocity is in the same direction as the
displacement away from equilibrium.
(b)
Velocity and acceleration are in the same direction half the time.
(c)
Acceleration is always opposite to the position vector, and never in the same direction.
Q12.6
We assume that the coils of the spring do not hit one another. The frequency will be higher than
f
by
the factor
2 . When the spring with two blocks is set into oscillation in space, the coil in the center
of the spring does not move. We can imagine clamping the center coil in place without affecting the
motion. We can effectively duplicate the motion of each individual block in space by hanging a
single block on a halfspring here on Earth. The halfspring with its center coil clamped—or its other
half cut off—has twice the spring constant as the original uncut spring, because an applied force of
the same size would produce only onehalf the extension distance. Thus the oscillation frequency in
space is
1
2
2
2
12
F
H
G
I
K
J
F
H
G
I
K
J
=
k
m
f
. The absence of a force required to support the vibrating system in
orbital free fall has no effect on the frequency of its vibration.
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Oscillatory Motion
Q12.7
No; Kinetic, Yes; Potential, No. For constant amplitude, the total energy
1
2
2
kA
stays constant. The
kinetic energy
1
2
2
mv
would increase for larger mass if the speed were constant, but here the greater
mass causes a decrease in frequency and in the average and maximum speed, so that the kinetic and
potential energies at every point are unchanged.
Q12.8
Since the acceleration is not constant in simple harmonic motion, none of the equations in Table 2.2
are valid.
Equation
Information given by equation
xt
A
t
af
bg
=+
cos
ωφ
position as a function of time
vt
A
t
=−
+
ωω
φ
sin
velocity as a function of time
vx
A
x
a
f
ej
=±
−
ω
22
12
velocity as a function of position
at
A
t
+
2
cos
acceleration as a function of time
ax
2
acceleration as a function of position
The angular frequency
appears in every equation. It is a good idea to figure out the value of angular
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 Fall '07
 MahaAshourAbdalla
 Energy, Simple Harmonic Motion

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