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1039
Chapter 14
Oscillations
Conceptual Problems
1
•
Determine the Concept
The acceleration of an oscillator of amplitude
A
and frequency
f
is zero when it is passing through its equilibrium position and is a maximum when it is
at its turning points.
When
v
=
v
max
:
0
=
a
When
x
=
x
max
:
A
f
A
a
2
2
2
4
π
ω
=
=
2
•
Determine the Concept
The condition for simple harmonic motion is that there be a linear
restoring force; i.e., that
F
=
−
kx.
Thus, the acceleration and displacement (when they are
not zero) are always oppositely directed.
v
and
a
can be in the same direction, as can
v
and
x
.
3
•
(
a
) False. In simple harmonic motion, the period is independent of the amplitude.
(
b
) True. In simple harmonic motion, the frequency is the reciprocal of the period which,
in turn, is independent of the amplitude.
(
c
) True. The condition that the acceleration of a particle is proportional to the
displacement and oppositely directed is equivalent to requiring that there be a linear
restoring force; i.e.,
F
=
−
kx
⇔
ma =
−
kx
or
a =
−
(
k/m
)
x.
*4
•
Determine the Concept
The energy of a simple harmonic oscillator varies as the square
of the amplitude of its motion. Hence, tripling the amplitude increases the energy by a
factor of 9.
5
••
Picture the Problem
The total energy of an object undergoing simple harmonic motion
is given by
,
2
2
1
tot
kA
E
=
where
k
is the stiffness constant and
A
is the amplitude of the
motion. The potential energy of the oscillator when it is a distance
x
from its equilibrium
position is
()
.
2
2
1
kx
x
U
=
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View Full DocumentChapter 14
1040
Express the ratio of the potential
energy of the object when it is 2 cm
from the equilibrium position to its
total energy:
( )
2
2
2
2
1
2
2
1
tot
A
x
kA
kx
E
x
U
=
=
Evaluate this ratio for
x
= 2 cm and
A
= 4 cm:
( )
( )
()
4
1
cm
4
cm
2
cm
2
2
2
tot
=
=
E
U
( )
( )
4
1
cm
4
cm
2
cm
2
2
2
tot
=
=
E
U
and
correct.
is
)
(
a
6
•
(
a
) True. The factors determining the period of the object, i.e., its mass and the spring
constant, are independent of the oscillator’s orientation.
(
b
) True. The factors determining the maximum speed of the object, i.e., its amplitude
and angular frequency, are independent of the oscillator’s orientation.
7
•
False.
In order for a simple pendulum to execute simple harmonic motion, the restoring
force must be linear. This condition is satisfied, at least approximately, for small initial
angular displacements.
8
•
True.
In order for a simple pendulum to execute periodic motion, the restoring force must
be linear. This condition is satisfied for any initial angular displacement.
*9
••
Determine the Concept
Assume that the first cart is given an initial velocity
v
by the
blow.
After the initial blow, there are no external forces acting on the carts, so their
center of mass moves at a constant velocity
v
/2.
The two carts will oscillate about their
center of mass in simple harmonic motion where the amplitude of their velocity is
v
/2.
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 Spring '07
 Licini
 Acceleration

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