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Chapter 13: Oscillations About Equilibrium Solutions to Problems
2.
Picture the Problem
: The rocking chair completes one full cycle or oscillation each time it returns back to its original
position.
Strategy:
The period is the time for one cycle.
The frequency is the inverse of the period, or the number of cycles per
second.
Solution:
1
. Divide the total time by the number of cycles to
determine the period:
21 s
1.75 s
12 cycles
t
T
n
=
=
=
2
.
Invert the period to determine the frequency:
1
1
0.57 Hz
1.75 s
f
T
=
=
=
Insight:
Since period and frequency are inverses of each other, when the period is greater than a second, the frequency
will be less than a hertz.
8.
Picture the Problem
:
The position of the mass oscillating on a spring is given by the equation of motion.
Strategy:
The oscillation period can be obtained directly from the argument of the cosine function. The mass is at one
extreme of its motion at
t
= 0, when the cosine is a maximum.
It then moves toward the center as the cosine approaches
zero.
The first zero crossing will occur when the cosine function first equals zero, that is, after onequarter period.
Solution:
1. (a)
Identify
T
with the time 0.58 s:
Since
2
2
cos
cos
0.58
t
t
T
s
π
=
, therefore T = 0.58 s.
2.
(b)
Multiply the period by onequarter to
find the first zero crossing:
( )
1
0.58 s
0.15 s .
4
t
=
=
Insight:
A cosine function is zero at ¼ and ¾ of a period. It has its greatest magnitude at 0 and
½ of a period.
10.
Picture the Problem
: The mass is attached to the spring and pulled down from equilibrium and released.
The stiffness
of the spring causes the mass to oscillate at the given frequency.
Strategy:
Since the mass starts at
x
=
A
at time
t
= 0, this is a cosine function given by
( )
cos
x
A
t
ω
=
.
From the data
given you need to identify the constants
A
and
.
Solution:
1.
Identify the amplitude as
A
:
A
= 6.4 cm
2.
Calculate
from the period:
2
2
0.83 s
T
=
=
3
. Substitute
A
and
into the cosine equation:
( )
2
6.4 cm cos
0.83 s
x
t
=
Insight:
A cosine function has a maximum amplitude at
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 Spring '08
 CROFT
 Physics

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