8.3 Periods and Frequencies 211
The period of oscillation,T, for the given system — or in other words, the time that it takes for
the mass to complete one cycle — is given by
(8.2)
Most of you have seen oscillating pendulums in clocks. The pendulum is another good
example of a periodic system. The period of oscillation for a pendulum is given by
(8.3)
whereLis the length of the pendulum ( m) andgis the acceleration due to gravity ( m/s
2
).
Note that the period of oscillation is independent of the mass of the pendulum. Not too
long ago, oil companies used measured changes in the period of an oscillating pendulum to
detect variations in acceleration due to gravity that could indicate an underground oil
reservoir.
An understanding of periods and frequencies is also important in the design of electrical
and electronic components. In general, excited mechanical systems have much lower frequencies
than electrical /electronic systems. Examples of frequencies of various electrical and electronic
systems are given in Table 8.1.
T 2 p
B
L
g
T
1
fn
TABLE 8.1 Examples of Frequencies of Various Electrical
and Electronic Systems
Application Frequency
Alternating current (USA) 60 Hz
AM radio 540 kHz –1.6 MHz
FM radio 88 –108 MHz
Emergency, fire, police 153 –159 MHz
Personal computer clocks (as of year 2010) up to 3 GHz
Wireless router (2009) 5 GHz
m = 2 kg
k = 5 N/mm = 5000 N/m
■Figure 8.5 A simple spring–mass system.
Example 8.1 Determine the natural frequency of the simple spring-mass system shown in Figure 8.5.
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