amplitude, Period, Frequency, and angular Frequency

Here are some terms that we’ll use in discussing periodic motions of all kinds:

The amplitude of the motion, denoted by A, is the maximum magnitude of
displacement from equilibrium—that is, the maximum value of 0 x 0 . It is always
positive. If the spring in Fig. 14.2 is an ideal one, the total overall range of the
motion is 2A. The SI unit of A is the meter. A complete vibration, or cycle, is one
complete round trip—say, from A to -A and back to A, or from O to A, back
through O to -A, and back to O. Note that motion from one side to the other (say,
-A to A) is a half-cycle, not a whole cycle.

The period, T, is the time to complete one cycle. It is always positive. The SI unit is the second, but it is sometimes expressed as “seconds per cycle.” The frequency, f, is the number of cycles in a unit of time. It is always positive. The SI unit of frequency is the hertz, named for the 19th-century German physicist Heinrich Hertz:

   1 hertz = 1 Hz = 1 cycle/s = 1 s^-1

The angular frequency,\(x = { \Omega}\)is 2p times the frequency:

We’ll learn shortly why v is a useful quantity. It represents the rate of change of an angular quantity (not necessarily related to a rotational motion) that is always measured in radians, so its units are rad>s. Since f is in cycle>s, we may regard the number 2p as having units rad/cycle. By definition, period and frequency are reciprocals of each other:

Also, from the definition of v,

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