Period and Frequency

The period is the duration of one cycle in a repeating event, while the frequency is the number of cycles per unit time.

Learning Objective

Practice converting between frequency and period

Key Points

Motion that repeats itself regularly is called periodic motion. One complete repetition of the motion is called a cycle. The duration of each cycle is the period.
The frequency refers to the number of cycles completed in an interval of time. It is the reciprocal of the period and can be calculated with the equation f=1/T.
Some motion is best characterized by the angular frequency (ω). The angular frequency refers to the angular displacement per unit time and is calculated from the frequency with the equation ω=2πf.

Terms

angular frequency
The angular displacement per unit time.
period
The duration of one cycle in a repeating event.
frequency
The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.

Example

A Victorian parlor trick was to listen to the pitch of a fly's buzz, reproduce the musical note on the piano, and announce how many times the fly's wings had flapped in one second. If the fly's wings flap, say, 200 times in one second, then the frequency of their motion is f=200/1 s=200 Hz. The period is one 200th of a second, T=1/f=(1/200) s=0.005 s.

Full Text

Period and Frequency

The usual physics terminology for motion that repeats itself over and over is periodic motion, and the time required for one repetition is called the period, often expressed as the letter T. (The symbol P is not used because of the possible confusion with momentum. ) One complete repetition of the motion is called a cycle. The frequency is defined as the number of cycles per unit time. Frequency is usually denoted by a Latin letter f or by a Greek letter ν (nu). Note that period and frequency are reciprocals of each other .

Sinusoidal Waves of Varying Frequencies

Sinusoidal waves of various frequencies; the bottom waves have higher frequencies than those above. The horizontal axis represents time.

$f = 1/T$

For example, if a newborn baby's heart beats at a frequency of 120 times a minute, its period (the interval between beats) is half a second. If you calibrate your intuition so that you expect large frequencies to be paired with short periods, and vice versa, you may avoid some embarrassing mistakes on physics exams.

Units

In SI units, the unit of frequency is the hertz (Hz), named after the German physicist Heinrich Hertz: 1 Hz indicates that an event repeats once per second. A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated RPM. 60 RPM equals one hertz (i.e., one revolution per second, or a period of one second). The SI unit for period is the second.

Angular Frequency

Often periodic motion is best expressed in terms of angular frequency, represented by the Greek letter ω (omega). Angular frequency refers to the angular displacement per unit time (e.g., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function.

Locomotive Wheels

The locomotive's wheels spin at a frequency of f cycles per second, which can also be described as ω radians per second. The mechanical linkages allow the linear vibration of the steam engine's pistons, at frequency f, to drive the wheels.

$y(t) = sin(\theta (t))=sin(\omega t)=sin(2\pi ft)$

$\omega =2\pi f$

Angular frequency is often represented in units of radians per second (recall there are 2π radians in a circle).

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