# 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.

#### 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

• The duration of one cycle in a repeating event.

• The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.

• The angular displacement per unit time.

#### Examples

• 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.

#### Figures

1. ##### Sinusoidal Waves of Varying Frequencies

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

2. ##### 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.

## 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 (Figure 1).

$f = 1/T$

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. (Figure 2)

$y(t) = sin(θ(t))=sin(ωt)=sin(2πft)$

$ω=2πf$

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

#### Key Term Glossary

angular
Relating to an angle or angles; having an angle or angles; forming an angle or corner; sharp-cornered; pointed; as in, an angular figure.
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angular frequency
The angular displacement per unit time.
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displacement
A vector quantity that denotes distance with a directional component.
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Displacement
The length and direction of a straight line between two objects.
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frequency
The quotient of the number of times n a periodic phenomenon occurs over the time t in which it occurs: f = n / t.
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hertz
In the International System of Units, the derived unit of frequency; one (period or cycle of any periodic event) per second. Symbol: Hz
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Hertz
Measurement of sound frequency.
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momentum
(of a body in motion) the product of its mass and velocity.
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motion
A change of position with respect to time.
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oscillation
the act of oscillating or the state of being oscillated
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period
The duration of one cycle in a repeating event.
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Period
The period is the duration of one cycle in a repeating event.
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phase
Any one point or portion in a recurring series of changes, as in the changes of motion of one of the particles constituting a wave or vibration; one portion of a series of such changes (in distinction from a contrasted portion) as the portion on one side of a position of equilibrium (in contrast with that on the opposite side).
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The angle subtended at the centre of a circle by an arc of the circle of the same length as the circle's radius.
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rotation
The act of turning around a centre or an axis.
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sinusoidal
In the form of a wave, especially one whose amplitude varies in proportion to the sine of some variable (such as time).
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SI units
International System of Units (abbreviated SI from French: Le Système international d'unités). It is the modern form of the metric system.
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wave
A moving disturbance in the energy level of a field.
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waveform
The shape of a physical wave, such as sound or electromagnetic radiation.  The shape can be any function that repeats in space.