Period & Frequency
- Define period and frequency and their reciprocal relationship
- Convert between period and frequency
- Interpret oscillation counts over a time interval
Two ways to describe a rhythm
Period (T) is the time for one complete cycle, in seconds. Frequency (f) is the number of cycles per second, in hertz (Hz). They describe the same oscillation from opposite angles and are reciprocals: f = 1/T. A pendulum with a 2 s period completes half a cycle each second, so its frequency is 0.5 Hz.
Counting cycles and angular frequency
If you count how many cycles happen in a stretch of time, frequency is simply (number of cycles) ÷ (time). Physicists also use angular frequency ω = 2πf, measured in rad/s, because one full cycle corresponds to 2π radians of phase. All three — T, f, and ω — carry the same information in different units.
An oscillator completes 30 cycles in 6 seconds. Find its frequency and its period.
- 1.Frequency is cycles per second: f = 30 ÷ 6 = 5 Hz.
- 2.Period is the reciprocal of frequency: T = 1 ÷ f.
- 3.Substitute: T = 1 ÷ 5.
- 4.Result: T = 0.2 s.
Period and frequency are reciprocals: if one is a small number, the other is large. A quick sanity check — a fast oscillation (high f) must have a short period (small T).
An oscillator completes 20 cycles in 4 seconds. What is its frequency?
An oscillator has a period of 0.25 s. What is its frequency?
Keep the units straight: period is seconds per cycle, frequency is cycles per second. When a problem gives "cycles in a time," divide cycles by time for frequency, then invert for period.
Answer the 2 checkpoints as you read.
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