Oscillatory Motion and Waves and Physics of Hearing.
117 Period and Frequency in Oscillations
Learning Objectives
- Observe the vibrations of a guitar string.
- Determine the frequency of oscillations.
When you pluck a guitar string, it produces a steady tone that lasts for a significant duration. This happens because the string undergoes regular, repeated vibrations. This kind of motion is known as periodic motion—motion that repeats itself at regular intervals.
The time it takes to complete one full oscillation is called the period and is denoted by [latex]T[/latex]. The standard unit for period is the second (s), but any time unit may be used depending on the context.
Closely related to period is the frequency of oscillation, which is the number of complete cycles that occur per unit time. For periodic motion, frequency [latex]f[/latex] is defined as:
The SI unit of frequency is the hertz (Hz), which corresponds to one cycle per second:
Note that a cycle refers to one complete oscillation. While vibrations may consist of single or multiple events, oscillations are typically understood to be repetitive and sustained over multiple cycles.
Example 117.1: Determine the Frequency of Two Oscillations: Medical Ultrasound and the Period of Middle C
We can use the formulas presented in this module to determine both the frequency based on known oscillations and the oscillation based on a known frequency. Let’s try one example of each. (a) A medical imaging device produces ultrasound by oscillating with a period of 0.400 µs. What is the frequency of this oscillation? (b) The frequency of middle C on a typical musical instrument is 264 Hz. What is the time for one complete oscillation?
Strategy
Both questions (a) and (b) can be answered using the relationship between period and frequency. In question (a), the period [latex]T[/latex] is given and we are asked to find frequency [latex]f[/latex]. In question (b), the frequency [latex]f[/latex] is given and we are asked to find the period [latex]T[/latex].
Solution a
- Substitute [latex]0\text{.}\text{400}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{μ}}\text{s}[/latex] for [latex]T[/latex] in [latex]f=\frac{1}{T}[/latex]:
[latex]f=\frac{1}{T}=\frac{1}{0\text{.}\text{400}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}s}.[/latex]
- Solve to find
[latex]f=2\text{.}\text{50}{\text{×}}{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{Hz.}[/latex]
Discussion a
The frequency of sound found in (a) is much higher than the highest frequency that humans can hear and, therefore, is called ultrasound. Appropriate oscillations at this frequency generate ultrasound used for noninvasive medical diagnoses, such as observations of a fetus in the womb.
Solution b
- Identify the known values:
The time for one complete oscillation is the period [latex]T[/latex]:
[latex]f=\frac{1}{T}.[/latex] - Solve for [latex]T[/latex]:
[latex]T=\frac{1}{f}.[/latex]
- Substitute the given value for the frequency into the resulting expression:
[latex]T=\frac{1}{f}=\frac{1}{\text{264}\phantom{\rule{0.25em}{0ex}}\text{Hz}}=\frac{1}{\text{264}\phantom{\rule{0.25em}{0ex}}\text{cycles/s}}=3\text{.}\text{79}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{s}=3\text{.}\text{79}\phantom{\rule{0.25em}{0ex}}\text{ms}.[/latex]
Discussion
The period found in (b) is the time per cycle, but this value is often quoted as simply the time in convenient units (ms or milliseconds in this case).
Check your Understanding
Identify an event in your life (such as receiving a paycheck) that occurs regularly. Identify both the period and frequency of this event.
I visit my parents for dinner every other Sunday. The frequency of my visits is 26 per calendar year. The period is two weeks.
Section Summary
- Periodic motion is a repetitious oscillation.
- The time for one oscillation is the period [latex]T[/latex].
- The number of oscillations per unit time is the frequency [latex]f[/latex].
- These quantities are related by
[latex]f=\frac{1}{T}.[/latex]
Problems & Exercises
- What is the period of [latex]\text{60}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{Hz}[/latex] electrical power?
- If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?
- Find the frequency of a tuning fork that takes [latex]2\text{.}\text{50}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{s}[/latex] to complete one oscillation.
- A stroboscope is set to flash every [latex]8\text{.}\text{00}×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}\text{s}[/latex]. What is the frequency of the flashes?
- A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at 30.0 m/s?
- Engineering Application Each piston of an engine makes a sharp sound every other revolution of the engine. (a) How fast is a race car going if its eight-cylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer? (b) At how many revolutions per minute is the engine rotating?
Glossary
- period
- time it takes to complete one oscillation
- periodic motion
- motion that repeats itself at regular time intervals
- frequency
- number of events per unit of time
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- vibrations
- repetitive back-and-forth or oscillatory motion of an object about an equilibrium position, often caused by a restoring force. Vibrations can occur in solids, liquids, or gases, and may produce waves such as sound.
motion that repeats itself at regular time intervals
time it takes to complete one oscillation
number of events per unit of time
Repetitive back-and-forth or oscillatory motion of an object about an equilibrium position, often caused by a restoring force. Vibrations can occur in solids, liquids, or gases, and may produce waves such as sound.
