163. Period and Frequency in OscillationsLearning Objectives
When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time. Each successive vibration of the string takes the same time as the previous one. We define The SI unit for frequency is the cycle per second, which is defined to be a hertz (Hz): $$\text{1 Hz}=1\frac{\text{cycle}}{\text{sec}}\text{or 1 Hz}=\frac{1}{\text{s}}$$A cycle is one complete oscillation. Note that a vibration can be a single or multiple event, whereas oscillations are usually repetitive for a significant number of cycles. Example 1: Determine the Frequency of Two Oscillations: Medical Ultrasound and the Period of Middle CWe 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 $T$ is given and we are asked to find frequency $f$. In question (b), the frequency $f$ is given and we are asked to find the period $T$. Solution a
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
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 UnderstandingIdentify an event in your life (such as receiving a paycheck) that occurs regularly. Identify both the period and frequency of this event. Show/Hide Solution SolutionI 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
Problems & ExercisesExercise 1What is the period of $\text{60}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{Hz}$ electrical power? Show/Hide Solution Solution16.7 ms Exercise 2If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds? Show/Hide Solution Solution$\mathrm{0.400\; s}/\text{beats}$ Exercise 3Find the frequency of a tuning fork that takes $2\text{.}\text{50}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{s}$ to complete one oscillation. Show/Hide Solution Solution400 Hz Exercise 4A stroboscope is set to flash every $8\text{.}\text{00}\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{s}$. What is the frequency of the flashes? Show/Hide Solution Solution12,500 Hz Exercise 5A 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? Show/Hide Solution Solution1.50 kHz Exercise 6Engineering 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 eightcylinder 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? Show/Hide Solution Solution(a) 93.8 m/s (b) $\text{11}\text{.}3\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{rev/min}$
