{"id":839,"date":"2017-10-27T16:30:54","date_gmt":"2017-10-27T16:30:54","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/period-and-frequency-in-oscillations\/"},"modified":"2017-11-08T03:25:37","modified_gmt":"2017-11-08T03:25:37","slug":"period-and-frequency-in-oscillations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/period-and-frequency-in-oscillations\/","title":{"raw":"Period and Frequency in Oscillations","rendered":"Period and Frequency in Oscillations"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Observe the vibrations of a guitar string.<\/li>\n<li>Determine the frequency of oscillations.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2420237\">\n<div class=\"bc-figcaption figcaption\">The strings on this guitar vibrate at regular time intervals. (credit: JAR)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2953405\" data-alt=\"The given figure shows a closed zoom view of the strings of a guitar. There are two slanting white colored strings in the picture. In the nearer string, the gaps between the circular threads of the string are visible, whereas the second white string at the back looks like a white thin stick.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_02_01a.jpg\" data-media-type=\"image\/jpeg\" alt=\"The given figure shows a closed zoom view of the strings of a guitar. There are two slanting white colored strings in the picture. In the nearer string, the gaps between the circular threads of the string are visible, whereas the second white string at the back looks like a white thin stick.\" width=\"225\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2639405\">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 <span data-type=\"term\" id=\"import-auto-id1910734\">periodic motion<\/span> to be a motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by an object on a spring moving up and down. The time to complete one oscillation remains constant and is called the <span data-type=\"term\" id=\"import-auto-id2999877\">period<\/span> [latex]T[\/latex]. Its units are usually seconds, but may be any convenient unit of time. The word period refers to the time for some event whether repetitive or not; but we shall be primarily interested in periodic motion, which is by definition repetitive. A concept closely related to period is the frequency of an event. For example, if you get a paycheck twice a month, the frequency of payment is two per month and the period between checks is half a month. <span data-type=\"term\" id=\"import-auto-id1907896\">Frequency<\/span>[latex]f[\/latex] is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]f=\\frac{1}{T}.[\/latex]<\/div>\n<p id=\"import-auto-id2402363\">The SI unit for frequency is the <em data-effect=\"italics\">cycle per second<\/em>, which is defined to be a <em data-effect=\"italics\">hertz<\/em> (Hz):<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{1 Hz}=1\\frac{\\text{cycle}}{\\text{sec}}\\text{or 1 Hz}=\\frac{1}{\\text{s}}[\/latex]<\/div>\n<p id=\"import-auto-id2980173\">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.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3397394\">\n<div data-type=\"title\" class=\"title\">Determine the Frequency of Two Oscillations: Medical Ultrasound and the Period of Middle C<\/div>\n<p>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\u2019s try one example of each. (a) A medical imaging device produces ultrasound by oscillating with a period of 0.400 \u00b5s. 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?<\/p>\n<p id=\"import-auto-id3037764\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3037765\">Both questions (a) and (b) can be answered using the relationship between period and frequency. In question (a), the period <em data-effect=\"italics\">[latex]T[\/latex]<\/em> 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 <em data-effect=\"italics\">[latex]T[\/latex]<\/em>.<\/p>\n<p id=\"import-auto-id1514315\"><strong>Solution a<\/strong><\/p>\n<ol id=\"fs-id3162993\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2206459\">Substitute [latex]0\\text{.}\\text{400}\\phantom{\\rule{0.25em}{0ex}}\\mathrm{\\text{\u03bc}}\\text{s}[\/latex] for <em data-effect=\"italics\">[latex]T[\/latex]<\/em> in [latex]f=\\frac{1}{T}[\/latex]:\n<div data-type=\"equation\" class=\"equation\">[latex]f=\\frac{1}{T}=\\frac{1}{0\\text{.}\\text{400}\u00d7{\\text{10}}^{-6}\\phantom{\\rule{0.25em}{0ex}}s}.[\/latex]<\/div>\n<p id=\"import-auto-id2979900\">Solve to find<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]f=2\\text{.}\\text{50}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{Hz}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2683550\"><strong>Discussion a<\/strong><\/p>\n<p id=\"import-auto-id2683551\">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.<\/p>\n<p id=\"import-auto-id3116694\"><strong>Solution b<\/strong><\/p>\n<ol id=\"fs-id2399993\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2659586\">Identify the known values:\n<p id=\"import-auto-id1994588\">The time for one complete oscillation is the period <em data-effect=\"italics\">[latex]T[\/latex]<\/em>:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]f=\\frac{1}{T}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1909739\">Solve for <em data-effect=\"italics\">[latex]T[\/latex]<\/em>:\n<div data-type=\"equation\" class=\"equation\">[latex]T=\\frac{1}{f}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1931109\">Substitute the given value for the frequency into the resulting expression:\n<div data-type=\"equation\" class=\"equation\">[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}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{s}=3\\text{.}\\text{79}\\phantom{\\rule{0.25em}{0ex}}\\text{ms}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id3077931\"><strong>Discussion <\/strong><\/p>\n<p id=\"import-auto-id3077932\">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).<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2401394\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3454521\">\n<p id=\"import-auto-id2639826\">Identify an event in your life (such as receiving a paycheck) that occurs regularly. Identify both the period and frequency of this event.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3105489\" data-print-placement=\"here\">\n<p id=\"import-auto-id3103253\">I visit my parents for dinner every other Sunday. The frequency of my visits is 26 per calendar year. The period is two weeks.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3112543\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2399846\">\n<li id=\"import-auto-id3437496\">Periodic motion is a repetitious oscillation.<\/li>\n<li id=\"import-auto-id1904875\">The time for one oscillation is the period [latex]T[\/latex].<\/li>\n<li id=\"import-auto-id2931325\">The number of oscillations per unit time is the frequency <em data-effect=\"italics\">[latex]f[\/latex]<\/em>.<\/li>\n<li id=\"import-auto-id2382777\">These quantities are related by\n<div data-type=\"equation\" class=\"equation\" id=\"eip-794\">[latex]f=\\frac{1}{T}.[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1941066\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2409820\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3046355\">\n<p id=\"import-auto-id1091158\">What is the period of [latex]\\text{60}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{Hz}[\/latex] electrical power?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2391328\">\n<p id=\"import-auto-id1941607\">16.7 ms<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2952945\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1413589\">\n<p>If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2053610\">\n[latex]0.400 s\/\\text{beats}[\/latex]\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1863946\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3105518\">Find the frequency of a tuning fork that takes [latex]2\\text{.}\\text{50}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex] to complete one oscillation.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3013159\">\n<p id=\"fs-id1917220\">400 Hz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1402085\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1404551\">\n<p id=\"import-auto-id701556\">A stroboscope is set to flash every [latex]8\\text{.}\\text{00}\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex]. What is the frequency of the flashes?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1869736\">\n<p id=\"fs-id2384713\">12,500 Hz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2421181\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3013679\">\n<p id=\"import-auto-id3229120\">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?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2442107\">\n<p id=\"import-auto-id3046355\">1.50 kHz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1011966\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3229120\">\n<p id=\"import-auto-id2674203\"><strong>Engineering Application<\/strong><\/p>\n<p id=\"eip-id1472446\">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?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3045334\">\n<p id=\"import-auto-id3135232\">(a) 93.8 m\/s<\/p>\n<p id=\"eip-id1625911\">(b) [latex]\\text{11}\\text{.}3\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{rev\/min}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2931652\">\n<dt>period<\/dt>\n<dd id=\"fs-id2674381\">time it takes to complete one oscillation<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2435584\">\n<dt>periodic motion<\/dt>\n<dd id=\"fs-id1438866\">motion that repeats itself at regular time intervals<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>frequency<\/dt>\n<dd id=\"fs-id2931175\">number of events per unit of time<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Observe the vibrations of a guitar string.<\/li>\n<li>Determine the frequency of oscillations.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2420237\">\n<div class=\"bc-figcaption figcaption\">The strings on this guitar vibrate at regular time intervals. (credit: JAR)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2953405\" data-alt=\"The given figure shows a closed zoom view of the strings of a guitar. There are two slanting white colored strings in the picture. In the nearer string, the gaps between the circular threads of the string are visible, whereas the second white string at the back looks like a white thin stick.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_02_01a.jpg\" data-media-type=\"image\/jpeg\" alt=\"The given figure shows a closed zoom view of the strings of a guitar. There are two slanting white colored strings in the picture. In the nearer string, the gaps between the circular threads of the string are visible, whereas the second white string at the back looks like a white thin stick.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2639405\">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 <span data-type=\"term\" id=\"import-auto-id1910734\">periodic motion<\/span> to be a motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by an object on a spring moving up and down. The time to complete one oscillation remains constant and is called the <span data-type=\"term\" id=\"import-auto-id2999877\">period<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. Its units are usually seconds, but may be any convenient unit of time. The word period refers to the time for some event whether repetitive or not; but we shall be primarily interested in periodic motion, which is by definition repetitive. A concept closely related to period is the frequency of an event. For example, if you get a paycheck twice a month, the frequency of payment is two per month and the period between checks is half a month. <span data-type=\"term\" id=\"import-auto-id1907896\">Frequency<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-85c83a1efb09a94e18c6c757182e15b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2402363\">The SI unit for frequency is the <em data-effect=\"italics\">cycle per second<\/em>, which is defined to be a <em data-effect=\"italics\">hertz<\/em> (Hz):<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0074efd47f146f3f04684d7ed617ac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#72;&#122;&#125;&#61;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#121;&#99;&#108;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#99;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#32;&#49;&#32;&#72;&#122;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"191\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2980173\">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.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3397394\">\n<div data-type=\"title\" class=\"title\">Determine the Frequency of Two Oscillations: Medical Ultrasound and the Period of Middle C<\/div>\n<p>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\u2019s try one example of each. (a) A medical imaging device produces ultrasound by oscillating with a period of 0.400 \u00b5s. 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?<\/p>\n<p id=\"import-auto-id3037764\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3037765\">Both questions (a) and (b) can be answered using the relationship between period and frequency. In question (a), the period <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/em> is given and we are asked to find frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/>. In question (b), the frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> is given and we are asked to find the period <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/em>.<\/p>\n<p id=\"import-auto-id1514315\"><strong>Solution a<\/strong><\/p>\n<ol id=\"fs-id3162993\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2206459\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2a18b85a3b5a3753318fe3648154fca3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> for <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/em> in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ad9a9cff0492c89c9bab3f85657421a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/>:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d6704cf4d5f6da9c7c5b60f4d16da681_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"152\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id2979900\">Solve to find<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b98c2c53f9718170f6855c2b4bc9d1da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#122;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2683550\"><strong>Discussion a<\/strong><\/p>\n<p id=\"import-auto-id2683551\">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.<\/p>\n<p id=\"import-auto-id3116694\"><strong>Solution b<\/strong><\/p>\n<ol id=\"fs-id2399993\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2659586\">Identify the known values:\n<p id=\"import-auto-id1994588\">The time for one complete oscillation is the period <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/em>:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-85c83a1efb09a94e18c6c757182e15b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1909739\">Solve for <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/em>:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-69f0ac52ba17e375b7d39e401a8c62d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#102;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1931109\">Substitute the given value for the frequency into the resulting expression:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-264cc2a517972496b7e8b4f133efdb2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#102;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#122;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#121;&#99;&#108;&#101;&#115;&#47;&#115;&#125;&#125;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"405\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id3077931\"><strong>Discussion <\/strong><\/p>\n<p id=\"import-auto-id3077932\">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).<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2401394\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3454521\">\n<p id=\"import-auto-id2639826\">Identify an event in your life (such as receiving a paycheck) that occurs regularly. Identify both the period and frequency of this event.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3105489\" data-print-placement=\"here\">\n<p id=\"import-auto-id3103253\">I visit my parents for dinner every other Sunday. The frequency of my visits is 26 per calendar year. The period is two weeks.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3112543\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2399846\">\n<li id=\"import-auto-id3437496\">Periodic motion is a repetitious oscillation.<\/li>\n<li id=\"import-auto-id1904875\">The time for one oscillation is the period <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>.<\/li>\n<li id=\"import-auto-id2931325\">The number of oscillations per unit time is the frequency <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/><\/em>.<\/li>\n<li id=\"import-auto-id2382777\">These quantities are related by\n<div data-type=\"equation\" class=\"equation\" id=\"eip-794\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-85c83a1efb09a94e18c6c757182e15b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#84;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1941066\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2409820\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3046355\">\n<p id=\"import-auto-id1091158\">What is the period of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8e1a1a23d8c672df945fe325803370df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/> electrical power?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2391328\">\n<p id=\"import-auto-id1941607\">16.7 ms<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2952945\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1413589\">\n<p>If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2053610\">\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1d173d537b05439af2fc609cf97699c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#52;&#48;&#48;&#32;&#115;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#101;&#97;&#116;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -5px;\" \/>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1863946\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3105518\">Find the frequency of a tuning fork that takes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cb04edb276128d7d9f96db6be9b092a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"79\" style=\"vertical-align: -1px;\" \/> to complete one oscillation.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3013159\">\n<p id=\"fs-id1917220\">400 Hz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1402085\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1404551\">\n<p id=\"import-auto-id701556\">A stroboscope is set to flash every <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8aa74f6a517ada8d803351c27fb826e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"79\" style=\"vertical-align: -1px;\" \/>. What is the frequency of the flashes?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1869736\">\n<p id=\"fs-id2384713\">12,500 Hz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2421181\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3013679\">\n<p id=\"import-auto-id3229120\">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?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2442107\">\n<p id=\"import-auto-id3046355\">1.50 kHz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1011966\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3229120\">\n<p id=\"import-auto-id2674203\"><strong>Engineering Application<\/strong><\/p>\n<p id=\"eip-id1472446\">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?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3045334\">\n<p id=\"import-auto-id3135232\">(a) 93.8 m\/s<\/p>\n<p id=\"eip-id1625911\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4958006bfcd40ba35004b1b7b6feeb52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#118;&#47;&#109;&#105;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2931652\">\n<dt>period<\/dt>\n<dd id=\"fs-id2674381\">time it takes to complete one oscillation<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2435584\">\n<dt>periodic motion<\/dt>\n<dd id=\"fs-id1438866\">motion that repeats itself at regular time intervals<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>frequency<\/dt>\n<dd id=\"fs-id2931175\">number of events per unit of time<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-839","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":826,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/839","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/839\/revisions"}],"predecessor-version":[{"id":840,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/839\/revisions\/840"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/826"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/839\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=839"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=839"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=839"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}