{"id":847,"date":"2017-10-27T16:30:55","date_gmt":"2017-10-27T16:30:55","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/simple-harmonic-motion-a-special-periodic-motion\/"},"modified":"2017-11-08T03:25:39","modified_gmt":"2017-11-08T03:25:39","slug":"simple-harmonic-motion-a-special-periodic-motion","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/simple-harmonic-motion-a-special-periodic-motion\/","title":{"raw":"Simple Harmonic Motion: A Special Periodic Motion","rendered":"Simple Harmonic Motion: A Special Periodic Motion"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe a simple harmonic oscillator.<\/li>\n<li>Explain the link between simple harmonic motion and waves.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2444604\">The oscillations of a system in which the net force can be described by Hooke\u2019s law are of special importance, because they are very common. They are also the simplest oscillatory systems. <span data-type=\"term\" id=\"import-auto-id1917029\">Simple Harmonic Motion<\/span> (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke\u2019s law, and such a system is called a <span data-type=\"term\">simple harmonic oscillator<\/span>. If the net force can be described by Hooke\u2019s law and there is no <em data-effect=\"italics\"><em data-effect=\"italics\"> damping<\/em> (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position, as shown for an object on a spring in <a href=\"#import-auto-id1428057\" class=\"autogenerated-content\">(Figure)<\/a>. The maximum displacement from equilibrium is called the <span data-type=\"term\" id=\"import-auto-id1993471\">amplitude<\/span> [latex]X[\/latex]. The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.<\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2402363\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment: SHM and the Marble<\/div>\n<p>Find a bowl or basin that is shaped like a hemisphere on the inside. Place a marble inside the bowl and tilt the bowl periodically so the marble rolls from the bottom of the bowl to equally high points on the sides of the bowl. Get a feel for the force required to maintain this periodic motion. What is the restoring force and what role does the force you apply play in the simple harmonic motion (SHM) of the marble?<\/p>\n<\/div>\n<p id=\"import-auto-id3042914\">\n<\/p><div class=\"bc-figure figure\" id=\"import-auto-id1428057\">\n<div class=\"bc-figcaption figcaption\">An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude [latex]X[\/latex] and a period [latex]T[\/latex]. The object\u2019s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the period [latex]T[\/latex]. The greater the mass of the object is, the greater the period [latex]T[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3254682\" data-alt=\"The figure a shows a spring on a frictionless surface attached to a bar or wall from the left side. On the right side of the spring, an object attached to it with mass m, its amplitude is given by X, and X is equal to zero at the equilibrium level. Force F is applied to it from the right side, shown with left direction pointed red arrow and velocity v is equal to zero. A direction point showing the north and west direction is also given alongside this figure as well as with other four figures. In figure b, after the force has been applied the object moves to the left compressing the spring a bit. And the displaced area of the object from its initial point is shown in sketched dots. The F here is equal to zero and the v is max in negative direction. In figure c, the spring has been compressed to the maximum level, and the amplitude is negative X. Now the direction of force changes to the rightward direction, shown with right direction pointed red arrow and the velocity v is zero. In figure d the spring is shown released from the compressed level and the object has moved toward the right side up to the equilibrium level. The F is zero, and the velocity v is maximum. In figure e the spring has been stretched loose to the maximum level and the object has moved to the far right. Now again the velocity here is equal to zero and the direction of force again is to the left hand side, shown here as F is equal to zero.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure a shows a spring on a frictionless surface attached to a bar or wall from the left side. On the right side of the spring, an object attached to it with mass m, its amplitude is given by X, and X is equal to zero at the equilibrium level. Force F is applied to it from the right side, shown with left direction pointed red arrow and velocity v is equal to zero. A direction point showing the north and west direction is also given alongside this figure as well as with other four figures. In figure b, after the force has been applied the object moves to the left compressing the spring a bit. And the displaced area of the object from its initial point is shown in sketched dots. The F here is equal to zero and the v is max in negative direction. In figure c, the spring has been compressed to the maximum level, and the amplitude is negative X. Now the direction of force changes to the rightward direction, shown with right direction pointed red arrow and the velocity v is zero. In figure d the spring is shown released from the compressed level and the object has moved toward the right side up to the equilibrium level. The F is zero, and the velocity v is maximum. In figure e the spring has been stretched loose to the maximum level and the object has moved to the far right. Now again the velocity here is equal to zero and the direction of force again is to the left hand side, shown here as F is equal to zero.\" width=\"500\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2402624\">What is so significant about simple harmonic motion? One special thing is that the period [latex]T[\/latex] and frequency [latex]f[\/latex] of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. Because the period is constant, a simple harmonic oscillator can be used as a clock.<\/p>\n<p id=\"import-auto-id3127573\">Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant [latex]k[\/latex], which causes the system to have a smaller period. For example, you can adjust a diving board\u2019s stiffness\u2014the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.<\/p>\n<p id=\"import-auto-id3028369\">In fact, the mass [latex]m[\/latex] and the force constant [latex]k[\/latex] are the <em data-effect=\"italics\">only<\/em> factors that affect the period and frequency of simple harmonic motion. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2929144\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Period of Simple Harmonic Oscillator<\/div>\n<p id=\"import-auto-id2409627\">The <em data-effect=\"italics\">period of a simple harmonic oscillator<\/em> is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-41\">[latex]T=2\\pi \\sqrt{\\frac{m}{k}}[\/latex]<\/div>\n<p id=\"import-auto-id1593873\">and, because [latex]f=1\/T[\/latex], the <em data-effect=\"italics\">frequency of a simple harmonic oscillator<\/em> is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]f=\\frac{1}{2\\pi }\\sqrt{\\frac{k}{m}}.[\/latex]<\/div>\n<p>Note that neither [latex]T[\/latex] nor <em data-effect=\"italics\">[latex]f[\/latex]<\/em> has any dependence on amplitude.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2692701\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment: Mass and Ruler Oscillations <\/div>\n<p id=\"import-auto-id2406935\">Find two identical wooden or plastic rulers. Tape one end of each ruler firmly to the edge of a table so that the length of each ruler that protrudes from the table is the same. On the free end of one ruler tape a heavy object such as a few large coins. Pluck the ends of the rulers at the same time and observe which one undergoes more cycles in a time period, and measure the period of oscillation of each of the rulers.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1366587\">\n<div data-type=\"title\" class=\"title\">Calculate the Frequency and Period of Oscillations: Bad Shock Absorbers in a Car <\/div>\n<p>If the shock absorbers in a car go bad, then the car will oscillate at the least provocation, such as when going over bumps in the road and after stopping (See <a href=\"#import-auto-id1352637\" class=\"autogenerated-content\">(Figure)<\/a>). Calculate the frequency and period of these oscillations for such a car <em data-effect=\"italics\">if the car\u2019s mass (including its load) is 900 kg and the force constant (<em data-effect=\"italics\">[latex]k[\/latex]<\/em>) of the suspension system is [latex]6\\text{.}\\text{53}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/m}[\/latex].<\/em><\/p>\n<p id=\"import-auto-id2618137\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3032062\">The frequency of the car\u2019s oscillations will be that of a simple harmonic oscillator as given in the equation [latex]f=\\frac{1}{2\\pi }\\sqrt{\\frac{k}{m}}[\/latex]. The mass and the force constant are both given.<\/p>\n<p id=\"import-auto-id2956915\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id3068651\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1576543\">Enter the known values of <em data-effect=\"italics\">k<\/em> and <em data-effect=\"italics\">m<\/em>:\n<div data-type=\"equation\" class=\"equation\">[latex]f=\\frac{1}{2\\pi }\\sqrt{\\frac{k}{m}}=\\frac{1}{2\\pi }\\sqrt{\\frac{6\\text{.}\\text{53}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/m}}{\\text{900}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}}}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1848789\">Calculate the frequency:\n<div data-type=\"equation\" class=\"equation\" id=\"eip-465\">[latex]\\frac{1}{2\\pi }\\sqrt{\\text{72.}6\/{\\text{s}}^{\u20132}}=1\\text{.}{\\text{3656}\/\\text{s}}^{\\text{\u20131}}\\approx 1\\text{.}{\\text{36}\/\\text{s}}^{\\text{\u20131}}=\\text{1.36 Hz}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1411067\">You could use [latex]T=2\\pi \\sqrt{\\frac{m}{k}}[\/latex] to calculate the period, but it is simpler to use the relationship [latex]T=1\/f[\/latex] and substitute the value just found for [latex]f[\/latex]:\n<div data-type=\"equation\" class=\"equation\" id=\"eip-207\">[latex]T=\\frac{1}{f}=\\frac{1}{1\\text{.}\\text{356}\\phantom{\\rule{0.25em}{0ex}}\\text{Hz}}=0\\text{.}\\text{738}\\phantom{\\rule{0.25em}{0ex}}\\text{s}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id3192561\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2679099\">The values of [latex]T[\/latex] and [latex]f[\/latex] both seem about right for a bouncing car. You can observe these oscillations if you push down hard on the end of a car and let go.  <\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2681584\">\n<h1 data-type=\"title\">The Link between Simple Harmonic Motion and Waves<\/h1>\n<p>If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in <a href=\"#import-auto-id1352637\" class=\"autogenerated-content\">(Figure)<\/a>. Similarly, <a href=\"#import-auto-id1177267\" class=\"autogenerated-content\">(Figure)<\/a> shows an object bouncing on a spring as it leaves a wavelike \"trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1352637\">\n<div class=\"bc-figcaption figcaption\">The bouncing car makes a wavelike motion. If the restoring force in the suspension system can be described only by Hooke\u2019s law, then the wave is a sine function. (The wave is the trace produced by the headlight as the car moves to the right.)<\/div>\n<p><span data-type=\"media\" data-alt=\"The figure shows the front right side of a running car on an uneven rough surface which also shows the driver in the driving seat. There is an oscillating sine wave drawn from left to the right side horizontally throughout the figure.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows the front right side of a running car on an uneven rough surface which also shows the driver in the driving seat. There is an oscillating sine wave drawn from left to the right side horizontally throughout the figure.\" width=\"300\"><\/span><\/p><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1177267\">\n<div class=\"bc-figcaption figcaption\">The vertical position of an object bouncing on a spring is recorded on a strip of moving paper, leaving a sine wave.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2391861\" data-alt=\"There are two iron paper roll bars standing vertically with a paper strip stitched from one bar to the other. There is a vertical hanging spring just over the middle of the two bars, perpendicular to the strip of the paper, having an object with mass m tied to it. There is a line graph with amplitude scale as X, zero and negative X on the left side of the paper strip, vertically over each other with their points marked. A perpendicular line is drawn through this amplitude scale toward the right with a point T marked over it, showing the time duration of the amplitude. This line has an oscillating wave drawn through it.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"There are two iron paper roll bars standing vertically with a paper strip stitched from one bar to the other. There is a vertical hanging spring just over the middle of the two bars, perpendicular to the strip of the paper, having an object with mass m tied to it. There is a line graph with amplitude scale as X, zero and negative X on the left side of the paper strip, vertically over each other with their points marked. A perpendicular line is drawn through this amplitude scale toward the right with a point T marked over it, showing the time duration of the amplitude. This line has an oscillating wave drawn through it.\" width=\"175\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3008498\">The displacement as a function of time <em data-effect=\"italics\">t<\/em> in any simple harmonic motion\u2014that is, one in which the net restoring force can be described by Hooke\u2019s law, is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-555\">[latex]x\\left(t\\right)=X\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\frac{2\\mathrm{\\pi t}}{T},[\/latex]<\/div>\n<p>where [latex]X[\/latex] is amplitude. At [latex]t=0[\/latex], the initial position is [latex]{x}_{0}=X[\/latex], and the displacement oscillates back and forth with a period [latex]T[\/latex]<em data-effect=\"italics\">.<\/em> (When [latex]t=T[\/latex], we get [latex]x=X[\/latex] again because [latex]\\text{cos}\\phantom{\\rule{0.25em}{0ex}}2\\pi =1[\/latex].). Furthermore, from this expression for <strong data-effect=\"bold\"><em data-effect=\"italics\">[latex]x[\/latex]<\/em>, the velocity [latex]v[\/latex] as a function of time is given by:<\/strong><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-468\">[latex]v\\left(t\\right)=-{v}_{\\text{max}}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\left(\\frac{2\\pi t}{T}\\right),[\/latex]<\/div>\n<p id=\"import-auto-id2666940\">where [latex]{v}_{\\text{max}}=2\\pi X\/T=X\\sqrt{k\/m}[\/latex]. The object has zero velocity at maximum displacement\u2014for example, [latex]v=0[\/latex] when [latex]t=0[\/latex], and at that time [latex]x=X[\/latex]. The minus sign in the first equation for [latex]v\\left(t\\right)[\/latex] gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton\u2019s second law. [Then we have [latex]x\\left(t\\right),\\phantom{\\rule{0.25em}{0ex}}v\\left(t\\right),\\phantom{\\rule{0.25em}{0ex}}t,[\/latex] and [latex]a\\left(t\\right)[\/latex], the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton\u2019s second law, the acceleration is [latex]a=F\/m=\\text{kx}\/m[\/latex]<em data-effect=\"italics\">.<\/em> So, [latex]a\\left(t\\right)[\/latex] is also a cosine function:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-987\">[latex]a\\left(t\\right)=-\\frac{\\text{kX}}{m}\\text{cos}\\frac{2\\pi t}{T}.[\/latex]<\/div>\n<p id=\"import-auto-id2397669\">Hence, [latex]a\\left(t\\right)[\/latex] is directly proportional to and in the opposite direction to [latex]x\\left(t\\right)[\/latex].<\/p>\n<p><a href=\"#import-auto-id2429266\" class=\"autogenerated-content\">(Figure)<\/a> shows the simple harmonic motion of an object on a spring and presents graphs of [latex]x\\left(t\\right),v\\left(t\\right),[\/latex] and [latex]a\\left(t\\right)[\/latex] versus time.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2429266\">\n<div class=\"bc-figcaption figcaption\">Graphs of [latex]x\\left(t\\right),\\phantom{\\rule{0.25em}{0ex}}v\\left(t\\right),[\/latex] and [latex]a\\left(t\\right)[\/latex] versus [latex]t[\/latex] for the motion of an object on a spring. The net force on the object can be described by Hooke\u2019s law, and so the object undergoes simple harmonic motion. Note that the initial position has the vertical displacement at its maximum value [latex]X[\/latex]; [latex]v[\/latex] is initially zero and then negative as the object moves down; and the initial acceleration is negative, back toward the equilibrium position and becomes zero at that point.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2404258\" data-alt=\"In the figure at the top there are ten springboards with objects of different mass values tied to them. This makes some springs highly compressed some as loosely stretched and some at equilibrium, which are shown as red spherical shaped. Alongside the figure there is a scale given for different amplitude values as x equal to positive X, zero and negative X. the upward and downward pointing arrows are shown with a few springboards.  In the second figure there are three graphs. The first graph shows distance covered in form of a sine wave starting from a point x units on positive y-axis. The height of the wave above x-axis is marked as amplitude. The gap between two consecutive crests is marked as T. Below first graph there is another graph showing velocity in form of a sine wave starting from the origin downward. In the third graph below the second one, acceleration is shown in the form of sine wave starting from x units on the negative y-axis upward. In the last figure three position of a spring are shown. The first position shows the unstretched length of a spring pendulum. A hand is holding the bob of the pendulum. In the second position the equilibrium position of the spring and bob is shown. This position is lower the first one. In the third case the up and down oscillations of the spring pendulum are shown. The bob is moving x units in upward and downward directions alternatively.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"In the figure at the top there are ten springboards with objects of different mass values tied to them. This makes some springs highly compressed some as loosely stretched and some at equilibrium, which are shown as red spherical shaped. Alongside the figure there is a scale given for different amplitude values as x equal to positive X, zero and negative X. the upward and downward pointing arrows are shown with a few springboards.  In the second figure there are three graphs. The first graph shows distance covered in form of a sine wave starting from a point x units on positive y-axis. The height of the wave above x-axis is marked as amplitude. The gap between two consecutive crests is marked as T. Below first graph there is another graph showing velocity in form of a sine wave starting from the origin downward. In the third graph below the second one, acceleration is shown in the form of sine wave starting from x units on the negative y-axis upward. In the last figure three position of a spring are shown. The first position shows the unstretched length of a spring pendulum. A hand is holding the bob of the pendulum. In the second position the equilibrium position of the spring and bob is shown. This position is lower the first one. In the third case the up and down oscillations of the spring pendulum are shown. The bob is moving x units in upward and downward directions alternatively.\" height=\"575\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2658069\">The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2056610\" 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-id1293481\">\n<p id=\"import-auto-id3149731\">Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. Describe what happens to the sound waves in terms of period, frequency and amplitude as the sound decreases in volume.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2680599\" data-print-placement=\"here\">\n<p id=\"import-auto-id2930765\">Frequency and period remain essentially unchanged. Only amplitude decreases as volume decreases.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2056612\" 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-id3122227\">\n<p id=\"import-auto-id3149771\">A babysitter is pushing a child on a swing. At the point where the swing reaches [latex]x[\/latex], where would the corresponding point on a wave of this motion be located?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1362944\" data-print-placement=\"here\">\n<p id=\"import-auto-id1447563\">[latex]x[\/latex] is the maximum deformation, which corresponds to the amplitude of the wave. The point on the wave would either be at the very top or the very bottom of the curve.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-432\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Masses and Springs<\/div>\n<p id=\"eip-id2906922\">A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2778789\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/f9c1f6148f21d37ec06b322249becbaa6da33296\/mass-spring-lab_en.jar\">Masses and Springs<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_17.3\" data-alt=\"\"><a href=\"\/resources\/f9c1f6148f21d37ec06b322249becbaa6da33296\/mass-spring-lab_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/jpg\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/jpg\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3209887\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul>\n<li>Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke\u2019s law. Such a system is also called a simple harmonic oscillator. <\/li>\n<li id=\"import-auto-id1488427\">Maximum displacement is the amplitude <em data-effect=\"italics\">[latex]X[\/latex]<\/em>. The period <em data-effect=\"italics\">[latex]T[\/latex]<\/em> and frequency [latex]f[\/latex] of a simple harmonic oscillator are given by\n<p id=\"import-auto-id3358885\">[latex]T=2\\pi \\sqrt{\\frac{m}{k}}[\/latex] and [latex]f=\\frac{1}{2\\pi }\\sqrt{\\frac{k}{m}}[\/latex], where [latex]m[\/latex] is the mass of the system. <\/p>\n<\/li>\n<li id=\"import-auto-id1860331\">Displacement in simple harmonic motion as a function of time is given by [latex]x\\left(t\\right)=X\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\frac{2\\pi t}{T}.[\/latex]<\/li>\n<li>The velocity is given by\n<p>[latex]v\\left(t\\right)=-{v}_{\\text{max}}\\text{sin}\\frac{2\\pi \\text{t}}{T}[\/latex], where  <\/p>\n<p>[latex]{v}_{\\text{max}}=\\sqrt{k\/m}X[\/latex].<\/p><\/li>\n<li id=\"import-auto-id954284\">The acceleration is found to be [latex]a\\left(t\\right)=-\\frac{\\mathrm{kX}}{m}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\frac{2\\pi t}{T}.[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3143102\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2017072\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3025808\">\n<p id=\"import-auto-id2673447\">What conditions must be met to produce simple harmonic motion?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1561901\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3228419\">(a) If frequency is not constant for some oscillation, can the oscillation be simple harmonic motion?<\/p>\n<p id=\"import-auto-id2423880\">(b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1571112\">\n<p>Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1888472\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2594604\">\n<p id=\"import-auto-id765532\">Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a spongy material.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3306170\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1815181\">\n<p id=\"import-auto-id2957194\">As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3449442\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2053170\">\n<p id=\"import-auto-id3047367\">Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id3077105\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1587278\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1422331\">\n<p id=\"import-auto-id1872785\">A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2398932\">\n<p id=\"import-auto-id3163897\">[latex]2\\text{.}\\text{37}\\phantom{\\rule{0.25em}{0ex}}\\text{N\/m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3397701\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id347645\">\n<p id=\"import-auto-id3080601\">If the spring constant of a simple harmonic oscillator is doubled, by what factor will the mass of the system need to change in order for the frequency of the motion to remain the same?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2697740\">\n<p id=\"import-auto-id1816394\">A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2618365\">\n<p id=\"import-auto-id2659905\">0.389 kg<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3062785\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3285510\">By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1931315\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2056751\">\n<p id=\"import-auto-id3306170\">Suppose you attach the object with mass [latex]m[\/latex] to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring\u2019s original rest length. (a) Show that the spring exerts an upward force of <\/p>\n[latex]2.00\\phantom{\\rule{0.25em}{0ex}}\\mathrm{mg}[\/latex]\n<p> on the object at its lowest point. (b) If the spring has a force constant of [latex]\\text{10}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{N\/m}[\/latex] and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations. (c) Find the maximum velocity. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1941088\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2010292\">\n<p id=\"import-auto-id1561696\">A diver on a diving board is undergoing simple harmonic motion. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver is a male whose period of simple harmonic oscillation is 1.05 s. What is his mass if the mass of the board is negligible? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2503482\">\n<p id=\"import-auto-id2659378\">94.7 kg<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3068943\">\n<p>Suppose a diving board with no one on it bounces up and down in a simple harmonic motion with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the simple harmonic motion of a 75.0-kg diver on the board? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3032062\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<div class=\"bc-figure figure\" id=\"import-auto-id2023197\">\n<div class=\"bc-figcaption figcaption\">This child\u2019s toy relies on springs to keep infants entertained. (credit: By Humboldthead, Flickr)\n<div data-type=\"newline\"><\/div>\n<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2687821\" data-alt=\"The figure shows a little kid, about ten to twelve months old, standing in a toy jolly jumper, which is tied to the ceiling hook by its four spring belts.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_02_02a.jpg\" data-media-type=\"image\/png\" alt=\"The figure shows a little kid, about ten to twelve months old, standing in a toy jolly jumper, which is tied to the ceiling hook by its four spring belts.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"eip-id1165652466883\">The device pictured in <a href=\"#import-auto-id2023197\" class=\"autogenerated-content\">(Figure)<\/a> entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring constant.<\/p>\n<p id=\"eip-id1165652459177\">(a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its spring constant?<\/p>\n<p>(b) What is the time for one complete bounce of this child? (c) What is the child\u2019s maximum velocity if the amplitude of her bounce is 0.200 m?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3035922\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2687937\">\n<p>A 90.0-kg skydiver hanging from a parachute bounces up and down with a period of 1.50 s. What is the new period of oscillation when a second skydiver, whose mass is 60.0 kg, hangs from the legs of the first, as seen in <a href=\"#import-auto-id1282324\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">The oscillations of one skydiver are about to be affected by a second skydiver. (credit: U.S. Army, www.army.mil)<\/div>\n<p><span data-type=\"media\" data-alt=\"The figure shows two skydivers midway through the air, with both with open having their parachutes open.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_02_03a.jpg\" data-media-type=\"image\/png\" alt=\"The figure shows two skydivers midway through the air, with both with open having their parachutes open.\" width=\"200\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1942411\">\n<p id=\"import-auto-id2056190\">1.94 s<\/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-id1439009\">\n<dt>amplitude<\/dt>\n<dd id=\"fs-id2648557\">the maximum displacement from the equilibrium position of an object oscillating around the equilibrium position<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3149576\">\n<dt>simple harmonic motion<\/dt>\n<dd id=\"fs-id2438369\">the oscillatory motion in a system where the net force can be described by Hooke\u2019s law<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3123906\">\n<dt>simple harmonic oscillator<\/dt>\n<dd id=\"fs-id2422256\">a device that implements Hooke\u2019s law, such as a mass that is attached to a spring, with the other end of the spring being connected to a rigid support such as a wall<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe a simple harmonic oscillator.<\/li>\n<li>Explain the link between simple harmonic motion and waves.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2444604\">The oscillations of a system in which the net force can be described by Hooke\u2019s law are of special importance, because they are very common. They are also the simplest oscillatory systems. <span data-type=\"term\" id=\"import-auto-id1917029\">Simple Harmonic Motion<\/span> (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke\u2019s law, and such a system is called a <span data-type=\"term\">simple harmonic oscillator<\/span>. If the net force can be described by Hooke\u2019s law and there is no <em data-effect=\"italics\"><em data-effect=\"italics\"> damping<\/em> (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position, as shown for an object on a spring in <a href=\"#import-auto-id1428057\" class=\"autogenerated-content\">(Figure)<\/a>. The maximum displacement from equilibrium is called the <span data-type=\"term\" id=\"import-auto-id1993471\">amplitude<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>. The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.<\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2402363\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment: SHM and the Marble<\/div>\n<p>Find a bowl or basin that is shaped like a hemisphere on the inside. Place a marble inside the bowl and tilt the bowl periodically so the marble rolls from the bottom of the bowl to equally high points on the sides of the bowl. Get a feel for the force required to maintain this periodic motion. What is the restoring force and what role does the force you apply play in the simple harmonic motion (SHM) of the marble?<\/p>\n<\/div>\n<p id=\"import-auto-id3042914\">\n<div class=\"bc-figure figure\" id=\"import-auto-id1428057\">\n<div class=\"bc-figcaption figcaption\">An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> and a 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;\" \/>. The object\u2019s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller 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;\" \/>. The greater the mass of the object is, the greater 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;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3254682\" data-alt=\"The figure a shows a spring on a frictionless surface attached to a bar or wall from the left side. On the right side of the spring, an object attached to it with mass m, its amplitude is given by X, and X is equal to zero at the equilibrium level. Force F is applied to it from the right side, shown with left direction pointed red arrow and velocity v is equal to zero. A direction point showing the north and west direction is also given alongside this figure as well as with other four figures. In figure b, after the force has been applied the object moves to the left compressing the spring a bit. And the displaced area of the object from its initial point is shown in sketched dots. The F here is equal to zero and the v is max in negative direction. In figure c, the spring has been compressed to the maximum level, and the amplitude is negative X. Now the direction of force changes to the rightward direction, shown with right direction pointed red arrow and the velocity v is zero. In figure d the spring is shown released from the compressed level and the object has moved toward the right side up to the equilibrium level. The F is zero, and the velocity v is maximum. In figure e the spring has been stretched loose to the maximum level and the object has moved to the far right. Now again the velocity here is equal to zero and the direction of force again is to the left hand side, shown here as F is equal to zero.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure a shows a spring on a frictionless surface attached to a bar or wall from the left side. On the right side of the spring, an object attached to it with mass m, its amplitude is given by X, and X is equal to zero at the equilibrium level. Force F is applied to it from the right side, shown with left direction pointed red arrow and velocity v is equal to zero. A direction point showing the north and west direction is also given alongside this figure as well as with other four figures. In figure b, after the force has been applied the object moves to the left compressing the spring a bit. And the displaced area of the object from its initial point is shown in sketched dots. The F here is equal to zero and the v is max in negative direction. In figure c, the spring has been compressed to the maximum level, and the amplitude is negative X. Now the direction of force changes to the rightward direction, shown with right direction pointed red arrow and the velocity v is zero. In figure d the spring is shown released from the compressed level and the object has moved toward the right side up to the equilibrium level. The F is zero, and the velocity v is maximum. In figure e the spring has been stretched loose to the maximum level and the object has moved to the far right. Now again the velocity here is equal to zero and the direction of force again is to the left hand side, shown here as F is equal to zero.\" width=\"500\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2402624\">What is so significant about simple harmonic motion? One special thing is that 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;\" \/> and 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;\" \/> of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. Because the period is constant, a simple harmonic oscillator can be used as a clock.<\/p>\n<p id=\"import-auto-id3127573\">Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/>, which causes the system to have a smaller period. For example, you can adjust a diving board\u2019s stiffness\u2014the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.<\/p>\n<p id=\"import-auto-id3028369\">In fact, the mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> and the force constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> are the <em data-effect=\"italics\">only<\/em> factors that affect the period and frequency of simple harmonic motion. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2929144\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Period of Simple Harmonic Oscillator<\/div>\n<p id=\"import-auto-id2409627\">The <em data-effect=\"italics\">period of a simple harmonic oscillator<\/em> is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-41\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-24366e860e66701342f06680db9da49c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#107;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1593873\">and, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2b4a38e1c644a9f0b1cf83f9725ad02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#49;&#47;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -5px;\" \/>, the <em data-effect=\"italics\">frequency of a simple harmonic oscillator<\/em> 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-6eedf1786829e0d26099dd44161b24de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#109;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"91\" style=\"vertical-align: -11px;\" \/><\/div>\n<p>Note that neither <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;\" \/> nor <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> has any dependence on amplitude.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2692701\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment: Mass and Ruler Oscillations <\/div>\n<p id=\"import-auto-id2406935\">Find two identical wooden or plastic rulers. Tape one end of each ruler firmly to the edge of a table so that the length of each ruler that protrudes from the table is the same. On the free end of one ruler tape a heavy object such as a few large coins. Pluck the ends of the rulers at the same time and observe which one undergoes more cycles in a time period, and measure the period of oscillation of each of the rulers.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1366587\">\n<div data-type=\"title\" class=\"title\">Calculate the Frequency and Period of Oscillations: Bad Shock Absorbers in a Car <\/div>\n<p>If the shock absorbers in a car go bad, then the car will oscillate at the least provocation, such as when going over bumps in the road and after stopping (See <a href=\"#import-auto-id1352637\" class=\"autogenerated-content\">(Figure)<\/a>). Calculate the frequency and period of these oscillations for such a car <em data-effect=\"italics\">if the car\u2019s mass (including its load) is 900 kg and the force constant (<em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/em>) of the suspension system is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c95661e6308a1d635c8963ee9dd558d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#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;&#78;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/>.<\/em><\/p>\n<p id=\"import-auto-id2618137\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3032062\">The frequency of the car\u2019s oscillations will be that of a simple harmonic oscillator as given in the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-78328a0bffd8eee4d15767f4c1669248_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#109;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"88\" style=\"vertical-align: -11px;\" \/>. The mass and the force constant are both given.<\/p>\n<p id=\"import-auto-id2956915\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id3068651\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1576543\">Enter the known values of <em data-effect=\"italics\">k<\/em> and <em data-effect=\"italics\">m<\/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-13fb9c977fb86e10924c17394042b69e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#109;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#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;&#78;&#47;&#109;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#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;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"234\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1848789\">Calculate the frequency:\n<div data-type=\"equation\" class=\"equation\" id=\"eip-465\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6060af838d49744ad480dc72370aca13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#50;&#46;&#125;&#54;&#47;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#53;&#54;&#125;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#125;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#125;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#51;&#54;&#32;&#72;&#122;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"371\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1411067\">You could use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-24366e860e66701342f06680db9da49c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#107;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -7px;\" \/> to calculate the period, but it is simpler to use the relationship <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a93839bbdda531d12e7e62a684983aa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#49;&#47;&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -5px;\" \/> and substitute the value just found for <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;\" \/>:\n<div data-type=\"equation\" class=\"equation\" id=\"eip-207\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c9fa1ca484d2016806da011e5bca3bca_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;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#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;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#51;&#56;&#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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"208\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id3192561\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2679099\">The values of <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;\" \/> and <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;\" \/> both seem about right for a bouncing car. You can observe these oscillations if you push down hard on the end of a car and let go.  <\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2681584\">\n<h1 data-type=\"title\">The Link between Simple Harmonic Motion and Waves<\/h1>\n<p>If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in <a href=\"#import-auto-id1352637\" class=\"autogenerated-content\">(Figure)<\/a>. Similarly, <a href=\"#import-auto-id1177267\" class=\"autogenerated-content\">(Figure)<\/a> shows an object bouncing on a spring as it leaves a wavelike &#8220;trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1352637\">\n<div class=\"bc-figcaption figcaption\">The bouncing car makes a wavelike motion. If the restoring force in the suspension system can be described only by Hooke\u2019s law, then the wave is a sine function. (The wave is the trace produced by the headlight as the car moves to the right.)<\/div>\n<p><span data-type=\"media\" data-alt=\"The figure shows the front right side of a running car on an uneven rough surface which also shows the driver in the driving seat. There is an oscillating sine wave drawn from left to the right side horizontally throughout the figure.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows the front right side of a running car on an uneven rough surface which also shows the driver in the driving seat. There is an oscillating sine wave drawn from left to the right side horizontally throughout the figure.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1177267\">\n<div class=\"bc-figcaption figcaption\">The vertical position of an object bouncing on a spring is recorded on a strip of moving paper, leaving a sine wave.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2391861\" data-alt=\"There are two iron paper roll bars standing vertically with a paper strip stitched from one bar to the other. There is a vertical hanging spring just over the middle of the two bars, perpendicular to the strip of the paper, having an object with mass m tied to it. There is a line graph with amplitude scale as X, zero and negative X on the left side of the paper strip, vertically over each other with their points marked. A perpendicular line is drawn through this amplitude scale toward the right with a point T marked over it, showing the time duration of the amplitude. This line has an oscillating wave drawn through it.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"There are two iron paper roll bars standing vertically with a paper strip stitched from one bar to the other. There is a vertical hanging spring just over the middle of the two bars, perpendicular to the strip of the paper, having an object with mass m tied to it. There is a line graph with amplitude scale as X, zero and negative X on the left side of the paper strip, vertically over each other with their points marked. A perpendicular line is drawn through this amplitude scale toward the right with a point T marked over it, showing the time duration of the amplitude. This line has an oscillating wave drawn through it.\" width=\"175\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3008498\">The displacement as a function of time <em data-effect=\"italics\">t<\/em> in any simple harmonic motion\u2014that is, one in which the net restoring force can be described by Hooke\u2019s law, is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-555\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-34ee28e7f3756a0f1e5f50b9c570b2d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#88;&#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;&#111;&#115;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#112;&#105;&#32;&#116;&#125;&#125;&#123;&#84;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is amplitude. At <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7b41acc5cb99fb07aaa07b445eb2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/>, the initial position is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1daf8586eb4e88054ddafa13b55ebd00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#61;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -3px;\" \/>, and the displacement oscillates back and forth with a 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;\" \/><em data-effect=\"italics\">.<\/em> (When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2962a9cbea029fd97ae13e2f0269181b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>, we get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76ecbaa38823100415d2fffdcee22a8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/> again because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-854155c822d2ec49f2bde05023c31f8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#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;&#50;&#92;&#112;&#105;&#32;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: -1px;\" \/>.). Furthermore, from this expression for <strong data-effect=\"bold\"><em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/em>, the velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> as a function of time is given by:<\/strong><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-468\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d03296b8883fc5a5ad08c35f9da8e999_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#97;&#120;&#125;&#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;&#105;&#110;&#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;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#112;&#105;&#32;&#116;&#125;&#123;&#84;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"186\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2666940\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8f7d52bddbcbaafaf9b876474fa19951_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#97;&#120;&#125;&#125;&#61;&#50;&#92;&#112;&#105;&#32;&#88;&#47;&#84;&#61;&#88;&#92;&#115;&#113;&#114;&#116;&#123;&#107;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"207\" style=\"vertical-align: -6px;\" \/>. The object has zero velocity at maximum displacement\u2014for example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-67b297dfc2c97b2e282dc2716c601577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7b41acc5cb99fb07aaa07b445eb2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/>, and at that time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76ecbaa38823100415d2fffdcee22a8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/>. The minus sign in the first equation for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5dbd503d889d025c31c055b7b26509b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\" \/> gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton\u2019s second law. [Then we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bb3376cc7cb2c66137347bf8278a461d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#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;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#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;&#116;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5adf8520ff4291e76485d2ff57d871c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\" \/>, the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton\u2019s second law, the acceleration is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ad24cbcaf6a8c31b0d0e58956b2b2e69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#70;&#47;&#109;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#120;&#125;&#47;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"137\" style=\"vertical-align: -5px;\" \/><em data-effect=\"italics\">.<\/em> So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5adf8520ff4291e76485d2ff57d871c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\" \/> is also a cosine function:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-987\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ad71f1856c387a1521820eb4642a11b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#88;&#125;&#125;&#123;&#109;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#112;&#105;&#32;&#116;&#125;&#123;&#84;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"143\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2397669\">Hence, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5adf8520ff4291e76485d2ff57d871c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\" \/> is directly proportional to and in the opposite direction to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0fdd4a3d40e945941a35495caf464f53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><a href=\"#import-auto-id2429266\" class=\"autogenerated-content\">(Figure)<\/a> shows the simple harmonic motion of an object on a spring and presents graphs of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7cea7654068779f9a1c596450293c5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5adf8520ff4291e76485d2ff57d871c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\" \/> versus time.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2429266\">\n<div class=\"bc-figcaption figcaption\">Graphs of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7887d72dfbb3927b1dcbba2f3fe1504b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#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;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5adf8520ff4291e76485d2ff57d871c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\" \/> versus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> for the motion of an object on a spring. The net force on the object can be described by Hooke\u2019s law, and so the object undergoes simple harmonic motion. Note that the initial position has the vertical displacement at its maximum value <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is initially zero and then negative as the object moves down; and the initial acceleration is negative, back toward the equilibrium position and becomes zero at that point.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2404258\" data-alt=\"In the figure at the top there are ten springboards with objects of different mass values tied to them. This makes some springs highly compressed some as loosely stretched and some at equilibrium, which are shown as red spherical shaped. Alongside the figure there is a scale given for different amplitude values as x equal to positive X, zero and negative X. the upward and downward pointing arrows are shown with a few springboards.  In the second figure there are three graphs. The first graph shows distance covered in form of a sine wave starting from a point x units on positive y-axis. The height of the wave above x-axis is marked as amplitude. The gap between two consecutive crests is marked as T. Below first graph there is another graph showing velocity in form of a sine wave starting from the origin downward. In the third graph below the second one, acceleration is shown in the form of sine wave starting from x units on the negative y-axis upward. In the last figure three position of a spring are shown. The first position shows the unstretched length of a spring pendulum. A hand is holding the bob of the pendulum. In the second position the equilibrium position of the spring and bob is shown. This position is lower the first one. In the third case the up and down oscillations of the spring pendulum are shown. The bob is moving x units in upward and downward directions alternatively.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_03_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"In the figure at the top there are ten springboards with objects of different mass values tied to them. This makes some springs highly compressed some as loosely stretched and some at equilibrium, which are shown as red spherical shaped. Alongside the figure there is a scale given for different amplitude values as x equal to positive X, zero and negative X. the upward and downward pointing arrows are shown with a few springboards.  In the second figure there are three graphs. The first graph shows distance covered in form of a sine wave starting from a point x units on positive y-axis. The height of the wave above x-axis is marked as amplitude. The gap between two consecutive crests is marked as T. Below first graph there is another graph showing velocity in form of a sine wave starting from the origin downward. In the third graph below the second one, acceleration is shown in the form of sine wave starting from x units on the negative y-axis upward. In the last figure three position of a spring are shown. The first position shows the unstretched length of a spring pendulum. A hand is holding the bob of the pendulum. In the second position the equilibrium position of the spring and bob is shown. This position is lower the first one. In the third case the up and down oscillations of the spring pendulum are shown. The bob is moving x units in upward and downward directions alternatively.\" height=\"575\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2658069\">The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2056610\" 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-id1293481\">\n<p id=\"import-auto-id3149731\">Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. Describe what happens to the sound waves in terms of period, frequency and amplitude as the sound decreases in volume.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2680599\" data-print-placement=\"here\">\n<p id=\"import-auto-id2930765\">Frequency and period remain essentially unchanged. Only amplitude decreases as volume decreases.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2056612\" 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-id3122227\">\n<p id=\"import-auto-id3149771\">A babysitter is pushing a child on a swing. At the point where the swing reaches <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, where would the corresponding point on a wave of this motion be located?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1362944\" data-print-placement=\"here\">\n<p id=\"import-auto-id1447563\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is the maximum deformation, which corresponds to the amplitude of the wave. The point on the wave would either be at the very top or the very bottom of the curve.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-432\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Masses and Springs<\/div>\n<p id=\"eip-id2906922\">A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2778789\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/f9c1f6148f21d37ec06b322249becbaa6da33296\/mass-spring-lab_en.jar\">Masses and Springs<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_17.3\" data-alt=\"\"><a href=\"\/resources\/f9c1f6148f21d37ec06b322249becbaa6da33296\/mass-spring-lab_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/jpg\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/jpg\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3209887\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul>\n<li>Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke\u2019s law. Such a system is also called a simple harmonic oscillator. <\/li>\n<li id=\"import-auto-id1488427\">Maximum displacement is the amplitude <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/><\/em>. 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> and 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;\" \/> of a simple harmonic oscillator are given by\n<p id=\"import-auto-id3358885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-24366e860e66701342f06680db9da49c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#107;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -7px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-78328a0bffd8eee4d15767f4c1669248_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#109;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"88\" style=\"vertical-align: -11px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> is the mass of the system. <\/p>\n<\/li>\n<li id=\"import-auto-id1860331\">Displacement in simple harmonic motion as a function of time is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-245f7d544a08239d870817512b3e1507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#88;&#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;&#111;&#115;&#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;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#112;&#105;&#32;&#116;&#125;&#123;&#84;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"134\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>The velocity is given by\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a2e04276bc4feb013fc223031f5a3385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#97;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#112;&#105;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#123;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"150\" style=\"vertical-align: -6px;\" \/>, where  <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-47690cdd14393f0e14c0c3998c738b0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#97;&#120;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#107;&#47;&#109;&#125;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"127\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/li>\n<li id=\"import-auto-id954284\">The acceleration is found to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-071fd38f7896e29045acd068728b6f2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#107;&#88;&#125;&#125;&#123;&#109;&#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;&#111;&#115;&#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;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#112;&#105;&#32;&#116;&#125;&#123;&#84;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"152\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3143102\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2017072\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3025808\">\n<p id=\"import-auto-id2673447\">What conditions must be met to produce simple harmonic motion?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1561901\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3228419\">(a) If frequency is not constant for some oscillation, can the oscillation be simple harmonic motion?<\/p>\n<p id=\"import-auto-id2423880\">(b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1571112\">\n<p>Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1888472\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2594604\">\n<p id=\"import-auto-id765532\">Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a spongy material.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3306170\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1815181\">\n<p id=\"import-auto-id2957194\">As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3449442\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2053170\">\n<p id=\"import-auto-id3047367\">Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id3077105\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1587278\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1422331\">\n<p id=\"import-auto-id1872785\">A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2398932\">\n<p id=\"import-auto-id3163897\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-21a5430e5c059a4e82fa436b6eeb1f96_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;&#51;&#55;&#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;&#78;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3397701\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id347645\">\n<p id=\"import-auto-id3080601\">If the spring constant of a simple harmonic oscillator is doubled, by what factor will the mass of the system need to change in order for the frequency of the motion to remain the same?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2697740\">\n<p id=\"import-auto-id1816394\">A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2618365\">\n<p id=\"import-auto-id2659905\">0.389 kg<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3062785\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id3285510\">By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1931315\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2056751\">\n<p id=\"import-auto-id3306170\">Suppose you attach the object with mass <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring\u2019s original rest length. (a) Show that the spring exerts an upward force of <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d234a273285ba84617aa92af2831e121_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#48;&#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;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -3px;\" \/><\/p>\n<p> on the object at its lowest point. (b) If the spring has a force constant of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a036b8a9eaeb2fea4bc0d630aa992ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#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;&#78;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -4px;\" \/> and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations. (c) Find the maximum velocity. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1941088\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2010292\">\n<p id=\"import-auto-id1561696\">A diver on a diving board is undergoing simple harmonic motion. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver is a male whose period of simple harmonic oscillation is 1.05 s. What is his mass if the mass of the board is negligible? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2503482\">\n<p id=\"import-auto-id2659378\">94.7 kg<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3068943\">\n<p>Suppose a diving board with no one on it bounces up and down in a simple harmonic motion with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the simple harmonic motion of a 75.0-kg diver on the board? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3032062\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<div class=\"bc-figure figure\" id=\"import-auto-id2023197\">\n<div class=\"bc-figcaption figcaption\">This child\u2019s toy relies on springs to keep infants entertained. (credit: By Humboldthead, Flickr)<\/p>\n<div data-type=\"newline\"><\/div>\n<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2687821\" data-alt=\"The figure shows a little kid, about ten to twelve months old, standing in a toy jolly jumper, which is tied to the ceiling hook by its four spring belts.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_02_02a.jpg\" data-media-type=\"image\/png\" alt=\"The figure shows a little kid, about ten to twelve months old, standing in a toy jolly jumper, which is tied to the ceiling hook by its four spring belts.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"eip-id1165652466883\">The device pictured in <a href=\"#import-auto-id2023197\" class=\"autogenerated-content\">(Figure)<\/a> entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring constant.<\/p>\n<p id=\"eip-id1165652459177\">(a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its spring constant?<\/p>\n<p>(b) What is the time for one complete bounce of this child? (c) What is the child\u2019s maximum velocity if the amplitude of her bounce is 0.200 m?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3035922\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2687937\">\n<p>A 90.0-kg skydiver hanging from a parachute bounces up and down with a period of 1.50 s. What is the new period of oscillation when a second skydiver, whose mass is 60.0 kg, hangs from the legs of the first, as seen in <a href=\"#import-auto-id1282324\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">The oscillations of one skydiver are about to be affected by a second skydiver. (credit: U.S. Army, www.army.mil)<\/div>\n<p><span data-type=\"media\" data-alt=\"The figure shows two skydivers midway through the air, with both with open having their parachutes open.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_02_03a.jpg\" data-media-type=\"image\/png\" alt=\"The figure shows two skydivers midway through the air, with both with open having their parachutes open.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1942411\">\n<p id=\"import-auto-id2056190\">1.94 s<\/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-id1439009\">\n<dt>amplitude<\/dt>\n<dd id=\"fs-id2648557\">the maximum displacement from the equilibrium position of an object oscillating around the equilibrium position<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3149576\">\n<dt>simple harmonic motion<\/dt>\n<dd id=\"fs-id2438369\">the oscillatory motion in a system where the net force can be described by Hooke\u2019s law<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3123906\">\n<dt>simple harmonic oscillator<\/dt>\n<dd id=\"fs-id2422256\">a device that implements Hooke\u2019s law, such as a mass that is attached to a spring, with the other end of the spring being connected to a rigid support such as a wall<\/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-847","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\/847","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\/847\/revisions"}],"predecessor-version":[{"id":848,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/847\/revisions\/848"}],"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\/847\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=847"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=847"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=847"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=847"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}