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data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Oscillations <a href="#" class="mark-page-favorite pull-right" data-pid="272" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../physics.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../444/oscillations-and-waves.html">Oscillations and waves</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Oscillations</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 20 minutes"><i class="fa fa-clock-o"></i> 20'</span> </ol> <article id="main-article"> <p> <img alt="" src="../../waves/dong-1.jpg" style="float: left; width: 166px; height: 250px;">An oscillation is a movement through a full cycle.</p> <p>One example would be from a central <em>equilibrium position</em> to a maximum positive displacement, back through the equilibrium to a maximum negative displacement and back to the equilibrium position once more.</p> <p>Another would be from either maximum displacement to the other and back again.</p> <p>There are many oscillating systems that we could investigate, but we will start with the simple pendulum.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Simple pendulum</p> </div> </div> <div class="panel-body"> <div> <p>A real pendulum can twist and turn, and swing in circles with large angles. The string can even bend if the bob is dropped in a particular way.</p> <p>To simplify things in physics, we make some assumptions:</p> <ul> <li>the mass is very small (ideally a point)</li> <li>the string is inelastic (doesn’t stretch)</li> <li>the swings are small (< 10°)</li> </ul> <p>The downside of these assumptions is that a real pendulum will not necessarily behave as predicted, but it is quite easy to make your mass on a string at least resemble these.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Oscillations</p> </div> </div> <div class="panel-body"> <div> <p>An <em>oscillation</em> could consist of one complete swing from A – B – A:</p> <p style="text-align: center;"><img alt="" src="../../../ib/physics/activities/SHM1.png" style="width: 208px; height: 335px;"></p> <p><strong>Amplitude (<em>x<sub>o</sub></em>)</strong><strong>: </strong>A and B are positions of <em>amplitude</em>, where amplitude refers to the maximum displacement within a given swing. Amplitude is measured in m.</p> <p>NB: From <a href="../218/kinematics.html" title="Kinematics">Kinematics</a> that displacement (<em>x</em>) is the linear distance travelled from a starting location</p> <p><strong>Equilibrium position:</strong> The position where the pendulum bob would come to rest.</p> <p><strong>Time period (<em>T</em>): </strong>The time for one cycle (measured in s).</p> <p><strong>Frequency (<em>f</em><u>)</u>: </strong>The number of cycles per second (measured in Hz, which is equivalent to s<sup>-1</sup>)</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(f={1\over T}\)</span></p> </div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Simple harmonic motion</p> </div> </div> <div class="panel-body"> <div> <p>Simple harmonic motion is defined as motion where the acceleration is directly proportional to the displacement from a fixed point and always acts towards that point:</p> <p style="text-align: center;"><span class="math-tex">\(a\propto -x\)</span></p> <p>A graph of acceleration plotted against displacement would be:</p> <ul> <li>a straight line</li> <li>of negative gradient</li> <li>through the origin</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Phase</p> </div> </div> <div class="panel-body"> <div> <p>When bodies oscillate in time with each other they are said to be <em>in phase</em> - they need not have the same amplitude or dimensions.</p> <p>A full cycle can be described mathematically as 2<em>π</em> radians.</p> <p>Phase difference (or phase angle), <em>φ</em>, is the angular shift in the displacement between two oscillations:</p> <ul> <li><em>y</em><sub>1</sub> = <em>r</em> sin(<em>ωt</em>) </li> <li><em>y</em><sub>2</sub> = <em>r</em> sin(<em>ωt</em> + <em>φ</em>)</li> </ul> <p style="text-align: center;"><img alt="" src="../../images/shmphase.png" style="width: 365px; height: 132px;"></p> <p>If the phase difference, <em>φ</em> = <em>π</em>, the systems are described as being in <em>antiphase</em>. If the phase difference is 0 or 2<em>π</em> (etc!), the systems are <em>in phase</em>. Anything in between is <em>out of phase</em>.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Calculating time period</p> </div> </div> <div class="panel-body"> <div> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> It is possible to calculate the time period for oscillations of a pendulum and spring. This is only required at Higher Level. Click to reveal.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> Time period of a pendulum,<img align="middle" alt="T space equals space 2 pi square root of L over g end root" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»§#x000A0;«/mo»«mo»=«/mo»«mo»§#x000A0;«/mo»«mn»2«/mn»«mi»§#x003C0;«/mi»«msqrt»«mfrac»«mi»L«/mi»«mi»g«/mi»«/mfrac»«/msqrt»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-42.php?formula=d7a35f4d474507d1d09611a0334d36e8.png"></p> <ul> <li><em>L</em> is the length of the string in m</li> <li><em>g</em> is gravitational field strength</li> </ul> <p>NB: The mass of the bob has no effect on time period, nor does amplitude!</p> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> Time period of a mass on a spring,<img align="middle" alt="T equals 2 pi square root of m over k end root" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mn»2«/mn»«mi»§#x003C0;«/mi»«msqrt»«mfrac»«mi»m«/mi»«mi»k«/mi»«/mfrac»«/msqrt»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-43.php?formula=6838fb36b8d116b5fc67ee85216bbac6.png"></p> <ul> <li><em>m</em> is the mass in kg</li> <li><em>k</em> is the spring constant in Nm<sup>-1</sup></li> </ul> </section> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>General equations</p> </div> </div> <div class="panel-body"> <div> <p>The equation for the variation of displacement with time is as follows:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(x=x_0 \sin (2\pi ft)\)</span><br> <span style="background-color: rgb(255, 255, 255);"></span></p> <p>or...</p> <p style="text-align: center;"><span style="background-color: rgb(255, 255, 255);"><span class="math-tex">\(x=x_0 \cos (2\pi ft)\)</span></span></p> </div> <p><span style="background-color: rgb(255, 255, 255);"></span></p> <p><img class="sibico" src="../../../img/sibico/exam-style.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Exam-style question"> Since sine and cosine graphs are translations of the other, the decision of which equation to use from above will be based on the starting point of the motion. 'cos' is the most common, as a pendulum is usually started from its amplitude.</p> <p>There are several equations for use in simple harmonic oscillations, all of which are available in your Data Booklet:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(x=x_0\sin {\omega t}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(v=\omega x_0 \cos {\omega t}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(a=-\omega^2 x_0 \sin {\omega t}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(v = \omega \sqrt{x_0^2-x^2} \)</span></p> <p style="text-align: center;"><span class="math-tex">\(a=-\omega ^2 x\)</span></p> <ul> <li><em>x</em> (m) = displacement</li> <li><em>x<sub>0</sub></em> (m) = amplitude</li> <li><em>ω</em> (Hz) = angular frequency = 2π<em>f</em></li> <li><em>t</em> (s) = time</li> <li><em>a</em> (ms<sup>-2</sup>) = acceleration</li> <li><em>v</em> (ms<sup>-1</sup>) = velocity</li> </ul> </div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Energy changes</p> </div> </div> <div class="panel-body"> <div> <p>As a pendulum swings or mass bounces on a spring, there is a continual exchange of energy from potential to kinetic and back to potential. The total energy remains constant:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(E_{total}=E_P+E_K\)</span></p> </div> <p>Kinetic energy varies with time as follows:</p> <p><span class="math-tex">\(E_K={1\over 2}m\omega^2{x_0}^2\cos^2\omega t\)</span></p> <p>Since energy is conserved, the PE at any time will be PE<sub>max</sub> - KE:</p> <p><span class="math-tex">\(E_P={1\over 2}m\omega^2{x_0}^2\sin^2\omega t\)</span></p> <p>Maths dictates that <span class="math-tex">\(\cos^2\alpha+\sin^2\alpha=1\)</span> so the total energy simplifies to become:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(E_{total}={1\over 2}m\omega^2{x_0}^2 \)</span></p> </div> <p>The total energy, kinetic energy and potential energy can be shown on a graph as follows:</p> <p style="text-align: center;"><img alt="" src="../../images/shmenergy.png" style="width: 421px; height: 255px;"></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><i>Use flashcards to practise your recall.</i></p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="1" data-mode="Normal" data-topics="524" data-subject-id="6" data-n-flashcards="13" style="text-align:center">Show flashcards</a></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-red"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Just for Fun</p> </div> </div> <div class="panel-body"> <div> <p><em>Check out this πg physics summary.</em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; 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