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href="../1124/thin-film-interference.html">Thin film interference</a></label></li></ul></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" 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"> Simple harmonic motion <a href="#" class="mark-page-favorite pull-right" data-pid="1117" 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><a href="../1128/ahl-waves.html">AHL Waves</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Simple harmonic motion</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/shm.jpg" style="float: left; width: 250px; height: 166px;">In the Standard Level waves section we established the definition of simple harmonic oscillations: the acceleration of an oscillating body is proportional to displacement but in the opposite direction.</p> <p>The AHL Waves requirements are more quantitative - displacement, velocity, acceleration, time period, kinetic energy and potential energy must now be calculated at any time.</p> <p>We recommend that you revise <a href="../272/oscillations.html" title="Oscillations">Oscillations</a> before studying this page.</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> <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;"><img alt="" src="../../waves/shm-graph.png" style="width: 200px; height: 207px;"></p> <p>This definition can be represented by an equation:</p> <p style="text-align: center;"><span class="math-tex">\(a = -ω^2x\)</span></p> <ul> <li><span class="math-tex">\(a\)</span> is acceleration (ms<sup>-2</sup>)</li> <li><span class="math-tex">\(\omega\)</span> is angular frequency: <span class="math-tex">\(\omega=2\pi f={2\pi\over T}\)</span> (Hz)</li> <li><span class="math-tex">\(x\)</span> is displacement (m)</li> </ul> <p>NB: <span class="math-tex">\(f={1\over T}\)</span></p> <div class="panel panel-turquoise panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Motion equations</p> </div> </div> <div class="panel-body"> <div> <p>If we assume that the clock starts when the object is released from maximum displacement (the amplitude): </p> <p style="text-align: center;"><span class="math-tex">\(x=x_0\cos{\omega t} \Rightarrow v=-\omega x_0\sin \omega t \)</span></p> <ul> <li><span class="math-tex">\(x_0\)</span> is amplitude (m)</li> <li><span class="math-tex">\(t\)</span> is time (s)</li> <li><span class="math-tex">\(v\)</span> is velocity (ms<sup>-1</sup>)</li> </ul> <p>This velocity is derived by differentiating the displacement equation. Velocity can be found at any time by finding the gradient of displacement. Here the combination is shown (but the maxima of the three would not, in reality, be the same).</p> <p style="text-align: center;"><img alt="" src="../../waves/three-graphs.png" style="width: 639px; height: 302px;"></p> <p><span class="math-tex">\(\)</span>Alternatively, time might 'commence' when the object passes through equililbrium:</p> <p style="text-align: center;"><span class="math-tex">\(x=x_0\sin{\omega t} \Rightarrow v=\omega x_0\cos \omega t\)</span></p> <p>Also check out: <a href="../274/equations-for-shm.html" title="Equations for SHM">Equations for SHM</a></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> <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>Mass on a spring</p> </div> </div> <div class="panel-body"> <div> <p>A mass oscillating on a spring is an example of simple harmonic motion.</p> <p style="text-align: center"><iframe allowfullscreen="" height="600" scrolling="no" src="https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html" width="800"></iframe></p> <p style="text-align: center;"><img alt="" src="../../waves/mass-on-spring.png" style="width: 258px; height: 325px;"></p> <p style="text-align: center;"><span class="math-tex">\(T=2\pi \sqrt{m\over k}\)</span></p> <ul> <li><span class="math-tex">\(T\)</span> is time period (s)</li> <li><span class="math-tex">\(m\)</span> is mass (kg)</li> <li><span class="math-tex">\(k\)</span> is spring constant (N m<sup>-1</sup>)</li> </ul> <p>Also check out: <a href="../276/mass-on-a-spring.html" title="Mass on a spring">Mass on a spring</a></p> </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>Simple pendulum</p> </div> </div> <div class="panel-body"> <div> <p>Simple pendula are another example.</p> <p style="text-align: center"><iframe allowfullscreen="" height="600" scrolling="no" src="https://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html" width="800"></iframe></p> <p style="text-align: center;"><img alt="" src="../../waves/pendulum(1).png" style="width: 208px; height: 335px;"></p> <p style="text-align: center;"><span class="math-tex">\(T=2\pi\sqrt{l\over g}\)</span></p> <ul> <li><span class="math-tex">\(T\)</span> is time period (s)</li> <li><span class="math-tex">\(l\)</span> is the length of the string</li> <li><span class="math-tex">\(g\)</span> is gravitational field strength (N kg<sup>-1</sup>)</li> </ul> <p>Also check out: <a href="../273/simple-pendulum.html" title="Simple pendulum">Simple pendulum</a></p> </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>The energies involved are kinetic and potential energy. The two sum to give the total energy.</p> <p style="text-align: center;"><span class="math-tex">\(E_{total}=E_P+E_K\)</span></p> <p style="text-align: center;"><img alt="" src="../../images/shmenergy.png"></p> <p style="text-align: center;"><iframe height="398px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/2558125/width/588/height/398/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" style="border:0px;" width="588px"></iframe></p> <p>To commence our quantitative understanding, we need to be able to calculate velocity at any point.</p> <p style="text-align: center;"><span class="math-tex">\(v=\pm \omega\sqrt{{x_0}^2-x^2}\)</span></p> <h3>Kinetic energy</h3> <p>The usual equation for kinetic energy, <span class="math-tex">\(E_k={1\over 2}mv^2\)</span>, still applies. Substituting the velocity from above:</p> <p style="text-align: center;"><span class="math-tex">\(E_k={1\over 2}m\omega^2({x_0}^2-x^2)\)</span></p> <h3>Total energy</h3> <p>Since the total energy is constant, we can find the total energy by maximising the kinetic energy (and thus minimising the potential energy to zero). We do so by considering the body at zero displacement.</p> <p style="text-align: center;"><span class="math-tex">\(x=0\)</span></p> <p style="text-align: center;"><span class="math-tex">\(E_T={1\over 2}m\omega^2{x_0}^2\)</span></p> <p>Alternatively we could consider the maximum possible potential energy with the displacement as the amplitude.</p> <p>Also check out: <a href="../275/shm-and-energy.html" title="SHM and energy">SHM and energy</a></p> <h3>Potential energy</h3> <p>The potential energy at any point is equal to the total energy minus the kinetic energy.</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-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><em>Use flashcards to practise your recall.</em></p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="1,3" data-mode="" data-topics="987" data-subject-id="6" data-n-flashcards="5" 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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <p><em>Online tutorials to help you solve original problems.</em></p> <iframe src="../../../pages/subjects/page-networks/index-47.htm?ticket=082b9c9c4ae3624d&network-id=81&sitename=physics&subject-id=6&fullscreen=false&node=start" style="width: 100%; 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