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href="../444/oscillations-and-waves.html">Oscillations and waves</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../272/oscillations.html">Oscillations</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Equations for SHM</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 15 minutes"><i class="fa fa-clock-o"></i> 15'</span> </ol> <article id="main-article"> <p><img alt="" src="../../waves/7-calculus.jpg" style="float: left; width: 250px; height: 143px;">It is not obvious that a sinusoidally-varying displacement is a consequence of the condition that the acceleration is proportional to displacement but in the opposite direction!</p> <p>However, we can verify that the condition for simple harmonic motion is obeyed by differentiating the original function for displacement twice with respect to time. </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>By analysing the motion of a pendulum we find the displacement is sinusoidally related to time. This means that the displacement equation with respect to time will contain either sin or cos.</p> <p>The velocity is found from the gradient of the displacement time graph. If, for example, the displacement is a <strong>sin</strong> curve then velocity will be a <strong>cos</strong> curve.</p> <p>Acceleration is the gradient of the velocity time graph. Sticking with velocity as a <strong>cos</strong> curve, acceleration will be a <strong>-sin</strong> curve. This is proportional to but in the opposition direction to the displacement time curve.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/168002851"></iframe></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> <p>Imagine watching circular motion from the side, with your eye on the same plane as the circle itself. You would see the object moving from side-to-side or up-and-down repeatedly. The equations of this motion are the same as simple harmonic motion.</p> <p>This can be used to show that <span class="math-tex">\(a=-\omega^2x\)</span>.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/168003487"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-purple"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p>The equation for the displacement of a body executing SHM is either:</p> <ul> <li><span class="math-tex">\(x = x_o\sin(2πft)\)</span></li> <li><span class="math-tex">\(x = x_o\cos(2πft)\)</span></li> </ul> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/168003967"></iframe></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 quizzes to practise application of theory.</i></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#f3130605"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="f3130605"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> SHM and the sin function <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="6-83-274" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The diagram represents 3 sin curves (horizontal axis is time)</p><p style="text-align: center;"><img alt="" src="../../images/shmq1.jpg" style="width: 217px; height: 127px;"></p><p>If the green line is displacement then the blue line is</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> velocity</label></p><p><label class="radio"><input type="radio"> -velocity</label></p><p><label class="radio"><input type="radio"> -acceleration</label></p><p><label class="radio"><input class="c" type="radio"> acceleration</label></p></div><div class="q-explanation"><p>Motion represented is SHM so a is proportional to -x</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram represents 3 sin curves (horizontal axis is time)</p><p style="text-align: center;"><img alt="" src="../../images/shmq1.jpg" style="width: 217px; height: 127px;"></p><p>If the green line is displacement the red line is</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> acceleration</label></p><p><label class="radio"><input type="radio"> velocity</label></p><p><label class="radio"><input class="c" type="radio"> -velocity</label></p><p><label class="radio"><input type="radio"> -acceleration</label></p></div><div class="q-explanation"><p>Velocity is the gradient of the dispacent- time graph. The red line is - gradient</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>This graph represents the displacement time graph for a body moving with SHM. The axis are in meters and seconds.</p><p style="text-align: center;"><img alt="" src="../../shmq2.jpg" style="width: 229px; height: 173px;"></p><p>The maximum velocity is approximately</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 19 ms-1</label></p><p><label class="radio"><input type="radio"> 0.7 ms-1</label></p><p><label class="radio"><input type="radio"> can´t be determined</label></p><p><label class="radio"><input type="radio"> 1.5 ms-1</label></p></div><div class="q-explanation"><p>The gradient gives the velocity. This is certainly bigger than 1.5</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A pendulum bob is released from the position shown</p><p style="text-align: center;"><img alt="" height="209" src="../../shmgraphs2.jpg" width="384"></p><p>Which graph represents the displacement</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> red</label></p><p><label class="radio"><input class="c" type="radio"> green</label></p><p><label class="radio"><input type="radio"> blue</label></p><p><label class="radio"><input type="radio"> black</label></p></div><div class="q-explanation"><p>left is negative</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A pendulum bob is released from the position shown</p><p style="text-align: center;"><img alt="" height="211" src="../../shmgraphs2.jpg" width="387"></p><p>Which graph represents the velocity</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> black</label></p><p><label class="radio"><input type="radio"> green</label></p><p><label class="radio"><input type="radio"> red</label></p><p><label class="radio"><input class="c" type="radio"> blue</label></p></div><div class="q-explanation"><p>Starts from zero moving to the right (+)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A pendulum bob is released from the position shown</p><p style="text-align: center;"><img alt="" src="../../shmgraphs2.jpg" style="width: 385px; height: 210px;"></p><p>Which graph represents the acceleration</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> green</label></p><p><label class="radio"><input class="c" type="radio"> black</label></p><p><label class="radio"><input type="radio"> blue</label></p><p><label class="radio"><input type="radio"> red</label></p></div><div class="q-explanation"><p>Starts max and positive direction</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the equation of this line</p><p style="text-align: center;"><img alt="" height="193" src="../../siewaveq4.jpg" width="267"></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> y = -sin θ</label></p><p><label class="radio"><input type="radio"> y = tan θ</label></p><p><label class="radio"><input type="radio"> y = cos θ</label></p><p><label class="radio"><input type="radio"> y = sin θ</label></p></div><div class="q-explanation"><p>just have to know</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The equation for the displacment of a body executing SHM is of the form</p><p style="text-align: center;">y = A sin (2πBt)</p><p>B in this equation is</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> wavelength</label></p><p><label class="radio"><input type="radio"> time period</label></p><p><label class="radio"><input type="radio"> amplitude</label></p><p><label class="radio"><input class="c" type="radio"> frequency</label></p></div><div class="q-explanation"><p>y = a sin (2πft)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The greph represents two oscillations</p><p style="text-align: center;"><img alt="" src="../../shmsine5.jpg" style="width: 312px; height: 218px;"></p><p>The equation of the black line is sin(2πft). The equation of the red line is.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> sin (2πft - π)</label></p><p><label class="radio"><input type="radio"> sin (2πft +π)</label></p><p><label class="radio"><input type="radio"> sin (2πft - π/2)</label></p><p><label class="radio"><input class="c" type="radio"> sin (2πft + π/2)</label></p></div><div class="q-explanation"><p>The red line is π/2 in front of the black.</p><p>2π is one cycle so π/2 is 1/4 of a cycle</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below represents a ball moving anticlockwise in a circle at a constant speed .</p><p style="text-align: center;"><img alt="" src="../../circleshmq1.jpg" style="width: 414px; height: 203px;"></p><p>Which line represents the horizontal displacement if the clock is started when the ball is at the position shown.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> black</label></p><p><label class="radio"><input type="radio"> blue</label></p><p><label class="radio"><input type="radio"> green</label></p><p><label class="radio"><input class="c" type="radio"> red</label></p></div><div class="q-explanation"><p>Strats with zero horizontal displacement and goes left of centre.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below represents a ball moving anticlockwise in a circle at constant speed</p><p style="text-align: center;"><img alt="" src="../../circleshmq2.jpg" style="width: 384px; height: 179px;"></p><p>If the graph repreents the vertical component of the motion where was the ball when time = 0s</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> D</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p></div><div class="q-explanation"><p>Starts with zero vertical displacement and goes down.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success 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