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href="../272/oscillations.html">Oscillations</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Mass on a spring</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 5 minutes"><i class="fa fa-clock-o"></i> 5'</span> </ol> <article id="main-article"> <p><img alt="" src="../../waves/1-mass-spring.jpg" style="float: left; width: 250px; height: 166px;">Pendula are not the only oscillating system. Another approximation for many systems is a mass hanging on a spring. Although the forces here are a bit easier to analyse, we do need to make the assumption that the mass and spring system is in space!</p> <p>This is another example of simple harmonic motion, as the acceleration is proportional to displacement and acting in the opposite direction. </p> <hr class="hidden-separator"> <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>We know that, for springs obeying Hooke's law, the force is proportional to extension. For a spring in space, the elastic force is the resultant force. The acceleration of a mass hanging from a spring is therefore proportional to its displacement from the equilibrium position.</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/168001850"></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="#61d2ce5d"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="61d2ce5d"> <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%;"> Mass on a spring <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="6-81-276" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Fill in the missing words to explain why a mass on a spring executes SHM</p><p>first, second, third, Hooke´s, extension, displacement</p></div><div class="q-answer"><p>When a spring is extended it obeys <input type="text" style="height: auto;" data-c="Hooke´s"> <span class="review"></span> law, this means that the restoring force is proportional to the <input type="text" style="height: auto;" data-c="extension"> <span class="review"></span> . According to Newtons <input type="text" style="height: auto;" data-c="second"> <span class="review"></span> law the acceleration of a mass attached to the spring will therefore be proportional to the <input type="text" style="height: auto;" data-c="displacement"> <span class="review"></span> from the equilibrium position, which complies with the definition of SHM.</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 spring extends 2cm when a mass of 100 g is hung from it.</p><p>What is the spring constant of the spring?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 50 Nm<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 0.02 Nm<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 2 Nm<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 5 Nm<sup>-1</sup></label></p></div><div class="q-explanation"><p>k = F/x = 1/0.2</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 spring extends 2cm when a mass of 100 g is hung from it.</p><p>What will the extension be with a mass of 0.3 kg</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 60 cm</label></p><p><label class="radio"><input type="radio"> 3 cm</label></p><p><label class="radio"><input type="radio"> 30 cm</label></p><p><label class="radio"><input class="c" type="radio"> 6 cm</label></p></div><div class="q-explanation"><p>k is proportional to force.</p><p>0.3 kg will exert 3x the force so 3x extension</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 graph represents the force against extension for a rubber band</p><p style="text-align: center;"><img alt="" src="../../images/rubber.jpg" style="width: 328px; height: 323px;"></p><p>A mass hanging on this sample of rubber:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> will execute SHM for oscillations where the extension is between 8 cm and 10 cm</label></p><p><label class="radio"><input type="radio"> will not execute SHM</label></p><p><label class="radio"><input type="radio"> will execute SHM for oscillations where the extension is between 4 cm and 6 cm</label></p><p><label class="radio"><input class="c" type="radio"> will execute SHM for oscillations where the extension is between 0 cm and 1 cm</label></p></div><div class="q-explanation"><p>Between 0 and 1 cm the force is approximately proportional to extension. The graph is approximately linear in the middle region but F = kx +c not kx</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 spring showing the unloaded length, the length when a 100g mass is hanging at rest and the length when the mass is pulled down.</p><p style="text-align: center;"><img alt="" src="../../images/extension.jpg" style="width: 350px; height: 346px;"></p><p>The spring constant is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 50 Nm<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 0.1 Nm<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 5 Nm<sup>-1</sup></label></p><p><label class="radio"><input class="c" type="radio"> 10 Nm<sup>-1</sup></label></p></div><div class="q-explanation"><p>K= F/x =1/0.1</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 spring showing the unloaded length, the length when a 100g mass is hanging at rest and the length when the mass is pulled down.</p><p style="text-align: center;"><img alt="" src="../../images/extension.jpg" style="width: 350px; height: 346px;"></p><p>The Extension of the spring in the equilibrium position is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 20 cm</label></p><p><label class="radio"><input type="radio"> 50 cm</label></p><p><label class="radio"><input type="radio"> 40cm</label></p><p><label class="radio"><input class="c" type="radio"> 10 cm</label></p></div><div class="q-explanation"><p>The middle spring is in the equilibrium position. The extension is from the original length to the new length.</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 spring showing the unloaded length, the length when a 100g mass is hanging at rest and the length when the mass is pulled down.</p><p style="text-align: center;"><img alt="" src="../../images/extension.jpg" style="width: 350px; height: 346px;"></p><p>If the mass is released it will undergo SHM. The ammplitude of the oscillation is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 20 cm</label></p><p><label class="radio"><input type="radio"> 20 cm</label></p><p><label class="radio"><input type="radio"> 5 cm</label></p><p><label class="radio"><input class="c" type="radio"> 10 cm</label></p></div><div class="q-explanation"><p>Amplitude is from the maximum displacement to the equilibrium position</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 spring showing the unloaded length, the length when a 100g mass is hanging at rest and the length when the mass is pulled down.</p><p style="text-align: center;"><img alt="" src="../../images/extension.jpg" style="width: 350px; height: 346px;"></p><p>The mass is replaced by a 50g mass then it is pulled down to the same position. The amplitude of the oscillation will be:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 15 cm</label></p><p><label class="radio"><input type="radio"> 20 cm</label></p><p><label class="radio"><input type="radio"> 30 cm</label></p><p><label class="radio"><input type="radio"> 10 cm</label></p></div><div class="q-explanation"><p>The equilibrium position will be at an extension of 5 cm so when pulled down the displacement will be 15 cm</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 spring showing the unloaded length the length when a 100g mass is hanging at rest and the length when the mass is pulled down.</p><p style="text-align: center;"><img alt="" src="../../images/extension.jpg" style="width: 350px; height: 346px;"></p><p>The mass is replaced by a 50 g mass and it is pulled down to the same position. What will be the effect on the amplitude and frequency?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Amplitude larger, frequency higher</label></p><p><label class="radio"><input type="radio"> Amplitude the same, frequency higher</label></p><p><label class="radio"><input type="radio"> Amplitude smaller, frequency higher</label></p><p><label class="radio"><input type="radio"> Amplitude smaller, frequency lower</label></p></div><div class="q-explanation"><p>The equilibrium position will be at an extension of 5 cm so the amplitude will be bigger. The frequency is inversely proportional to √m</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 spring showing the unloaded length, the length when a 100g mass is hanging at rest and the length when the mass is pulled down.</p><p style="text-align: center;"><img alt="" src="../../images/extension.jpg" style="width: 350px; height: 346px;"></p><p>The time period of the oscillation is given by the equation <span class="tib-mathml"><math display="block"> <semantics> <mrow> <mi>T</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt> <mrow> <mfrac> <mi>m</mi> <mi>k</mi> </mfrac> </mrow> </msqrt> </mrow> <annotation encoding="MathType-MTEF">MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2da9iaaikdacqaHapaCdaGcaaqaamaalaaabaGaamyBaaqaaiaadUgaaaaaleqaaaaa@347E@</annotation> </semantics></math></span> The time period is:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 0.2 π</label></p><p><label class="radio"><input type="radio"> 20 π</label></p><p><label class="radio"><input type="radio"> 0.02 π</label></p><p><label class="radio"><input type="radio"> 2 π</label></p></div><div class="q-explanation"><p>k = 10 Nm<sup>-1</sup><br>m = 0.1 kg<br><span class="tib-mathml"><math display="block"> </math></span> <semantics><mrow><mi>T</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt> <mrow> <mfrac> <mrow> <mn>0.1</mn></mrow> <mrow> <mn>10</mn></mrow> </mfrac> </mrow> </msqrt> <mo>=</mo><mn>0.1</mn><mo>×</mo><mn>2</mn><mi>π</mi></mrow> <annotation encoding="MathType-MTEF">MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2da9iaaikdacqaHapaCdaGcaaqaamaalaaabaGaaGimaiaac6cacaaIXaaabaGaaGymaiaaicdaaaaaleqaaOGaeyypa0JaaGimaiaac6cacaaIXaGaey41aqRaaGOmaiabec8aWbaa@3DFF@</annotation> </semantics></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"> 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