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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="../443/circular-motion-and-gravity.html">Circular motion and gravity</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../264/circular-motion.html">Circular motion</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Centripetal force</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="../../circular-motion/6-skate.jpg" style="float: left; width: 250px; height: 166px;">If a body moves in a circle there must be a resultant force acting towards the centre. This is because the change in direction implies that the body is accelerating.</p> <p>Deriving the equation is not so simple though... </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-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>Perpendicular force</p> </div> </div> <div class="panel-body"> <div> <p>A force acting at right angles to the direction of motion causes the body to travel in a circular path.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>We can model this using a ramp made from a circular cut-out in which the normal reaction force is perpendicular to the velocity. When the force stops, the object moves along the <em>tangent</em> to the circle.</p> <p style="text-align: center;"><img alt="" height="213" class="gifffer" data-gifffer="/media/physics/networks-2/perpendicularf.gif" width="383"></p> </section> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Here is another example:</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/164056736"></iframe></div> </section> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Relationship between ω and v</p> </div> </div> <div class="panel-body"> <div> <p>ω is the angular velocity = <span class="math-tex">\(2\pi \over T\)</span><br> v is the speed = <span class="math-tex">\(2\pi r \over T\)</span></p> <p>From this we can deduce that <span class="math-tex">\(v = ωr\)</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/164057169"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </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>When deriving the equation for centripetal acceleration we start by considering a small part of the motion we can deduce that</p> <p style="text-align: center;"><span class="math-tex">\(a = {mv^2\over r}\)</span></p> <p>This is the acceleration.</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/164057680"></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="#2044e8d0"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="2044e8d0"> <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%;"> Circular motion 1 <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="6-67-265" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Fill in the blanks of this justification that the force causing a body to move around a circle at constant speed must be towards the centre.</p><p>Word list: 1st, 2nd 3rd, direction , perpendicular, parallel, force, tangential, radial, kinetic, potential, work, acceleration</p></div><div class="q-answer"><p>When a body travels in a circle its <input type="text" style="height: auto;" data-c=" direction "> <span class="review"></span> is always changing so it must have <input type="text" style="height: auto;" data-c=" acceleration "> <span class="review"></span> . According to Newtons <input type="text" style="height: auto;" data-c=" 1st "> <span class="review"></span> law there must be an unbalanced force acting on it.</p><p>If the speed is constant then the <input type="text" style="height: auto;" data-c=" kinetic "> <span class="review"></span> energy must be constant which means no <input type="text" style="height: auto;" data-c=" work "> <span class="review"></span> is being done on the body.</p><p> <input type="text" style="height: auto;" data-c=" work "> <span class="review"></span> done = <input type="text" style="height: auto;" data-c=" force "> <span class="review"></span> x distance moved in direction of <input type="text" style="height: auto;" data-c="force"> <span class="review"></span> so if no work is done the <input type="text" style="height: auto;" data-c="force"> <span class="review"></span> must be <input type="text" style="height: auto;" data-c=" perpendicular "> <span class="review"></span> to the direction of motion, i.e. <input type="text" style="height: auto;" data-c=" radial"> <span class="review"></span> .</p></div><div class="q-explanation"></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 mass on a string completes 3 revolutions in 2 seconds.</p><p>The angular velocity in radians s<sup>-1</sup> is:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 3π</label></p><p><label class="radio"><input type="radio"> 6π</label></p><p><label class="radio"><input type="radio"> π</label></p><p><label class="radio"><input type="radio"> 2π</label></p></div><div class="q-explanation"><p>3 revs is 6π rads so ω = 6π/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 mass travels in a circle on the end of a spring at constant speed</p><p style="text-align: center;"><img alt="" src="../../images/circq1.png" style="width: 297px; height: 225px;"></p><p>Which of the following statements is true</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> the spring is pulled out by the centripetal force</label></p><p><label class="radio"><input type="radio"> the mass does work on the spring</label></p><p><label class="radio"><input type="radio"> the mass travels with constant velocity</label></p><p><label class="radio"><input class="c" type="radio"> the forces on the ball are unbalanced</label></p></div><div class="q-explanation"><p>centripetal force is the name given to the force that acts towards the centre so doesn't pull the spring out, however the spring pulls the ball in so according to newt 3 the ball must pull the spring out.</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 ball moves in a circle of radius 2 m with constant speed 2 ms<sup>-1</sup>.</p><p>The time to complete 1 revolution is:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 2π s</label></p><p><label class="radio"><input type="radio"> 2 s</label></p><p><label class="radio"><input type="radio"> 1 s</label></p><p><label class="radio"><input type="radio"> π s</label></p></div><div class="q-explanation"><p>distance travelled in 1 revolution = 2πr</p><p>t = d/v = 2π x 2/2</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>10° in radians is about</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 1 rad</label></p><p><label class="radio"><input type="radio"> 10 rad</label></p><p><label class="radio"><input type="radio"> 10π rad</label></p><p><label class="radio"><input class="c" type="radio"> 0.2 rad</label></p></div><div class="q-explanation"><p>10/360 x 2π approx 6/36 = 1/6</p><p>process of elimination, all the others are way too big</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 ball on a string moves in a circle as shown</p><p style="text-align: center;"><img alt="" src="../../images/circq2.png" style="width: 200px; height: 178px;"></p><p>The magnitude of change in velocity is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 10ms<sup>-1</sup></label></p><p><label class="radio"><input class="c" type="radio"> √50 ms<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 0 ms<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> 50 ms<sup>-1</sup></label></p></div><div class="q-explanation"><p>Must calculate change vectorially so use Pythagoras.</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 mass travels in a circle of radius 2 m with constant speed 3 ms<sup>-1</sup>.</p><p>The acceleration of the mass is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 18 ms<sup>-2</sup></label></p><p><label class="radio"><input class="c" type="radio"> 4.5 ms<sup>-2</sup></label></p><p><label class="radio"><input type="radio"> 1.5 ms<sup>-2</sup></label></p><p><label class="radio"><input type="radio"> 0 ms<sup>-2</sup></label></p></div><div class="q-explanation"><p>a = v<sup>2</sup>/r</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>If a = v<sup>2</sup>/r and v = ωr then:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> a = r<sup>2</sup>/ω</label></p><p><label class="radio"><input class="c" type="radio"> a = ω<sup>2</sup>r</label></p><p><label class="radio"><input type="radio"> a = ω/r<sup>2</sup></label></p><p><label class="radio"><input type="radio"> a = ω<sup>2</sup>/r</label></p></div><div class="q-explanation"><p>a = ω<sup>2</sup>r<sup>2</sup>/r =ω<sup>2</sup>r<label class="radio"></label></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 2 kg mass travels in a circle of radius 5m with time period π s.</p></div><div class="q-answer"><p>Angular velocity = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> rads<sup>-1</sup></p><p>Centripetal acceleration = <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> rads<sup>-2</sup></p><p>Centripetal force = <input type="text" style="height: auto;" data-c="40"> <span class="review"></span> N</p></div><div class="q-explanation"><p>ω = 2π/T</p><p>a = ω<sup>2</sup>r</p><p>F = ma</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 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