1Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121, April 8, 2008.Harmonic Motion.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.April 8, 2008.• Course Information• Topics to be discussed today:• Simple Harmonic Motion (Review).• Simple Harmonic Motion: Example Systems.• Damped Harmonic Motion• Driven Harmonic MotionFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.April 8, 2008.• Homework set # 8 is due on Saturday morning, April 12, at8.30 am.• Homework set # 9 will be available on Saturday morning at8.30 am, and will be due on Saturday morning, April 19, at8.30 am.• Requests for regarding part of Exam # 1 and # 2 need to begiven to me by April 17. You need to write down what Ishould look at and give me your written request and yourblue exam booklet(s).2Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterHarmonic motion (a quick review).Motion that repeats itself at regular intervals.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (a quick review). x(t) = xmcos(! t + ")AmplitudePhase ConstantAngular FrequencyFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (a quick review).• Other variables frequently used todescribe simple harmonicmotion:• The period T: the time required tocomplete one oscillation. Theperiod T is equal to 2π/ω.• The frequency of the oscillation isthe number of oscillations carriedout per second:ν = 1/TThe unit of frequency is the Hertz(Hz). Per definition, 1 Hz = 1 s-1.3Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (a quick review).What forces are required?• Using Newton’s second law we can determine the forceresponsible for the harmonic motion:F = ma = -mω2x• The total mechanical energy of a system carrying out simpleharmonic motion is constant.• A good example of a force that produces simple harmonicmotion is the spring force: F = -kx. The angular frequencydepends on both the spring constant k and the mass m:ω = √( k/m)Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM).The torsion pendulum.• What is the angular frequency ofthe SHM of a torsion pendulum:• When the base is rotated, it twiststhe wire and a the wire generateda torque which is proportional tothe the angular twist:τ = -KθThe torque generates an angularacceleration α:α = d2θ/dt2 = τ/I = -(K/I) θThe resulting motion is harmonicmotion with an angular frequencyω = √(K/I).Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM). The simple pendulum.• Calculate the angular frequencyof the SHM of a simplependulum.• A simple pendulum is a pendulumfor which all the mass is located ata single point at the end of amassless string.• There are two forces acting on themass: the tension T and thegravitational force mg.• The tension T cancels the radialcomponent of the gravitationalforce.4Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM). The simple pendulum.• The net force acting on he mass isdirected perpendicular to thestring and is equal toF = - mg sinθThe minus sign indicates that theforce is directed opposite to theangular displacement.• When the angle θ is small, we canapproximate sinθ by θ:F = - mgθ = - mg x/L• Note: the force is againproportional to the displacement.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM). The Simple Pendulum.• The equation of motion for thependulum is thusF = m d2x/dt2 = -(mg/L) xor d2x/dt2 = - (g/L) x• The equation of motion is thesame as the equation of motionfor a SHM, and the pendulum willthus carry out SHM with anangular frequency ω = √( g/L).• The period of the pendulum isthus 2π/ω = 2π √( L/g). Note: theperiod is independent of the massof the pendulum.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM).The physical pendulum.• In a realistic pendulum, not allmass is located at a single point.• The motion carried out by thisrealistic pendulum around itsrotation point O can bedetermined by determining thetotal torque with respect to thispoint:• If the angle θ is small, we canapproximate the torque by!= "mgh sin#!= "mgh#5Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM).The physical pendulum.• The angular acceleration α isrelated to the torque:• The equation of motion for theangular acceleration α is givenby• This again is an equation forSHM with an angular frequencyω where!= I"!=d2"dt2=#I= $mghI"!2=mghIFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM).The physical pendulum.• The period of the physicalpendulum is equal to• We can double check our answerby requiring that the simplependulum is a special case of thephysical pendulum (h = L,I#=#mL2):T =2!"= 2!ImghT = 2!Imgh= 2!mL2mgL= 2!LgFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.Quiz lecture 21.• The quiz today will have 3 questions!6Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM).The equation of motion.• All examples of SHM were derived from he followingequation of motion:• The most general solution to the equation isd2xdt2= !"2xx t( )= A cos!t +"( )+ B sin!t +#( )Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterSimple Harmonic Motion (SHM).The equation of motion.• If A = Bwhich is SHM.x t( )= A cos!t +"( )+ B sin!t +#( )== A sin12$%!t %"&'()*++ sin!t +#( )&'()*+== 2 A sin14$+#2%"2&'()*+cos14$%!t %#2%"2&'()*+Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterDamped Harmonic Motion.• Consider what happens when in addition to the restoringforce a damping force (such as the drag force) is acting onthe system:• The equation of motion is now