Section 1 Simple Harmonic Chapter 11 Motion
Chapter 11Section 1 Simple HarmonicMotionPreview Objectives Hooke’s Law Sample Problem Simple Harmonic Motion The Simple Pendulum Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionObjectives Identify the conditions of simple harmonic motion. Explain how force, velocity, and acceleration changeas an object vibrates with simple harmonic motion. Calculate the spring force using Hooke’s law. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionHooke’s Law One type of periodicmotion is the motion ofa mass attached to aspring. The direction of theforce acting on themass (Felastic) is alwaysopposite the directionof the mass’sdisplacement fromequilibrium (x 0). Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionHooke’s Law, continuedAt equilibrium: The spring force and the mass’s accelerationbecome zero. The speed reaches a maximum.At maximum displacement: The spring force and the mass’s acceleration reacha maximum. The speed becomes zero. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionHooke’s Law, continued Measurements show that the spring force, orrestoring force, is directly proportional to thedisplacement of the mass. This relationship is known as Hooke’s Law:Felastic –kxspring force –(spring constant displacement) The quantity k is a positive constant called thespring constant. Houghton Mifflin Harcourt Publishing Company
Section 1 Simple HarmonicMotionChapter 11Spring ConstantClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionSample ProblemHooke’s LawIf a mass of 0.55 kg attached to a vertical springstretches the spring 2.0 cm from its original equilibriumposition, what is the spring constant? Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionSample Problem, continued1. DefineGiven:m 0.55 kgx –2.0 cm –0.02 mg 9.81 m/s2Unknown:k ? Houghton Mifflin Harcourt Publishing CompanyDiagram:
Chapter 11Section 1 Simple HarmonicMotionSample Problem, continued2. PlanChoose an equation or situation: When the massis attached to the spring,the equilibrium positionchanges. At the new equilibrium position, the netforce acting on the mass is zero. So the spring force(given by Hooke’s law) must be equal and oppositeto the weight of the mass.Fnet 0 Felastic FgFelastic –kxFg –mg–kx – mg 0 Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionSample Problem, continued2. Plan, continuedRearrange the equation to isolate the unknown:kx mg 0kx mgmgk x Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionSample Problem, continued3. CalculateSubstitute the values into the equation andsolve:mg(0.55 kg)(9.81 m/s2 )k x–0.020 mk 270 N/m4. EvaluateThe value of k implies that 270 N of force isrequired to displace the spring 1 m. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionSimple Harmonic Motion The motion of a vibrating mass-spring system is anexample of simple harmonic motion. Simple harmonic motion describes any periodicmotion that is the result of a restoring force that isproportional to displacement. Because simple harmonic motion involves a restoringforce, every simple harmonic motion is a backand-forth motion over the same path. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionSimple Harmonic MotionClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionForce and Energy in Simple Harmonic MotionClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionThe Simple Pendulum A simple pendulum consists ofa mass called a bob, which isattached to a fixed string. At any displacement fromequilibrium, the weight of thebob (Fg) can be resolved intotwo components. The x component (Fg,x Fg sin ) is the only force acting on thebob in the direction of its motionand thus is the restoring force. Houghton Mifflin Harcourt Publishing CompanyThe forces acting on thebob at any point are theforce exerted by thestring and thegravitational force.
Chapter 11Section 1 Simple HarmonicMotionThe Simple Pendulum, continued The magnitude of the restoring force(Fg,x Fg sin ) is proportional to sin . When the maximum angle ofdisplacement is relatively small( 15 ), sin is approximately equal to in radians. As a result, the restoring force is very nearlyproportional to the displacement. Thus, the pendulum’s motion is an excellentapproximation of simple harmonic motion. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionRestoring Force and Simple PendulumsClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 1 Simple HarmonicMotionSimple Harmonic Motion Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionPreview Objectives Amplitude, Period, and Frequency in SHM Period of a Simple Pendulum in SHM Period of a Mass-Spring System in SHM Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionObjectives Identify the amplitude of vibration. Recognize the relationship between period andfrequency. Calculate the period and frequency of an objectvibrating with simple harmonic motion. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionAmplitude, Period, and Frequency in SHM In SHM, the maximum displacement from equilibriumis defined as the amplitude of the vibration.– A pendulum’s amplitude can be measured by the anglebetween the pendulum’s equilibrium position and itsmaximum displacement.– For a mass-spring system, the amplitude is the maximumamount the spring is stretched or compressed from itsequilibrium position. The SI units of amplitude are the radian (rad) andthe meter (m). Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionAmplitude, Period, and Frequency in SHM The period (T) is the time that it takes a completecycle to occur.– The SI unit of period is seconds (s). The frequency (f) is the number of cycles orvibrations per unit of time.– The SI unit of frequency is hertz (Hz).– Hz s–1 Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionAmplitude, Period, and Frequency in SHM,continued Period and frequency are inversely related:11f or T Tf Thus, any time you have a value for period orfrequency, you can calculate the other value. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionMeasures of Simple Harmonic MotionClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionMeasures of Simple Harmonic Motion Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionPeriod of a Simple Pendulum in SHM The period of a simple pendulum depends on thelength and on the free-fall acceleration.LT 2 aglengthperiod 2 free-fall acceleration The period does not depend on the mass of the bobor on the amplitude (for small angles). Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 2 Measuring SimpleHarmonic MotionPeriod of a Mass-Spring System in SHM The period of an ideal mass-spring systemdepends on the mass and on the spring constant.mT 2 kmassperiod 2 spring constant The period does not depend on the amplitude. This equation applies only for systems in which thespring obeys Hooke’s law. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesPreview Objectives Wave Motion Wave Types Period, Frequency, and Wave Speed Waves and Energy Transfer Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesObjectives Distinguish local particle vibrations from overallwave motion. Differentiate between pulse waves and periodicwaves. Interpret waveforms of transverse and longitudinalwaves. Apply the relationship among wave speed,frequency, and wavelength to solve problems. Relate energy and amplitude. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesWave Motion A wave is the motion of a disturbance. A medium is a physical environment through which adisturbance can travel. For example, water is themedium for ripple waves in a pond. Waves that require a medium through which to travelare called mechanical waves. Water waves andsound waves are mechanical waves. Electromagnetic waves such as visible light do notrequire a medium. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesWave Types A wave that consists of a single traveling pulse iscalled a pulse wave. Whenever the source of a wave’s motion is a periodicmotion, such as the motion of your hand moving upand down repeatedly, a periodic wave is produced. A wave whose source vibrates with simple harmonicmotion is called a sine wave. Thus, a sine wave is aspecial case of a periodic wave in which the periodicmotion is simple harmonic. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesRelationship Between SHM and WaveMotionAs the sine wave created by this vibrating blade travels to theright, a single point on the string vibrates up and down withsimple harmonic motion. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesWave Types, continued A transverse wave is a wave whose particles vibrateperpendicularly to the direction of the wave motion. The crest is the highest point above the equilibrium position,and the trough is the lowest point below the equilibriumposition. The wavelength (l) is the distance between two adjacentsimilar points of a wave. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesTransverse WavesClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesWave Types, continued A longitudinal wave is a wave whose particles vibrate parallelto the direction the wave is traveling. A longitudinal wave on a spring at some instant t can berepresented by a graph. The crests correspond to compressedregions, and the troughs correspond to stretched regions. The crests are regions of high density and pressure (relativeto the equilibrium density or pressure of the medium), and thetroughs are regions of low density and pressure. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesLongitudinal WavesClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesPeriod, Frequency, and Wave Speed The frequency of a wave describes the number ofwaves that pass a given point in a unit of time. The period of a wave describes the time it takes fora complete wavelength to pass a given point. The relationship between period and frequency inSHM holds true for waves as well; the period of awave is inversely related to its frequency. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesCharacteristics of a WaveClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesPeriod, Frequency, and Wave Speed, continued The speed of a mechanical wave is constant forany given medium. The speed of a wave is given by the followingequation:v flwave speed frequency wavelength This equation applies to both mechanical andelectromagnetic waves. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 3 Properties of WavesWaves and Energy Transfer Waves transfer energy by the vibration of matter. Waves are often able to transport energy efficiently. The rate at which a wave transfers energy dependson the amplitude.– The greater the amplitude, the more energy awave carries in a given time interval.– For a mechanical wave, the energy transferred isproportional to the square of the wave’s amplitude. The amplitude of a wave gradually diminishes overtime as its energy is dissipated. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsPreview Objectives Wave Interference Reflection Standing Waves Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsObjectives Apply the superposition principle. Differentiate between constructive and destructiveinterference. Predict when a reflected wave will be inverted. Predict whether specific traveling waves will producea standing wave. Identify nodes and antinodes of a standing wave. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsWave Interference Two different material objects can never occupy thesame space at the same time. Because mechanical waves are not matter but ratherare displacements of matter, two waves can occupythe same space at the same time. The combination of two overlapping waves is calledsuperposition. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsWave Interference, continuedIn constructive interference, individual displacementson the same side of the equilibrium position are addedtogether to form the resultant wave. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsWave Interference, continuedIn destructive interference, individual displacementson opposite sides of the equilibrium position are addedtogether to form the resultant wave. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsComparing Constructive and DestructiveInterferenceClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsReflection What happens tothe motion of awave when itreaches aboundary? At a freeboundary, wavesare reflected. At a fixedboundary, wavesare reflected andFree boundaryinverted. Houghton Mifflin Harcourt Publishing CompanyFixed boundary
Section 4 Wave InteractionsChapter 11Standing WavesClick below to watch the Visual Concept.Visual Concept Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsStanding Waves A standing wave is a wave pattern that results whentwo waves of the same frequency, wavelength, andamplitude travel in opposite directions and interfere. Standing waves have nodes and antinodes.– A node is a point in a standing wave that maintainszero displacement.– An antinode is a point in a standing wave, halfwaybetween two nodes, at which the largestdisplacement occurs. Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsStanding Waves, continued Only certain wavelengthsproduce standing wave patterns. The ends of the string must benodes because these pointscannot vibrate. A standing wave can be producedfor any wavelength that allowsboth ends to be nodes. In the diagram, possiblewavelengths include 2L (b), L (c),and 2/3L (d). Houghton Mifflin Harcourt Publishing Company
Chapter 11Section 4 Wave InteractionsStanding WavesThis photographshows fourpossible standingwaves that canexist on a givenstring. Thediagram showsthe progressionof the secondstanding wavefor one-half of acycle. Houghton Mifflin Harcourt Publishing Company
Simple Harmonic Motion The motion of a vibrating mass-spring system is an example of simple harmonic motion. Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement. Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
Lesson 14: Simple harmonic motion, Waves (Sections 10.6-11.9) Lesson 14, page 1 Circular Motion and Simple Harmonic Motion The projection of uniform circular motion along any axis (the x-axis here) is the same as simple harmonic motion. We use our understanding of uniform circular motion to arrive at the equations of simple harmonic motion.
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
Chapter 11: Harmonic Motion 170 Learning Goals In this chapter, you will: DLearn about harmonic motion and how it is fundamental to understanding natural processes. DUse harmonic motion to keep accurate time using a pendulum. DLearn how to interpret and make graphs of harmonic motion. DConstruct simple oscillators.
Simple Harmonic Motion, SHM Simple harmonic motion . Simple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F - kx. A .
Simple harmonic motion (SHM) Simple Harmonic Oscillator (SHO) When the restoring force is directly proportional to the displacement from equilibrium, the resulting motion is called simple harmonic motion (SHM). An ideal spring obeys Hooke’s law, so the restoring force is F x –kx, which results in simple harmonic motion.
SIMPLE PENDULUM AND PROPERTIES OF SIMPLE HARMONIC MOTION Purpose a. To investigate the dependence of time period of a simple pendulum on the length of the pendulum and the acceleration of gravity. b. To study properties of simple harmonic motion. Theory A simple pendulum is a small object that is suspended at the end of a string.
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