Lecture5 Simple Harmonic Motion

A P P h y s i c s B–C o u r s e S y l l a b u sA. COURSE OVERVIEWAdvance Placement Physics B, the third course in the accelerated science program, is designed for the student who has advanced skills in math and science and intends to pursue a post-secondary education in the fields of Science, Pre-Medical, Engineering or Mathematics. This is a first-year course in physics. Topics covered include mechanics, electricity and magnetism, sound and light. The student should be concurrently enrolled in Honors Precalculus and have the approval of the Science department. Evaluation is based upon homework, tests, quizzes, laboratory work, midyear and final exams.B. METHOD OF INSTRUCTIONClass meetings will generally take three common forms, lab/activity, interactive lecture discussions, or problem solving/review. The design as such will allow students to experience and engage the subject conceptually, actively, and analytically. Individual classes may contain multiple elements of these models to suit the topic. Classes meet each weekday for 47 minutes. Every fourth day will be a double length period allowing for longer labs/activities.Lab activities will be of two varieties: investigation or application. Investigation labs and activities will allow students to do just that – investigate a physical phenomenon, and draw conclusions from their measurements and observations. Investigation labs or activities may take place before any reading, or formal in-class discussion on the topic has begun in order to allow students to explore the subject and discover the principles via their own inquiry and collaborative group effort. Much of the course content will be initially discovered using this “workshop physics” approach.Application labs and activities will provide students the opportunity to conduct experiments that involve the concepts they are studying as well as apply understanding of physics to solve practical problems. These labs will frequently be open-ended or contain an open-ended component challenging students to solve a problem by utilizing both their understanding of the topic as well as their critical thinking skills. Individual labs may contain both application and investigation elements. Nearly all units will involve some hands-on lab component. Some activites will consist of a self-contained packet, while others will require the student take their own notes and write their own procedure, observations, data, conclusions etc. There will be at least one formal lab report per quarter. All lab materials are to be kept in a notebook for reference.Interactive lecture discussions will contain elements of a traditional lecture, where concepts are formally presented to students and problem solving is modeled. However, these sessions should also lead to a conversation between students and instructor where the observations from investigations are considered and generalized as well as considering students experience of the concepts from their lives and their interests. Classes will often begin with a starter exercise, which may be a problem or a demonstration of a discrepant event may be presented, and students will be asked to come up with a written explanation. Problem solving and review sessions may involve problems solving strategy and concepts to be reviewed by the class as a whole, or smaller group workshop sessions enabling peer interactive learning, facilitated by the instructor.C. COURSE OBJECTIVES1. To utilize real-world experience to understand physical phenomena2. To utilize controlled laboratory experience to understand physical phenomena3. To gain an understanding of the workings of our physical world and be able to express that understanding interms of:a) written/spoken languageb) graphical diagramsc) mathematical analysis4. To develop observational problem solving and critical thinking skills that will benefit you for any vocationD. TEXTBOOKS AND SOFTWAREPrimary Textbook: James S. Walker, Physics, AP* Edition, 3rd ed., Prentice Hall, Upper Saddle River New Jersey, 2007.Secondary Textbook: Douglas C. Giancoli, Physics – Principles with Applications 5th ed., Prentice Hall, Upper Saddle River New Jersey, 1998.Data Collection/Analysis Software: Logger Pro, Vernier SoftwareE. COURSE CONTENT AREAS0. The Study of Physics — Chapter 1A. Scientific Method and PhilosophyB. Measurement and MathematicsI. Newtonian mechanicsA. Kinematics1. Motion in one dimension — Chapter 22. Uses of Vectors — Chapter 33. Motion in two dimensions — Chapter 4B. Newton’s laws of motion — Chapters 5 & 61. Static equilibrium (1st law)2. Dynamics of a single particle (2nd law)3. Systems of two or more bodies (3rd law)4. Uniform Circular MotionC. Work, energy and power — Chapters 7 & 81. Work and the work-energy theorem2. Power3. Conservative forces and potential energy4. Conservation of energyD. Systems of particles, linear momentum — Chapter 91. Impulse and momentum2. Conservation of linear momentum, collisions3. Center of MassF. Circular Motion and Rotation — Chapters 10 & 111. Angular position, velocity, and acceleration2. Torque and rotational statics3. Rotational kinematics and dynamics4. Angular momentumE. Gravitation — Chapter 121. Newton’s law of gravity2. Orbits of planets and satellitesa. Circularb. GeneralII. Oscillations, Waves and SoundA. Oscillations about equilibrium — Chapter 133. Simple harmonic motion (dynamics and energy relationships)4. Mass on a spring5. Pendulum and other oscillationsB. Wave motion — Chapter 141. Traveling Waves2. Wave Propagation3. Standing Waves4. SuperpositionIII. Fluid Mechanics and Thermal PhysicsA. Fluid Mechanics — Chapter 151. Hydrostatic pressure2. Buoyancy3. Fluid flow continuity4. Bernoulli’s equationB. Temperature and heat — Chapter 161. Mechanical equivalent of heat2. Heat transfer and thermal expansionC. Kinetic Theory and Thermodynamics1. Ideal gases — Chapter 17a. Kinetic modelb. Ideal gas law2. Laws of thermodynamics — Chapter 18a. First law (PV diagrams)b. Second Law (heat engines)c. Third Law (entropy)IV. Electricity and MagnetismA. Electrostatics — Chapter 191. Charge and Coloumb’s Law2. Electric field and electric potential (including point charges)3. Gauss’s Law4. Fields and potentials for charge distributionsB. Conductors and capacitors — Chapter 201. Electrostatics with conductors2. Capacitorsa. Capacitanceb. Parallel platec. Spherical and cylindrical3. DielectricsC. Electric circuits — Chapter 211. Current, resistance, power2. Steady-state direct current circuits with batteries and resistors only3. Capacitors in circuitsa. Steady Stateb. Transients in RC circuitsD. Magnetic Fields — Chapter 221. Forces on moving charges in magnetic fields2. Forces on current carrying wires in magnetic fields3. Fields of long current carrying wires4. Biot-Savart law and Ampere’s LawE. Electromagnetism — Chapter 231. Electromagnetic induction (including Faraday’s law and Lenz’s law)2. Inductance (including LR and LC circuits)3. Maxwell’s equationsV. Electromagnetic Waves and OpticsA. Physical Optics — Chapters 25 & 281. Interference and Diffraction2. Dispersion of Light and the electromagnetic spectrumB. Geometric optics — Chapters 26 & 271. Reflection and refraction2. Mirrors3. LensesVI. Atomic and Nuclear PhysicsA. Atomic physics and quantum effects — Chapter 301. Photons and the photoelectric effect2. Atomic energy levels3. Wave particle dualityB. Nuclear physics — Chapters 31, 32, and 291. Nuclear reactions (including conservation of mass number and charge)2. Mass-energy equivalenceF. PROPOSED LAB EXPERIMENTSThe following is a list of proposed lab experiments. There may be other investigative activities, demonstrations, and virtual labs in addition to those listed below.# Lab Title Notes Type1 Experimental Accuracy and Precision Introduce good lab practice, the concepts of accuracyand precision in measurement and calculationHands-on2 Galileo’s Experiment Study uniformly accelerated motion on an inclinedplaneHands-on3 One dimensional motion Use a motion detector to observe one dimensionalmotion in terms of position, displacement, velocity and accelerationHands-on4 Acceleration due to Gravity Determine the acceleration due to gravity by examiningposition at set time intervals using a ticker tapeHands-on5 Composition and Resolution ofForcesUse a force table to graphically and analytically addand subtract force vectorsHands-on6 Two dimensional motion Use a bowling ball on a level surface with regularlymarked positions to visualize and measure twodimensional motion / Plot two dimensional motionusing video analysisWholeclasshands-on /virtual7 Bull’s Eye Predict the landing location of a projectile based on measurement and calculationHands-on8 Coefficient of Friction Determine the coefficient of static and kinetic frictionof various objects including a student’s sneakerHands-on9 Atwood’s Machine and Friends Examining Newton’s second law in several dynamicsystems involving changing direction of tension forcesusing pulleys. Friction on the system will also beinvestigatedHands-on10 Work-energy theorem and energyconservationExploring conservation of energy and work on anumber of systems including cart on an inclined plane,human motion and a “popper”Hands-on11 Collisions and Explosions Conservation of momentum in collisions andexplosions in one dimension on a motion track, and intwo dimensions using video analysisHands-on/ virtual12 Torques and Rotational Equilibriumof a Rigid BodyUsing a meter stick with lever knives to determinecenter of gravity, and determine unknown mass / videoanalysis of an irregular object in two dimensionalmotion about center of gravityHands-on/ virtual13 Simple Harmonic Motion – Mass on aSpringDynamics and conservation of energy for a mass on aspring, including damping using a motion detectorHands-on14 Simple Harmonic Motion – Pendulum Conservation of energy, period, variation of mass andlength of a simple pendulum examinedHands-on15 Properties of Sound Examination of the wave properties of various soundsusing a microphone and wave visualization software, determination of the speed of sound using resonancetubesHands-on16 Buoyancy To explore Archimedes’ Principle and the principle ofFlotation and create the lightest boat that can carry themost mass without sinkingHands-on17 Specific Heat of Metals Use of calorimetry to identify unknown metals basedon specific heatHands-on18 Linear Thermal Expansion Determination of the linear coefficient of thermalexpansion for several metals by direct measurement oftheir expansion when heatedHands-on19 The Ideal Gas Law Boyle’s law and Charles’s law investigated using ahomemade apparatus made from a plastic syringeHands-on20 Coloumb’s Law Determination of charge on objects based on indirect measurement on electrostatic forcesHands-on21 Equipotentials and Electric Fields Mapping of equipotentials around charged conducting electrodes, construction of electric field lines,quantitative evaluation of the dependence of theelectric field on distance for a line of chargeHands-on22 Circuit Challenge Construction of series and parallel circuits based onfunctional requirementsHands-on23 Ohm’s Law Exploring the relationship between voltage, current,and resistance for ohmic and non-ohmic materialsHands-on24 RC Circuits Determination of the RC time constant using avoltmeter as circuit resistance, finding an unknown capacitance, finding an unknown resistanceHands-on25 Magnetic Fields Mapping the magnetic field around a permanentmagnetHands-on26 Magnetic Induction of a currentcarrying wireDetermination of the induced emf in a coil as ameasure of the magnetic field from an alternatingcurrent in a long straight wireHands-on27 Interference – Light as a wave Determination of the wavelength of a source of light byusing a double slit, determination of grating spacingbased on a known wavelength of lightHands-on28 Reflection Establish the law of reflection, determine the focallength and radius of curvature of cylindrical mirrorsusing the ray box. Determination of focal length andradius of curvature of spherical mirrors using imageheight and object distanceHands-on29 Snell’s Law Determination of the index of refraction of a Luciteblock and gelatin. Discovery of phenomenon of totalinternal reflection as an extension of Snell’s LawHands-on30 Bohr Theory of Hydrogen Comparison of the measured values of the wavelengthsof hydrogen spectrum with Bohr theory to determinethe Rydberg constantHands-on31 Radioactive Decay and Half - life Simulation of radioactive decay using dice as ananalog, Geiger counter measurement of the half-life of137BaHands-on。

The Simple Pendulumby Dr. James E. ParksDepartment of Physics and Astronomy401 Nielsen Physics BuildingThe University of TennesseeKnoxville, Tennessee 37996-1200Copyright June, 2000 by James Edgar Parks**All rights are reserved. No part of this publication may be reproduced or transmitted in any form or byany means, electronic or mechanical, including photocopy, recording, or any information storage orretrievalObjectivesThe purposes of this experiment are: (1) to study the motion of a simple pendulum, (2) tostudy simple harmonic motion, (3) to learn the definitions of period, frequency, andamplitude, (4) to learn the relationships between the period, frequency, amplitude andlength of a simple pendulum and (5) to determine the acceleration due to gravity usingthe theory, results, and analysis of this experiment.TheoryA simple pendulum may be described ideally as a point mass suspended by a masslessstring from some point about which it is allowed to swing back and forth in a place. Asimple pendulum can be approximated by a small metal sphere which has a small radiusand a large mass when compared relatively to the length and mass of the light string fromwhich it is suspended. If a pendulum is set in motion so that is swings back and forth, itsmotion will be periodic. The time that it takes to make one complete oscillation isdefined as the period T. Another useful quantity used to describe periodic motion is thefrequency of oscillation. The frequency f of the oscillations is the number of oscillationsthat occur per unit time and is the inverse of the period, f = 1/T. Similarly, the period isthe inverse of the frequency, T = l/f. The maximum distance that the mass is displacedfrom its equilibrium position is defined as the amplitude of the oscillation.When a simple pendulum is displaced from its equilibrium position, there will be arestoring force that moves the pendulum back towards its equilibrium position. As themotion of the pendulum carries it past the equilibrium position, the restoring forcechanges its direction so that it is still directed towards the equilibrium position. If the restoring force F G is opposite and directly proportional to the displacement x from theequilibrium position, so that it satisfies the relationshipF = - k xG G (1)then the motion of the pendulum will be simple harmonic motion and its period can becalculated using the equation for the period of simple harmonic motionT = 2πIt can be shown that if the amplitude of the motion is kept small, Equation (2) will besatisfied and the motion of a simple pendulum will be simple harmonic motion, andEquation (2) can be used.θFigure 1. Diagram illustrating the restoring force for a simple pendulum.The restoring force for a simple pendulum is supplied by the vector sum of thegravitational force on the mass. mg, and the tension in the string, T. The magnitude ofthe restoring force depends on the gravitational force and the displacement of the massfrom the equilibrium position. Consider Figure 1 where a mass m is suspended by astring of length l and is displaced from its equilibrium position by an angle θ and adistance x along the arc through which the mass moves. The gravitational force can beresolved into two components, one along the radial direction, away from the point ofsuspension, and one along the arc in the direction that the mass moves. The componentof the gravitational force along the arc provides the restoring force F and is given byF = - mg sin θ (3)where g is the acceleration of gravity, θ is the angle the pendulum is displaced, and theminus sign indicates that the force is opposite to the displacement. For small amplitudeswhere θ is small, sinθ can be approximated by θ measured in radians so thatEquation (3) can be written asF = - mg θ. (4)The angle θ in radians is xl, the arc length divided by the length of the pendulum or theradius of the circle in which the mass moves. The restoring force is then given byxF = - mgl(5)and is directly proportional to the displacement x and is in the form of Equation (1) wheremgk =l. Substituting this value of k into Equation (2), the period of a simple pendulumcan be found byT = 2π (6) andT = 2π. (7)Therefore, for small amplitudes the period of a simple pendulum depends only on itslength and the value of the acceleration due to gravity.ApparatusThe apparatus for this experiment consists of a support stand with a string clamp, a smallspherical ball with a 125 cm length of light string, a meter stick, a vernier caliper, and atimer. The apparatus is shown in Figure 2.Procedure1. The simple pendulum is composed of a small spherical ball suspended by a long, lightstring which is attached to a support stand by a string clamp. The string should beapproximately 125 cm long and should be clamped by the string clamp between thetwo flat pieces of metal so that the string always pivots about the same point.Figure 2. Apparatus for simple pendulum.2. Use a vernier caliper to measure the diameter d of the spherical ball and from thiscalculate its radius r. Record the values of the diameter and radius in meters.3. Prepare an Excel spreadsheet like the example shown in Figure 3. Adjust the lengthof the pendulum to about .10 m. The length of the simple pendulum is the distance from the point of suspension to the center of the ball. Measure the length of the string s l from the point of suspension to the top of the ball using a meter stick. Make the following table and record this value for the length of the string. Add the radius of the ball to the string length s l and record that value as the length of the pendulum s l l r =+.4. Displace the pendulum about 5º from its equilibrium position and let it swing backand forth. Measure the total time that it takes to make 50 complete oscillations. Record that time in your spreadsheet.5. Increase the length of the pendulum by about 0.10 m and repeat the measurementsmade in the previous steps until the length increases to approximately 1.0 m.6. Calculate the period of the oscillations for each length by dividing the total time bythe number of oscillations, 50. Record the values in the appropriate column of your data table.Figure 3. Example of Excel spreadsheet for recording and analyzing data.7. Graph the period of the pendulum as a function of its length using the chart feature ofExcel. The length of the pendulum is the independent variable and should be plotted on the horizontal axis or abscissa (x axis). The period is the dependent variable and should be plotted on the vertical axis or ordinate (y axis).8. Use the trendline feature to draw a smooth curve that best fits your data. To do this,from the main menu, choose Chart and then Add Trendline . . . from the dropdown menu. This will bring up a Add Trendline dialog window. From the Trend tab, choose Power from the Trend/Regression type selections. Then click on the Options tab and select Display equations on chart option.9. Examine the power function equation that is associated with the trendline. Does itsuggest the relationship between period and length given by Equation (7)?10. Examine your graph and notice that the change in the period per unit length, the slopeof the curve, decreases as the length increases. This indicates that the period increases with the length at a rate less than a linear rate. The theory and Equation (7) predict that the period depends on the square root of the length. If both sides ofEquation 7 are squared then224πT =g l . (8) If the theory is correct, a graph of T 2 versus l should result in a straight line.11. Square the values of the period measured for each length of the pendulum and recordyour results in the spreadsheet.12. Use the chart feature again to graph the period squared, T 2, as a function of the lengthof the pendulum l . The period squared is the dependent variable and should be plotted on the y axis. The length is the independent variable and should be plotted on the x axis.13. Examine your graph of T 2 versus l and check to see if there is a linear relationshipbetween T 2 and l so that the data points lie along a line.14. Use the trendline feature to perform a linear regression to find a straight line that bestfits your data points. This time from the Add Trendline dialog window. choose Linear from the Trend/Regression type selections. Click on the Options tab and once again select the Display equations on chart option. This should draw a straight line that best fits the data and should display the equation for this straight line.15. Equation (8), 224πT =g l , is of the form y=ax+b where y=T 2, 24πa = g, x=l , and b=0. A graph of T 2 versus l should therefore result in a straight line whose slope, a, is equal to 24πg. From the equation for the trendline, record the value for the slope, a, and from the equation a=24πgfind g, the acceleration due to gravity.16. Compare your result with the accepted value of the acceleration due to gravity9.8 m/s 2. Calculate the percent difference in your result and the accepted result.% Difference = [(your result - accepted value)/accepted value] x 100%17. Using the accepted value of the acceleration due to gravity and Equation 7 calculatethe period of a simple pendulum whose length is equal to the longest length measured in Table 1. Compare this theoretical result with the measured experimental result and calculate the percent difference.% Difference = [(Experimental Result - Theoretical Result)/Theoretical Result] x 100%.18. The equation for the period of a simple period, Equation (7), was developed byassuming that the amplitude is small. The range of amplitudes over which Equation(7) is valid is to be determined by measuring the period of a simple pendulum withdifferent amplitudes.19. Adjust the length of the pendulum to about 0.6 m. Measure the period of thependulum when it is displaced 5°, 10°, 15°, 20°, 25°, 30°, 40°, 50°, and 60° from its equilibrium position. Make a table to record the period T as a function of the amplitude A.20. Using your data, make a graph of the period versus the amplitude.21. Measure the length of the pendulum and use Equation (7) to calculate the period ofthe pendulum. Add this theoretical point to your graph for the period with zero amplitude.22. Examine your graph for the behavior of the period with amplitude. What conclusionscan you draw from your data regarding the range of amplitudes over which Equation(7) is valid?Questions1. How would the period of a simple pendulum be affected if it were located on themoon instead of the earth?2. What effect would the temperature have on the time kept by a pendulum clock if thependulum rod increases in length with an increase in temperature?3. What kind of graph would result if the period T were graphed as a function of the.4. What effect does the mass of the ball have on the period of a simple pendulum?What would be the effect of replacing the steel ball with a wooden ball, a lead ball, and a ping pong ball of the same size?。

Physics (Lecture Notes) (by: Dr. Yu)Simple Harmonic Motion (SHM)Key Words:oscillatory motion: 振动Simple Harmonic Motion (SHM): 简谐运动Simple Harmonic Oscillation (SHO): 简谐振动amplitude: 振幅frequency: 频率period: 周期angular frequency: 角频率phase constant: 初相，相角，相差equilibrium position: 平衡点pendulum: 摆Simple Harmonic Motion (SHM), or Simple Harmonic Oscillation (SHO)For a spring problem:The Restoring Force: F kx =-k : Spring Constant (N/m)The Equation of Motion:ma kx =-2 =k a x x m ω⇒=--， where ω=The Solution for x (t):cos()x A t ωφ=+The Period: 2T πω= The Frequency: 12f T ωπ==sin()v A t ωωφ=-+2cos()a A t ωωφ=-+(Note that: 2=a x ω-)The Angular Frequency: 2f ωπ=The Total Energy: 221122E mv kx =+= constant 212kA =v ∴= or, v =±Pendulum:sin g g a L Lθθθ=-≅-2a θωθ=-, where ω=2 2T πω∴==Multiple Choice1. A mass on a spring undergoes SHM. It goes through 8 complete oscillations in 4.0 s. What is the period?a. 0.031 sb. 0.50 sc. 2.0 sd. 32 s2. An object in simple harmonic motion obeys the following position versus time equation:0.50sin 2y t π⎛⎫= ⎪⎝⎭, in meters. What is the amplitude of vibration? a. 0.25 m b. 0.50 m c. 0.75 m d.1.0 m3. Consider the question above. What is the maximum speed of the object?a. 0.79 m/sb. 1.25 m/sc. 1.65 m/sd. 2.33 m/s4. Consider the question above. What is the maximum acceleration of the object?a. 0.252/m sb. 0.50 2/m sc. 1.232/m sd. 2.462/m s5. A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a minimum?a. At either A or Bb. Midway between A and Bc. One-fourth of the way between A and Bd. None of the above6. Doubling only the amplitude of a vibrating mass-and-spring system produces what effect on the system's mechanical energy?a. Increases the energy by a factor of two.b. Increases the energy by a factor of three.c. Increases the energy by a factor of four.d. Produces no change.7. A mass m = 0.2 kg is attached to a spring. It oscillates on a horizontal frictionless surface at a frequency of (4/π) Hz when displaced a distance of 2.0 cm from equilibrium and released. What is the maximum velocity attained by the mass?a. 0.02 m/sb. 0.04 m/sc. 0.08 m/sd. 0.16 m/sAnswer: bbaca,cdWritten Problems1.The displacement of a particle at is given by the expression 4.0cos(3.0)=+, where x isx tππin meters and t is in seconds.a)Determine the frequency and period of the motion.b)Find the amplitude of the motion,c)Find the phase constant, andd)the displacement of the particle at 0.25=.t s2. A particle moving along the x axis in simple harmonic motion starts from its equilibriumposition, the origin, at 0t= and moves to the right. The amplitude of its motion is 2.00cm, and the frequency is 1.50Hz.a)Show that the displacement of the particle is given by 2.00sin(3.00)=cm.x tπb)Determine the maximum speed and the earliest time(0)t>at which the particle has this speed.c)Find the maximum acceleration, and the earliest time (0)t>at which the particle has this acceleration.d)Find the total distance traveled between 0=.t= and 1.00t s3. A 0.500-kg mass attached to a spring with a force constant of 8.00N/m vibrates in simpleharmonic motion with amplitude of 10.0 cm.a)Calculate the maximum value of its speed and acceleration.b)Find the speed and acceleration when the mass is 6.00 cm from the equilibrium position.c)Calculate the time it takes the mass to move from0x= cm.x= to8.004. A 7.00-kg mass is hung from the bottom end of a vertical spring fastened to an overheadbeam. The mass is set into vertical oscillations with a period of 2.60 s. Find the force constant of the spring.5.An automobile having a mass of 1000 kg is driven into a brick wall in a safety test. Thebumper behaves as a spring of constant 6⨯ and compresses 3.16 cm as the car is5.0010/N mbrought to rest. What was the speed of the car before impact, assuming that no energy is lost during impact with the wall?6. A simple pendulum has a mass of 0.250kg and a length of 1.00 m. It is displaced through anangle of15.0 and then released.a)What is the maximum speed?b)Find the maximum angular acceleration, andc)the maximum restoring force?7. A particle of mass m slides without friction inside a hemispherical bowl of radius R. Showthat, if it starts from rest with a small displacement from equilibrium, the particle moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum oflength R, that is,ω=8. A simple pendulum with a length of 2.23 m and mass of 6.74 kg is given an initial speed of2.06 m/s at its equilibrium position. Assume that it undergoes simple harmonic motion anddetermine its (a) period, (b) total energy and (c) maximum angular displacement.9. A ball of mass m is connected to two rubber bands of length L, each under tension T, The ballis displaced by a small distance y perpendicular to the length of the rubber bands. Assuming that the tension does not change, show that (a) the restoring force is – (2T/L) y and (b) thesystem exhibits simple harmonic motion with an angular frequencyω=。