DOCSLIB.ORG

Department of Physics United States Naval Academy Lecture 36: Simple Harmonic Motion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Department of Physics United States Naval Academy Lecture 36: Simple Harmonic Motion Department of Physics United States Naval Academy Lecture 36: Simple Harmonic Motion Learning Objectives • Distinguish simple harmonic motion from other types of periodic motion; analyze SHM by identifying the various rela- tionships between amplitude, period, frequency, phase constant, position, velocity and acceleration. Oscillations: refers to the repetitive or periodic motion of a particle about an equilibrium position. In an oscillating system, the traditional variables x, v, t, and a still apply to motion, but it is more advantageous to introduce new variables that describe the periodic nature of the motion: amplitude, period, and frequency. Amplitude: In an oscillatory motion, a particle or object goes back and forth between two extreme points; the points of maximum displacement from the equilibrium point. Frequency, f of periodic or oscillatory motion is the number of oscillations per second (per unit time). In the SI system, it is measured in hertz: 1 Hz = 1 s−1. Period, T is the time required for one complete oscillation or cycle. It is measured in seconds (s) and is related to the frequency via 1 T = f Simple Harmonic Motion: Is a special kind of oscillatory motion in which the net restoring force acting on an object is proportional to the negative of the displacement from some equilibrium position. A system exhibiting simple harmonic motion is called a Simple Harmonic Oscillator. The displacement, x(t) of a particle from its equilibrium position is described by the equation x(t) = xm cos(wt + f) where xm is the amplitude of the displacement, wt + f is the phase of the motion (this is the argument of the Cosine function and defines the position or location of a point in time), and f is the phase constant. The angular frequency w is related to the period and frequency of the motion by 2p w = = 2p f T The velocity as a function of time of a particle in SHM is obtained by differentiating the displacement equation Figure 1: (a) The displacement x(t) of a particle oscillating in SHM with v(t) = −wxm sin(wt + f) phase angle f equal to zero. The pe- Similarly, the acceleration of a particle in SHM is obtained by differentiating the riod T marks one complete oscillation. velocity equation (b) The velocity v(t) of the particle. (c) a(t) = −w2x cos(wt + f) = −w2 x(t) m The acceleration a(t) of the particle. 2 The quantities, wxm and w xm are the amplitude of the velocity, vm and amplitude of the acceleration, am, respectively; i.e vm = wxm 2 am = w xm © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD The Force Law for Simple Harmonic Motion: The restoring force in simple har- monic motion can be written, according to Newton’s second law, as F = ma = m(−w2x) = −(mw2)x The restoring force always acts in a direction opposite the direction of displace- ment (hence, the negative sign). For the block-spring system, the restoring force is defined by Hooke’s law F = −kx Comparing the two force equations shows that the spring constant can be written as k = mw2 or equivalently, r k w = m w m Figure 2: (Top) A linear simple har- allowing for the calculation of the angular frequency if the mass, and spring’s k monic oscillator on a frictionless sur- constants are known. The period of the motion is then rm face. (bottom) As position changes, T = 2p k the energy shifts between the two types, but the total is constant. Energy in Simple Harmonic Motion: A particle in simple harmonic motion has, 1 2 1 2 at any time, kinetic energy K = 2 mv and potential energy U = 2 kx . If no friction is present, the mechanical energy E = K +U remains constant even though K and U change. For a block-spring system, 1 2 E = K +U = 2 kxm where xm is the amplitude of the spring’s displacement. © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 1.0 A 2.3 kg mass oscillates back and forth from the end of a spring of spring constant 120 N/m. At t = 0, the position of the block is x = 0:13 m and its velocity is vx = −3:4 m/s. (a) What is the angular frequency of the block? (b) What is the mechanical energy of this block-spring system? (c) What is the amplitude of the oscillation? (d) What is the maximum speed of the block and where is this experienced over the motion? (e) What is the phase constant? (Choose a cosine function to describe the motion.) (f) What is the position of block as a function of time? (g) What is the maximum acceleration of the block and where is this experienced over the motion? x = 0:13 m vx = −3:4 m/s © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 2.0 What is the phase constant for the harmonic oscillator with the velocity function v(t) given in the figure below if the posi- tion function x(t) has the form x = xm cos(wt + f)? The vertical axis scale is set by vs = 4:0 cm/s. © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 3.0 A 10 g particle undergoes SHM with an amplitude of 2.0 mm, a maximum acceleration of magnitude 8:0×103 m/s2, and an unknown phase constant f. What are (a) the period of the motion, (b) the maximum speed of the particle, and (c) the total mechanical energy of the oscillator? What is the magnitude of the force on the particle when the particle is at (d) its maximum displacement and (e) half its maximum displacement? © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 4.0 If the phase angle for a block-spring system in SHM is p=6 rad and the block’s position is given by x = xm cos(wt + f), what is the ratio of the kinetic energy to the potential energy at time t = 0? © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD.
Load more

Recommended publications
  • Simple Harmonic Motion
    [SHIVOK SP211] October 30, 2015 CH 15 Simple Harmonic Motion I. Oscillatory motion A. Motion which is periodic in time, that is, motion that repeats itself in time. B. Examples: 1. Power line oscillates when the wind blows past it 2. Earthquake oscillations move buildings C. Sometimes the oscillations are so severe, that the system exhibiting oscillations break apart. 1. Tacoma Narrows Bridge Collapse "Gallopin' Gertie" a) http://www.youtube.com/watch?v=j‐zczJXSxnw II. Simple Harmonic Motion A. http://www.youtube.com/watch?v=__2YND93ofE Watch the video in your spare time. This professor is my teaching Idol. B. In the figure below snapshots of a simple oscillatory system is shown. A particle repeatedly moves back and forth about the point x=0. Page 1 [SHIVOK SP211] October 30, 2015 C. The time taken for one complete oscillation is the period, T. In the time of one T, the system travels from x=+x , to –x , and then back to m m its original position x . m D. The velocity vector arrows are scaled to indicate the magnitude of the speed of the system at different times. At x=±x , the velocity is m zero. E. Frequency of oscillation is the number of oscillations that are completed in each second. 1. The symbol for frequency is f, and the SI unit is the hertz (abbreviated as Hz). 2. It follows that F. Any motion that repeats itself is periodic or harmonic. G. If the motion is a sinusoidal function of time, it is called simple harmonic motion (SHM).
    [Show full text]
  • Position, Displacement, Velocity Big Picture
    Position, Displacement, Velocity Big Picture I Want to know how/why things move I Need a way to describe motion mathematically: \Kinematics" I Tools of kinematics: calculus (rates) and vectors (directions) I Chapters 2, 4 are all about kinematics I First 1D, then 2D/3D I Main ideas: position, velocity, acceleration Language is crucial Physics uses ordinary words but assigns specific technical meanings! WORD ORDINARY USE PHYSICS USE position where something is where something is velocity speed speed and direction speed speed magnitude of velocity vec- tor displacement being moved difference in position \as the crow flies” from one instant to another Language, continued WORD ORDINARY USE PHYSICS USE total distance displacement or path length traveled path length trav- eled average velocity | displacement divided by time interval average speed total distance di- total distance divided by vided by time inter- time interval val How about some examples? Finer points I \instantaneous" velocity v(t) changes from instant to instant I graphically, it's a point on a v(t) curve or the slope of an x(t) curve I average velocity ~vavg is not a function of time I it's defined for an interval between instants (t1 ! t2, ti ! tf , t0 ! t, etc.) I graphically, it's the \rise over run" between two points on an x(t) curve I in 1D, vx can be called v I it's a component that is positive or negative I vectors don't have signs but their components do What you need to be able to do I Given starting/ending position, time, speed for one or more trip legs, calculate average
    [Show full text]
  • Ch 11 Vibrations and Waves Simple Harmonic Motion Simple Harmonic Motion
    Ch 11 Vibrations and Waves Simple Harmonic Motion Simple Harmonic Motion A vibration (oscillation) back & forth taking the same amount of time for each cycle is periodic. Each vibration has an equilibrium position from which it is somehow disturbed by a given energy source. The disturbance produces a displacement from equilibrium. This is followed by a restoring force. Vibrations transfer energy. Recall Hooke’s Law The restoring force of a spring is proportional to the displacement, x. F = -kx. K is the proportionality constant and we choose the equilibrium position of x = 0. The minus sign reminds us the restoring force is always opposite the displacement, x. F is not constant but varies with position. Acceleration of the mass is not constant therefore. http://www.youtube.com/watch?v=eeYRkW8V7Vg&feature=pl ayer_embedded Key Terms Displacement- distance from equilibrium Amplitude- maximum displacement Cycle- one complete to and fro motion Period (T)- Time for one complete cycle (s) Frequency (f)- number of cycles per second (Hz) * period and frequency are inversely related: T = 1/f f = 1/T Energy in SHOs (Simple Harmonic Oscillators) In stretching or compressing a spring, work is required and potential energy is stored. Elastic PE is given by: PE = ½ kx2 Total mechanical energy E of the mass-spring system = sum of KE + PE E = ½ mv2 + ½ kx2 Here v is velocity of the mass at x position from equilibrium. E remains constant w/o friction. Energy Transformations As a mass oscillates on a spring, the energy changes from PE to KE while the total E remains constant.
    [Show full text]
  • Euler Equation and Geodesics R
    Euler Equation and Geodesics R. Herman February 2, 2018 Introduction Newton formulated the laws of motion in his 1687 volumes, col- lectively called the Philosophiae Naturalis Principia Mathematica, or simply the Principia. However, Newton’s development was geometrical and is not how we see classical dynamics presented when we first learn mechanics. The laws of mechanics are what are now considered analytical mechanics, in which classical dynamics is presented in a more elegant way. It is based upon variational principles, whose foundations began with the work of Eu- ler and Lagrange and have been refined by other now-famous figures in the eighteenth and nineteenth centuries. Euler coined the term the calculus of variations in 1756, though it is also called variational calculus. The goal is to find minima or maxima of func- tions of the form f : M ! R, where M can be a set of numbers, functions, paths, curves, surfaces, etc. Interest in extrema problems in classical mechan- ics began near the end of the seventeenth century with Newton and Leibniz. In the Principia, Newton was interested in the least resistance of a surface of revolution as it moves through a fluid. Seeking extrema at the time was not new, as the Egyptians knew that the shortest path between two points is a straight line and that a circle encloses the largest area for a given perimeter. Heron, an Alexandrian scholar, deter- mined that light travels along the shortest path. This problem was later taken up by Willibrord Snellius (1580–1626) after whom Snell’s law of refraction is named.
    [Show full text]
  • Chapter 1 Chapter 2 Chapter 3
    Notes CHAPTER 1 1. Herbert Westren Turnbull, The Great Mathematicians in The World of Mathematics. James R. Newrnan, ed. New York: Sirnon & Schuster, 1956. 2. Will Durant, The Story of Philosophy. New York: Sirnon & Schuster, 1961, p. 41. 3. lbid., p. 44. 4. G. E. L. Owen, "Aristotle," Dictionary of Scientific Biography. New York: Char1es Scribner's Sons, Vol. 1, 1970, p. 250. 5. Durant, op. cit., p. 44. 6. Owen, op. cit., p. 251. 7. Durant, op. cit., p. 53. CHAPTER 2 1. Williarn H. Stahl, '' Aristarchus of Samos,'' Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 1, 1970, p. 246. 2. Jbid., p. 247. 3. G. J. Toorner, "Ptolerny," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 11, 1975, p. 187. CHAPTER 3 1. Stephen F. Mason, A History of the Sciences. New York: Abelard-Schurnan Ltd., 1962, p. 127. 2. Edward Rosen, "Nicolaus Copernicus," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 3, 1971, pp. 401-402. 3. Mason, op. cit., p. 128. 4. Rosen, op. cit., p. 403. 391 392 NOTES 5. David Pingree, "Tycho Brahe," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 2, 1970, p. 401. 6. lbid.. p. 402. 7. Jbid., pp. 402-403. 8. lbid., p. 413. 9. Owen Gingerich, "Johannes Kepler," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 7, 1970, p. 289. 10. lbid.• p. 290. 11. Mason, op. cit., p. 135. 12. Jbid .. p. 136. 13. Gingerich, op. cit., p. 305. CHAPTER 4 1.
    [Show full text]
  • Oscillations
    CHAPTER FOURTEEN OSCILLATIONS 14.1 INTRODUCTION In our daily life we come across various kinds of motions. You have already learnt about some of them, e.g., rectilinear 14.1 Introduction motion and motion of a projectile. Both these motions are 14.2 Periodic and oscillatory non-repetitive. We have also learnt about uniform circular motions motion and orbital motion of planets in the solar system. In 14.3 Simple harmonic motion these cases, the motion is repeated after a certain interval of 14.4 Simple harmonic motion time, that is, it is periodic. In your childhood, you must have and uniform circular enjoyed rocking in a cradle or swinging on a swing. Both motion these motions are repetitive in nature but different from the 14.5 Velocity and acceleration periodic motion of a planet. Here, the object moves to and fro in simple harmonic motion about a mean position. The pendulum of a wall clock executes 14.6 Force law for simple a similar motion. Examples of such periodic to and fro harmonic motion motion abound: a boat tossing up and down in a river, the 14.7 Energy in simple harmonic piston in a steam engine going back and forth, etc. Such a motion motion is termed as oscillatory motion. In this chapter we 14.8 Some systems executing study this motion. simple harmonic motion The study of oscillatory motion is basic to physics; its 14.9 Damped simple harmonic motion concepts are required for the understanding of many physical 14.10 Forced oscillations and phenomena. In musical instruments, like the sitar, the guitar resonance or the violin, we come across vibrating strings that produce pleasing sounds.
    [Show full text]
  • Chapter 10: Elasticity and Oscillations
    Chapter 10 Lecture Outline 1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 10: Elasticity and Oscillations •Elastic Deformations •Hooke’s Law •Stress and Strain •Shear Deformations •Volume Deformations •Simple Harmonic Motion •The Pendulum •Damped Oscillations, Forced Oscillations, and Resonance 2 §10.1 Elastic Deformation of Solids A deformation is the change in size or shape of an object. An elastic object is one that returns to its original size and shape after contact forces have been removed. If the forces acting on the object are too large, the object can be permanently distorted. 3 §10.2 Hooke’s Law F F Apply a force to both ends of a long wire. These forces will stretch the wire from length L to L+L. 4 Define: L The fractional strain L change in length F Force per unit cross- stress A sectional area 5 Hooke’s Law (Fx) can be written in terms of stress and strain (stress strain). F L Y A L YA The spring constant k is now k L Y is called Young’s modulus and is a measure of an object’s stiffness. Hooke’s Law holds for an object to a point called the proportional limit. 6 Example (text problem 10.1): A steel beam is placed vertically in the basement of a building to keep the floor above from sagging. The load on the beam is 5.8104 N and the length of the beam is 2.5 m, and the cross-sectional area of the beam is 7.5103 m2.
    [Show full text]
  • Exact Solution for the Nonlinear Pendulum (Solu¸C˜Aoexata Do Pˆendulon˜Aolinear)
    Revista Brasileira de Ensino de F¶³sica, v. 29, n. 4, p. 645-648, (2007) www.sb¯sica.org.br Notas e Discuss~oes Exact solution for the nonlinear pendulum (Solu»c~aoexata do p^endulon~aolinear) A. Bel¶endez1, C. Pascual, D.I. M¶endez,T. Bel¶endezand C. Neipp Departamento de F¶³sica, Ingenier¶³ade Sistemas y Teor¶³ade la Se~nal,Universidad de Alicante, Alicante, Spain Recebido em 30/7/2007; Aceito em 28/8/2007 This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is di®erent from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75¼ (135±) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ! in terms of the complete elliptic integral of the ¯rst kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude).
    [Show full text]
  • Distance, Displacement, and Position
    Distance, Displacement, and Position Introduction: What is the difference between distance, displacement, and position? Here's an example: A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below. What is the total distance traveled by the bee? What is the displacement of the bee? What is the position of the bee? total distance = displacement = position = 1 Warm-up A particle moves along the x-axis so that the acceleration at any time t is given by: At time , the velocity of the particle is and at time , the position is . (a) Write an expression for the velocity of the particle at any time . (b) For what values of is the particle at rest? (c) Write an expression for the position of the particle at any time . (d) Find the total distance traveled by the particle from to . 2 Warm-up Answers (a) (b) (c) (d) Total Distance = 3 Now, using the equation from the warm-up find the following (WITHOUT A CALCULATOR): (a) the total distance from 0 to 4 (b) the displacement from 0 to 4 4 To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement. Displacement = To find the distance traveled we have to use absolute value. Distance traveled = To find the distance traveled by hand you must: Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and x-axis.
    [Show full text]
  • The Effect of Spring Mass on the Oscillation Frequency
    The Effect of Spring Mass on the Oscillation Frequency Scott A. Yost University of Tennessee February, 2002 The purpose of this note is to calculate the effect of the spring mass on the oscillation frequency of an object hanging at the end of a spring. The goal is to find the limitations to a frequently-quoted rule that 1/3 the mass of the spring should be added to to the mass of the hanging object. This calculation was prompted by a student laboratory exercise in which it is normally seen that the frequency is somewhat lower than this rule would predict. Consider a mass M hanging from a spring of unstretched length l, spring constant k, and mass m. If the mass of the spring is neglected, the oscillation frequency would be ω = k/M. The quoted rule suggests that the effect of the spring mass would beq to replace M by M + m/3 in the equation for ω. This result can be found in some introductory physics textbooks, including, for example, Sears, Zemansky and Young, University Physics, 5th edition, sec. 11-5. The derivation assumes that all points along the spring are displaced linearly from their equilibrium position as the spring oscillates. This note will examine more general cases for the masses, including the limit M = 0. An appendix notes how the linear oscillation assumption breaks down when the spring mass becomes large. Let the positions along the unstretched spring be labeled by x, running from 0 to L, with 0 at the top of the spring, and L at the bottom, where the mass M is hanging.
    [Show full text]
  • Lecture I, Aug25, 2014 Newton, Lagrange and Hamilton's Equations of Classical Mechanics
    Lecture I, Aug25, 2014 Newton, Lagrange and Hamilton’s Equations of Classical Mechanics Introduction What this course is about... BOOK Goldstein is a classic Text Book Herbert Goldstein ( June 26, 1922 January 12, 2005) PH D MIT in 1943; Then at Harvard and Columbia first edition of CM book was published in 1950 ( 399 pages, each page is about 3/4 in area compared to new edition) Third edition appeared in 2002 ... But it is an old Text book What we will do different ?? Start with Newton’s Lagrange and Hamilton’s equation one after the other Small Oscillations: Marion, why is it important ??? we start right in the beginning talking about small oscillations, simple limit Phase Space Plots ( Marian, page 159 ) , Touch nonlinear physics and Chaos, Symmetries Order in which Chapters are covered is posted on the course web page Classical Encore 1-4pm, Sept 29, Oct 27, Dec 1 Last week of the Month: Three Body Problem.. ( chaos ), Solitons, may be General Relativity We may not cover scattering and Rigid body dynamics.. the topics that you have covered in Phys303 and I think there is less to gain there.... —————————————————————————— Newton’s Equation, Lagrange and Hamilton’s Equations Beauty Contest: Write Three equations and See which one are the prettiest?? (Simplicity, Mathematical Beauty...) Same Equations disguised in three different forms (I)Lagrange and Hamilton’s equations are scalar equations unlike Newton’s equation.. (II) To apply Newtons’ equation, Forces of constraints are needed to describe constrained motion (III) Symmetries are best described in the Lagrangian formulation (IV)For rectangular coordinates, Newtons’ s Eqn are the easiest.
    [Show full text]
  • The Harmonic Oscillator
    Appendix A The Harmonic Oscillator Properties of the harmonic oscillator arise so often throughout this book that it seemed best to treat the mathematics involved in a separate Appendix. A.1 Simple Harmonic Oscillator The harmonic oscillator equation dates to the time of Newton and Hooke. It follows by combining Newton’s Law of motion (F = Ma, where F is the force on a mass M and a is its acceleration) and Hooke’s Law (which states that the restoring force from a compressed or extended spring is proportional to the displacement from equilibrium and in the opposite direction: thus, FSpring =−Kx, where K is the spring constant) (Fig. A.1). Taking x = 0 as the equilibrium position and letting the force from the spring act on the mass: d2x M + Kx = 0. (A.1) dt2 2 = Dividing by the mass and defining ω0 K/M, the equation becomes d2x + ω2x = 0. (A.2) dt2 0 As may be seen by direct substitution, this equation has simple solutions of the form x = x0 sin ω0t or x0 = cos ω0t, (A.3) The original version of this chapter was revised: Pages 329, 330, 335, and 347 were corrected. The correction to this chapter is available at https://doi.org/10.1007/978-3-319-92796-1_8 © Springer Nature Switzerland AG 2018 329 W. R. Bennett, Jr., The Science of Musical Sound, https://doi.org/10.1007/978-3-319-92796-1 330 A The Harmonic Oscillator Fig. A.1 Frictionless harmonic oscillator showing the spring in compressed and extended positions where t is the time and x0 is the maximum amplitude of the oscillation.
    [Show full text]