Physics
Simple Harmonic Oscillator
A simple harmonic oscillator is a system that exhibits a specific type of periodic motion, where the restoring force is directly proportional to the displacement from equilibrium. This results in a sinusoidal motion characterized by a constant frequency and amplitude. Examples of simple harmonic oscillators include a mass on a spring and a pendulum.
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11 Key excerpts on "Simple Harmonic Oscillator"
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
When an arrow is shot from a bow, the feathers at the end of the arrow manage to snake around the bow staff without hitting it because the arrow oscillates. When a coin drops into a metal collection plate, the coin oscillates with such a familiar ring that the coin’s denomination can be determined from the sound. When a rodeo cowboy rides a bull, the cow- boy oscillates wildly as the bull jumps and turns (at least the cowboy hopes to be oscillating). The study and control of oscillations are two of the primary goals of both physics and engineering. In this chapter we discuss a basic type of oscillation called simple harmonic motion. Heads Up. This material is quite challenging to most students. One reason is that there is a truckload of definitions and symbols to sort out, but the main reason is that we need to relate an object’s oscillations (something that we can see or even experience) to the equations and graphs for the oscillations. Relating the real, vis- ible motion to the abstraction of an equation or graph requires a lot of hard work. Simple Harmonic Motion Figure 15.1.1 shows a particle that is oscillating about the origin of an x axis, repeatedly going left and right by iden- tical amounts. The frequency f of the oscillation is the number of times per sec- ond that it completes a full oscillation (a cycle) and has the unit of hertz (abbrevi- ated Hz), where 1 hertz = 1 Hz = 1 oscillation per second = 1 s −1 . (15.1.1) The time for one full cycle is the period T of the oscillation, which is T = 1 __ f . (15.1.2) Any motion that repeats at regular intervals is called periodic motion or har- monic motion. However, here we are interested in a particular type of periodic motion called simple harmonic motion (SHM). Such motion is a sinusoidal func- tion of time t. That is, it can be written as a sine or a cosine of time t. Here we arbitrarily choose the cosine function and write the displacement (or position) of the particle in Fig.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
When a coin drops into a metal collection plate, the coin oscillates with such a familiar ring that the coin’s denomination can be determined from the sound. When a rodeo cowboy rides a bull, the cow- boy oscillates wildly as the bull jumps and turns (at least the cowboy hopes to be oscillating). FCP The study and control of oscillations are two of the primary goals of both physics and engineering. In this chapter we discuss a basic type of oscillation called simple harmonic motion. Heads Up. This material is quite challenging to most students. One reason is that there is a truckload of definitions and symbols to sort out, but the main reason is that we need to relate an object’s oscillations (something that we can see or even experience) to the equations and graphs for the oscillations. Relating the real, vis- ible motion to the abstraction of an equation or graph requires a lot of hard work. Simple Harmonic Motion Figure 15.1.1 shows a particle that is oscillating about the origin of an x axis, repeatedly going left and right by identical amounts. The frequency f of the oscillation is the number of times per second that it completes a full oscillation (a cycle) and has the unit of hertz (abbreviated Hz), where 1 hertz = 1 Hz = 1 oscillation per second = 1 s −1 . (15.1.1) The time for one full cycle is the period T of the oscillation, which is T = 1 __ f . (15.1.2) Any motion that repeats at regular intervals is called periodic motion or har- monic motion. However, here we are interested in a particular type of periodic motion called simple harmonic motion (SHM). Such motion is a sinusoidal func- tion of time t. That is, it can be written as a sine or a cosine of time t. Here we arbitrarily choose the cosine function and write the displacement (or position) of the particle in Fig. 15.1.1 as +x m –x m x 0 Figure 15.1.1 A particle repeatedly oscillates left and right along an x axis, between extreme points x m and −x m .
eBook - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Simple harmonic motion involves a displacement that changes sinusoidally in time. We will study the behavior of two systems that undergo SHM—an object hanging from a spring (a mass-spring system) and a simple pendulum that oscillates at small angles. Pendula and masses on springs are two common examples of periodic systems that oscillate with SHM (or at least approximately SHM). Simple harmonic motion is quite common in the physical world and helps us describe such diverse phenomena as the behavior of the tiniest fundamental particles, how clocks work (both analog and digital), and the periodic signals emitted by pulsars. We will devise ways to describe oscillating systems in general and then apply these descriptions to SHM. Questions we will address include: What is periodic motion and how can it be characterized? What factors do the motions of a mass on a spring and a simple pendulum depend on? What mathematical behavior is required for a motion to be considered simple harmonic? How do Newton’s laws allow us to predict the motion of a mass on a spring or a pendulum oscillating at small angles? UNIT 14: SIMPLE HARMONIC MOTION 461 OSCILLATING SYSTEMS 14.2 CHARACTERISTICS OF PERIODIC SYSTEMS A mass on a spring, a simple pendulum, and a point on a wheel rotating with uniform rotational velocity undergo periodic motions that are quite similar (see Fig. 14.1). To observe the experiments in this section, you will need the following: • 1 pendulum bob • 1 string • 1 spring • 1 mass pan • 1 mass set, 100 g, 200 g, 500 g, 1 kg, 2 kg, etc. • 1 rotating disk, with a pin (or some other marker) on its outer rim • 1 variable speed motor (to drive the disk) • 1 table clamp • 2 rods • 1 right angle clamp • 1 stopwatch (or use the one on your phone) • 1 ruler • 1 protractor XII VI III IV V VII VIII IX X XI II I Fig. 14.1. A pendulum bob, a mass on a spring, and a peg on a rotating disk as oscillating objects.
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
THE HARMONIC OSCILLATOR 11 11.1 Introduction 412 11.2 Simple Harmonic Motion: Review 412 11.2.1 Nomenclature 413 11.2.2 Energy of the Harmonic Oscillator 414 11.3 The Damped Harmonic Oscillator 414 11.3.1 Energy Dissipation in the Damped Oscillator 416 11.3.2 The Q of an Oscillator 418 11.4 The Driven Harmonic Oscillator 421 11.4.1 Energy Stored in a Driven Harmonic Oscillator 423 11.4.2 Resonance 424 11.5 Transient Behavior 425 11.6 Response in Time and Response in Frequency 427 Note 11.1 Complex Numbers 430 Note 11.2 Solving the Equation of Motion for the Damped Oscillator 431 Note 11.3 Solving the Equation of Motion for the Driven Harmonic Oscillator 434 Problems 435 412 THE HARMONIC OSCILLATOR 11.1 Introduction The harmonic oscillator plays a loftier role in physics than one might guess from its humble origin: a mass bouncing at the end of a spring. The harmonic oscillator underlies the creation of sound by musical in-struments, the propagation of waves in media, the analysis and control of vibrations in machinery and airplanes, and the time-keeping crystals in digital watches. Furthermore, the harmonic oscillator arises in numer-ous atomic and optical quantum scenarios, in quantum systems such as lasers, and it is a recurrent motif in advanced quantum field theories. In short, if there were a competition for a logo for the universality of physics, the harmonic oscillator would make a pretty strong contender. We encountered simple harmonic motion—the periodic motion of a mass attached to a spring—in Chapter 3 . The treatment there was highly idealized because it neglected friction and the possibility of a time-dependent driving force. It turns out that friction is essential for the analysis to be physically meaningful and that the most interesting ap-plications of the harmonic oscillator generally involve its response to a driving force.
eBook - PDFQuantum Mechanics I
A Problem Text
- David DeBruyne, Larry Sorensen(Authors)
- 2018(Publication Date)
- Sciendo(Publisher)
Chapter 10 The Simple Harmonic Oscillator The infinite square well is useful to illustrate many concepts including energy quantization, but the infinite square well is a limiting case of a realistic potential. The Simple Harmonic Oscillator (SHO), in contrast, is a realistic and commonly encountered potential energy function. It is one of the most important problems in quantum mechanics and physics in general. It is often used as a first approximation to more complex phenomena. It is dominantly popular in modeling a multitude of cooperative phenomena. The electrical bonds between the atoms or molecules in a crystal lattice are often modeled as “little springs,” so group phenomena is modeled by a system of coupled SHO’s. The phonons of solid state physics and the quantum mechanical description of electromagnetic fields in free space use multiple coupled phonons and photons modeled by Simple Harmonic Oscillators. The rudiments are the same as classical mechanics; small oscillations in a smooth potential are modeled well by the SHO. If a particle is confined in any potential, it demonstrates the same basic qualitative behavior as a particle confined to an infinite square well. Energy is quantized in all bound systems. The energy levels of the SHO are different than an infinite square well because the “geometry” of the potential energy function is different. Look for other similarities in these two systems. For instance, compare the shapes of the eigenfunctions between the infinite square well and the SHO. Part 1 outlines the basic concepts and focuses on the arguments of linear algebra using raising and lowering operators and matrix operators. This approach is more elegant than brute force solutions of differential equations in position space, and uses and reinforces Dirac notation, which depends upon the arguments of linear algebra.
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
Its solution, which we describe in the next section, is a function x(t ) that describes the position of the oscillator as a function of the time, in analogy with Fig. 17-2a, which represents the variation of position with time of a different oscillator. The Simple Harmonic Oscillator problem is important for two reasons. First, many problems involving mechanical vi- brations at small amplitudes reduce to that of the Simple Harmonic Oscillator, or to a combination of such oscillators. This is equivalent to saying that if we consider a small enough portion of a restoring force curve near the equilib- rium position, Fig. 17-3a, for instance, it becomes arbitrar- ily close to a straight line, which, as Fig. 17-4a shows, is characteristic of simple harmonic motion. Or, in other words, the potential energy curve of Fig. 17-3b is very nearly parabolic near the equilibrium position. Second, as we have indicated, equations like Eq. 17-4 occur in many physical problems in acoustics, optics, me- chanics, electrical circuits, and even atomic physics. The Simple Harmonic Oscillator exhibits features common to many physical systems. 17-3 SIMPLE HARMONIC MOTION Let us now solve the equation of motion of the simple har- monic oscillator, (17-4) We derived Eq. 17-4 for a spring force F x kx (where the force constant k is a measure of the stiffness of the spring) acting on a particle of mass m. We shall see later that other oscillating systems are governed by similar equa- tions of motion, in which the constant k is related to other physical features of the system. We can use the oscillating mass – spring system as our prototype. Equation 17-4 gives a relation between a function of the time x(t ) and its second time derivative d 2 x / dt 2 . Our goal is d 2 x dt 2 k m x 0. d 2 x dt 2 k m x 0. kx m d 2 x dt 2 to find a function x(t ) that satisfies this relation.
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
Note that the period of the Simple Harmonic Oscillator is independent of the amplitude (or total energy) ; a system which exhibits this property is said to be isochronous. The plane pendulum is an example of an oscillator that is approximately isochronous, if the amplitude of oscillation is small. (In Section 7.4 we shall treat the general case of finite amplitudes.) If Θ represents the angle that the pendulum makes with the vertical, then for small Θ (i.e., sin θ = Θ) the equation of motion is the familiar expression θ + ωΐθ = 0 (6.14) where ω £ Ξ | (6.15) * Henceforth we shall adhere to the convention of denoting angular frequencies by ω and frequencies by v. Usually, ω will be referred to as a frequency for brevity, although angular frequency is to be understood. 132 6 · OSCILLATORY MOTION Thus, τ = 2π^ 1 -(6.16) The period is therefore independent of the mass of the pendulum bob and depends only on the length / and the gravitational acceleration g. 6.3 Damped Harmonic Motion The motion represented by the Simple Harmonic Oscillator [see Eq. (6.5)] is termed a free oscillation ; once set into oscillation, the motion would never cease. This is, of course, an oversimplification of the actual physical case in which dissipative or frictional forces would eventually damp the motion to the point that the oscillations would cease. It is possible to incorporate into the differential equation which describes the motion a term which represents the damping force. Such forces which are encountered in Nature frequently depend upon some power of the instantaneous velocity.* One of the more important cases is that in which the damping force is directly proportional to the velocity and can therefore be described by a term — bv, where b > 0. The traditional example of such a damping force is that encountered by a particle moving with a relatively low velocity through a viscous fluid.
eBook - PDF- Stephen Lee(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
It follows that any vibration which can be written as a function of t can be reproduced by adding simple harmonic vibrations. Fourier accompanied Napoleon to Egypt in 1798 and was made a baron ten years later. He discovered this theorem while working on the flow of heat. 14.5 Oscillating mechanical systems There are very many mechanical systems which can be modelled using SHM. Two of these are the spring–mass oscillator and the simple pendulum. The motion of the simple pendulum approximates to SHM for small angles as you will see in the next section. 326 AN INTRODUCTION TO MATHEMATICS FOR ENGINEERS : MECHANICS The simple pendulum A simple pendulum consists of a bob suspended on the end of a light inelastic string as illustrated by the apparatus in figure 14.23. Figure 14.23 Figure 14.24 The forces acting on the bob are the tension in the string and the force of gravity mg, where m is the mass of the bob as shown in figure 14.24. It swings through a small arc of a circle of radius l where l is the length of the string. There is no motion in the radial direction. In the transverse direction, the acceleration, l .. , is given by mg sin ml .. ⇒ .. g l sin . When the angle is measured in radians, sin for small angles (up to about 0.3 rad for accuracy correct to 2 d.p.). In this case: .. g l . This is the standard equation for SHM, x .. 2 x , with x replaced by and 2 replaced by g l . A pendulum is usually set in motion by pulling the bob to one side, say to an angle , and then releasing it from rest. If this is the case, and . 0 when t 0. O positive direction The weight may be resolved into components: radial: mg cos θ transverse: mg sin θ P mg acceleration mg sin θ mg cos θ l θ T l θ l θ 2 string bob board with angles marked as shown 10° 10° 20° 20° SIMPLE HARMONIC MOTION 327 The appropriate form of the SHM equation is cos g l t .
eBook - PDFFundamentals of Physics
Mechanics, Relativity, and Thermodynamics
- R. Shankar(Author)
- 2014(Publication Date)
- Yale University Press(Publisher)
chapter 17 Simple Harmonic Motion We’re now going to study what are called small oscillations, or simple harmonic motion. Take any mechanical system that is in a state of equi-librium. Equilibrium means the forces on the body add up to zero. It has no desire to move. If you give it a little kick, a push away from the equilib-rium point, what will happen? There are two main possibilities. Imagine a marble on top of a hill. That is in unstable equilibrium because if you give the marble a nudge, it will roll downhill and never return to you. The other possibility involves stable equilibrium : if you push the system away from equilibrium, there are forces bringing it back. The standard example is a marble in a bowl: when it is shaken from its position at the bottom, it will rock back and forth until it settles again. A rod hanging vertically from the ceiling from a pivot, when pulled to the side and released, will swing back and forth. These are examples of simple harmonic motion, which results whenever any system is slightly disturbed from stable equilibrium. The example that we’re going to consider is a mass m , resting on a table, connected to a spring, which in turn is connected to the wall. The spring is not stretched or contracted; the mass is at rest, as shown in Figure 17.1. That’s what I mean by equilibrium. Now let it be displaced by 275 276 Simple Harmonic Motion Figure 17.1 The mass m rests on a table and is connected to a spring of force constant k , which is anchored to the wall. The displacement from equilibrium is denoted by x . It is positive in the figure but it could also be negative if the mass were to be displaced the other way. an amount x from this point of equilibrium. The spring force is F = − kx and Newton’s law says m d 2 x dt 2 = − kx . (17.1) If the mass strays to the right, x is positive and − kx is to the left, so as to send it back toward its equilibrium position.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
The common abbre- viation for this unit is the Hertz (Hz): 1 Hz = 1 cycle/s I N T E R A C T I V E F E A T U R E Periodic Motion and Simple Harmonic Motion | 283 Animated Figure 11.1.1 illustrates the meaning of frequency and its relationship to period. Four small balls bounce back and forth between immovable barriers. When a ball hits a barrier it reverses direction, maintaining the same speed. The numerical values of the period and frequency are shown. Use these values in Equation 11.1.1 to make sure they are consistent and correct. Animated Figure 11.1.1 Periodic motion for four different frequencies is illustrated. The period is the reciprocal of the frequency: T f 1 = / . I N T E R A C T I V E F E A T U R E Animated Figure 11.1.2 An object rotates in uniform circular motion in a vertical circle. The shadow it casts on the x axis performs simple harmonic motion. I N T E R A C T I V E F E A T U R E In Practice Problem 11.1.1, the frequency depends on the length of travel during a cycle. There is a special kind of periodic motion—simple harmonic motion—for which the frequency and period are independent of the total length of travel during a complete cycle. Simple Harmonic Motion Simple harmonic motion is a special kind of periodic motion. Here we will answer two questions—namely, what is simple harmonic motion and what causes simple harmonic motion? The first question is answered with the help of Animated Figure 11.1.2. A small object (the dark green dot) rotates in uniform circular motion in a vertical circle of radius A. With light from above, the object casts a shadow on a horizontal x axis that is parallel to the shadow’s motion and whose origin ( x 0 = ) is directly below the center of the circle. At time t 0 = , the shadow is at x A = + and the motion begins with counterclockwise rotation at angular speed ω. Activate the animation and watch the motion. (The period of this motion is 2.0 s, so the frequency is 0.50 Hz.) I N T E R A C T I V E F E A T U R E
eBook - PDF- A. Douglas Davis(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
3 THE HARMONIC OSCILLATOR 3.1 Introductio n We hav e alread y see n harmoni c oscillator s in tw o forms—th e driven , dampe d harmoni c oscillato r tha t w e solve d numericall y in Chapte r 1 an d th e simpl e harmoni c oscillato r tha t w e solve d explicitl y in Chapte r 2 usin g energ y considerations . W e shal l no w loo k a t harmoni c oscillator s in mor e detail . Harmoni c oscillator s ar e importan t for severa l reasons . First , the y occu r throughou t nature . Indeed , any motio n abou t a stabl e equilibriu m is har -moni c a s lon g a s it is small . Second , th e equation s involve d offe r challengin g yet solvabl e example s o f second-orde r differentia l equations . Third , a s w e shal l se e later , th e sam e equations tha t describ e a mechanica l bod y under oscillation s als o describ e electrica l circuits . F = ma , wher e a = d 2 x/dt 2 , is a n exampl e o f a second-orde r differentia l equation ; entir e book s an d course s ar e devote d t o th e solutio n o f suc h equations . Let' s begi n her e b y lookin g a t th e simple harmoni c oscillator— a mas s attache d t o a sprin g in a friction/es s environment . W e shal l assum e a sprin g tha t obey s Hooke' s La w tha t th e stretc h o f th e spring , x , an d th e forc e of th e spring , F , ar e proportional . W e labe l th e proportionalit y constan t k an d cal l it th e spring constant. Thus , F = mx = —kx (3.1.1 ) SECTIO N 3. 1 / INTRODUCTIO N 5 1 By definition , a second-orde r differentia l equatio n involve s takin g two derivative s o f th e position , x . Ever y tim e a derivativ e is taken , som e informatio n is los t sinc e th e derivativ e involve s onl y th e slope . Thus , th e function s f ± = x 2 an d f 2 = x 2 + 7 bot h hav e th e same derivative , 2x . If w e star t wit h th e derivativ e df/dx = 2x , an d wis h t o reconstruc t th e origina l function , w e kno w it mus t b e x 2 + C.
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