Physics
Periodic Motion
Periodic motion refers to the repetitive back-and-forth movement of an object around a central position. This type of motion follows a regular pattern, with the object returning to its original position after a fixed time interval. Examples of periodic motion include the swinging of a pendulum, the vibrations of a guitar string, and the oscillations of a spring.
Written by Perlego with AI-assistance
Related key terms
1 of 5
9 Key excerpts on "Periodic Motion"
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
In Chapter 5, we studied uniform circular motion, in which a centripetal force creates a centripetal acceleration that is always directed toward the center of the circular path. Circular motion is one case of Periodic Motion, which is a motion that repeats itself. Here in Section 11.1, we draw a connection between circular motion and a special kind of Periodic Motion called simple harmonic motion. Periodic Motion Periodic Motion is a motion that is repeated at equal time intervals. The interval of time required for one complete cycle of the motion is called the period: 11.1 Periodic Motion AND SIMPLE HARMONIC MOTION Learning Objectives Definition of Period When a motion repeats itself at equal time intervals, the period T is the time required for one complete cycle of the motion. A cycle is a complete execution of a periodically repeated phenomenon. The SI unit of period is seconds (s). [Sometimes we say seconds per cycle, but cycles (being a pure number having no dimensions) are omitted in calculations.] Examples of Periodic Motion include the rotation of the Earth, with a period of 24 h; the vibrations of the tines of a “middle C” tuning fork with a period of 3.822 ms; and the opening and closing of an atrioventricular valve in a beating heart, with a period of 1.0 s (corresponding to a pulse of 60 beats per minute). Concept Check 11.1.1 contains an animation of Periodic Motion that will teach you how to identify the period. Periodic events are also characterized by their frequency, which is simply the recipro- cal of the period: Definition of Frequency When a motion repeats itself at equal time intervals, the frequency f of the motion is the number of complete cycle per unit time interval, and is the reciprocal of the period: f T 1 = (11.1.1) The SI unit of frequency is inverse seconds (s 1 − ) or cycles per second. 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
eBook - PDFApplied Structural and Mechanical Vibrations
Theory and Methods, Second Edition
- Paolo L. Gatti(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
In practical cases, it is often the ability to reproduce the data by a controlled experiment that pro-vides a general criterion to distinguish between the two. 1.3 SOME DEFINITIONS AND METHODS As stated in the introduction, the particular behaviour of a particle, a body or a complex system that moves about an equilibrium position is called oscil-latory motion. It is then natural to try a description of such a particle, body or system by an appropriate function of time x ( t ), whose physical meaning depends on the scope of the investigation and, as often happens in practice, on the available measuring instrumentation: it might be displacement, veloc-ity, acceleration, stress or strain in structural dynamics; pressure or density in acoustics; current or voltage in electronics or any other time-varying quantity. A function that repeats itself exactly after certain intervals of time is called periodic . The simplest case of Periodic Motion is called harmonic (or sinusoidal) and is mathematically represented by a sine or cosine function; for example x t X t ( 29 = -( 29 cos ω θ (1.4) where: X is the maximum , or peak amplitude (in the appropriate units) ω t − θ is the phase angle (in radians) ω is the angular frequency (in rad/s) θ is the initial phase angle (in radians), which, in turn, depends on the choice of the time origin and can be taken equal to zero if there is no rela-tive reference to other sinusoidal functions The time between two identical conditions of motion is the period T . It is measured in seconds and is the inverse of the frequency ν = ω /2 π , which is expressed in hertz (Hz, with dimensions of s −1 ). As is probably well known to the reader, frequency represents the number of cycles per unit time and for the harmonic function (Equation 1.4) we have the relations ω πν π ω = = = 2 1 2 , T ν (1.5) A plot of Equation 1.4, amplitude versus time, is illustrated in Figure 1.1 where the peak amplitude is taken as X = 1 and the initial phase angle is θ = 0.
- Christof M. Aegerter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
67 Periodic Motions: Oscillations ✲ t ✻ t E v(t) v 0 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✠ v 0 − a 0 t t 0 t ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏✏ ↓ Max. height Figure 3.5 Velocity profile for a constant acceleration. The dashed area underneath the curve corresponds the the distance traveled between time t 0 and time t . t E marks the end of the motion, which for the armadillo in Figure 3.1 is when it reaches ground again. The jumping and falling heights are the same, which implies that the areas of the triangles above and below the axis are equal. determine the integral graphically. As Figure 3.5 shows, the area underneath the curve for the velocity corresponds to the travelled path. 3.2 Periodic Motions: Oscillations An oscillation is a very special form of movement, namely one which repeats itself after a certain time. This time after which the motion repeats itself is called the period of the oscillation and is the most important variable that describes an oscillation. For some types of oscillation, harmonic oscillations, this period, together with the amplitude, i.e., the distance that the object moves maximally, is sufficient for a complete description of the movement. In general, we speak of an oscillation when a physical quantity changes in time around an average, resting value. These general oscillations can also have increasing or decaying (damped) amplitudes in time; see Section 3.8.1. In that case, they are no longer strictly periodic. Oscillations can run by themselves or be forced externally, which we will discuss in depth in Section 4.2. Further examples of oscillating systems are found in the central and inner ear, where the eardrum, the hammer, and the basilar membranes begin to vibrate according to the 68 Motions and Oscillations arriving sound waves, thus allowing the sensors in the ear to send appropriate signals to the brain.
eBook - PDF- Stephen Lee(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Simple harmonic motion Backwards and forwards half her length With a short uneasy motion. Samuel Coleridge Taylor The Rime of the Ancient Mariner The guitar, the clock and water ripples all involve oscillations or vibrations of particles or bodies. ● In the guitar, the strings vibrate in a controllable way, and the instrument transmits them as sound waves (vibrations of air molecules). ● In the clock, the pendulum oscillates in the familiar swinging pattern, and this regular motion is used to operate the clock mechanism. ● Ripples are created when the surface of water (or another liquid) is disturbed. The fluid particles vibrate up and down in a regular wave pattern. The same pattern is visible in ocean waves, or in the wakes of boats. The remarkable thing about all of these vibrations, and many others that occur in natural and man-made systems, is that they are essentially of the same form. The vibrations don’t go on for ever, but over a reasonable interval, you can plot the displacement of a vibrating particle against time for any of these systems and you will obtain a sine wave. 14 14.1 Oscillating motion The graph in figure 14.1 shows the displacement of an oscillating particle against time. Figure 14.1 From the graph you can see a number of important features of such motion. ● The particle oscillates about a central position , O. ● The particle moves between two points with displacements a and a . The distance a is called the amplitude of the motion. ● The motion repeats itself in a cyclic fashion. The number of cycles per second is called the frequency , and is usually denoted by , the Greek letter ‘nu’. ● The motion repeats itself after a time T. The time interval T is called the period : it is the time for one complete cycle of the motion. The frequency and period are reciprocals. For example a period of 1 1 0 of a second corresponds to a frequency of 10 cycles per second.
eBook - PDFAperiodic Structures in Condensed Matter
Fundamentals and Applications
- Enrique Macia Barber(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
1 Orderings of matter 1.1 Periodic thinking in physical sciences The notion of periodicity allows one to easily grasp the basic order underlying certain patterns and rhythms in Nature. The essence of periodicity relies on a basic motif which is inde…nitely repeated, along with a set of basic rules prescribing the way such a repetition process takes place. Periodicity can occur in time, space, or simultaneously in both of them. Periodicity in time guarantees that what is known to occur now will also occur later, and can be asserted to have already occurred before, provided that a certain relationship between those di/erent instants is ful…lled. Let t be a real number measuring the passage of time. Then, a function satisfying the condition f ( t T ) = f ( t ) is periodic in time with a period T , since its value is preserved (i.e., it is invariant) under transformations describing the set of translations generated back and forth by the arrow of time by the real number T . The existence of cyclic processes in Nature accurately obeying such a pe-riodicity condition is the basis for the possible adoption of physical clocks (characterized by their T value). In fact, from the galactic scale down to atomic and subatomic scales, the natural world has plenty of physical sys-tems exhibiting nearly exact periodicity in time. Most of these systems can be described, at least as a …rst approximation, in terms of dynamic equations of the form d 2 f dt 2 + ! 2 f =0 ; (1.1) which is usually referred to as the harmonic oscillator equation, where f is some physical magnitude (e.g., a position coordinate, the intensity of an elec-tric or magnetic …eld, or the chemical concentration of a substance) and !; the so-called natural frequency, is a quantity which depends on characteristic physical parameters of the system.
eBook - PDF- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
1 Oscillations and Waves 1.1 Introduction Objects subjected to restoring forces when displaced from their normal positions and released, perform to and from or vibrating motions. They move back and forth along a path, repeating over and over again, a series of motions. Such motion of constant frequency is called Periodic Motion or harmonic motion and objects performing such type of motion are called harmonic oscillators. In this book, it is tacitly assumed that there is a linear relationship between force and displacement; frequency remains constant throughout the motion. In real systems however, the linear behavior, implicit in simple harmonic motion, is rarely obeyed. If the frequency of the oscillatory system is not constant, then it is called anharmonic motion – its study is beyond the scope of this book due to its mathematical complexities. In oscillatory systems, it is not necessarily the bodies themselves who execute oscillations; bodies may be at rest. If the physical properties of a system undergo changes in an oscillatory manner, the system will also be called an oscillatory system. The electromagnetic energy transfer between the capacitor and inductor in a tank circuit used in electronic gadgets, variation of pressure in air due to propagation of sound waves, vibration of the diaphragm of a speaker in sound systems, flow of alternating current, variation of electric and magnetic vectors during propagation of electromagnetic waves, etc., are examples of oscillatory systems. 1.1.1 Parameters of an oscillatory system i. Mean position The position of the oscillating body when there is no oscillation is called the mean position or equilibrium position. This is the rest position of the oscillating body. 2 Principles of Engineering Physics 1 ii. Amplitude ( r ) It is the absolute value of the maximum displacement of the oscillating particle from its mean position or equilibrium position.
eBook - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Name Section Date UNIT 14: SIMPLE HARMONIC MOTION Alaettin YILDIRIM/Shutterstock.com The swing of this antique clock pendulum is one of many examples of what physicists call simple harmonic motion. Due to the pendulum motion, each tick of the clock takes the same amount of time, and so it serves as a reliable timekeeper. And if the clock is running slow or fast, one can adjust the pendulum to compensate. By the end of this unit, you should have an understanding of why a pendulum undergoes simple harmonic motion. 460 WORKSHOP PHYSICS ACTIVITY GUIDE UNIT 14: SIMPLE HARMONIC MOTION OBJECTIVES 1. To learn about the quantities used to describe periodic (and harmonic) motion, such as period, frequency, amplitude, and phase. 2. To understand the basic properties of simple harmonic motion, in which the displacement of an object varies sinusoidally in time, and to experi- mentally measure such motion in different systems. 3. To use Newton’s laws to theoretically explore the factors that influence the motion of both a mass-spring system and a simple pendulum. 14.1 OVERVIEW Any motion that repeats itself regularly is known as Periodic Motion. The pendulum in a grandfather clock, molecules in a crystal, the vibrations of a car after it encounters a pothole on the road, and the rotation of Earth around the sun are examples of Periodic Motion. In this unit we will be especially interested in a type of Periodic Motion known as simple harmonic motion, which is often abbreviated as SHM. 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).
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 8 Maimonides Harmonic Motion Cherish striving for wisdom more than wisdom itself 8.1 Introduction Periodic Motions and processes occur on all scales, from the microscopic to the galactic, including biological systems. Harmonic motion is a special, rather restricted case of Periodic Motion, but of great importance. We begin by discussing periodic, though not necessarily harmonic, motion. We then narrow and deepen our treatment and treat simple, damped and forced harmonic motion. For simplicity we restrict ourselves to one dimension. Periodic Motion is represented by a displacement which is a periodic function fit) of time, satisfying the periodicity condition f(t + T) = f(t) (8.1) where the period T is the shortest time for which (8.1) is satisfied. Of course, integer multiples of T also satisfy (8.1). Examples of periodic functions are: x(t) = Asm(m + 0), x(t) = Acos 2 (a>t) + B, Asin(cot+6), Acos 2 (cot) + B, 5tan(30+2sin(7r) +12cos(9f) + 6 The diagram shows a couple of examples of periodic functions, often arising in electronics. If a periodic function f(f) is continuous and differentiable, then by (8.1) all its derivatives fit) Ait) are also periodic, thus fit) = ft+T), etc. Therefore, if / is a displacement periodic in time, the velocity and the acceleration are also periodic. Example 1 The period. Find the period of the functions x{t) = Asinicot+8) and Acos 2 icot)+B, where A, co , 6 and B are known constants. 243 244 Classical and Relativistic Mechanics Solution Asin((o(t + T) + 0) = Asm[(6)t + 0) + (dr] = Asm(cDt + 0) by (8.1). Since the sine function has period 2K rad, we have coT= 2K and T = 2x1 co (8.2) independent of A and 6 (called the amplitude and phase, respectively, to be discussed later). For the function Acos 2 (cot)+B, clearly B is just a shift in the value of the function, without influencing the period. Use the identity cos 2 a = (l+cos2a)/2, and let a = cot.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Taller and shorter buildings remained standing. John T. Barr/Getty Images, Inc. 434 CHAPTER 15 OSCILLATIONS Frequency The frequency f of periodic, or oscillatory, motion is the number of oscillations per second. In the SI system, it is mea- sured in hertz: 1 hertz = 1 Hz = 1 oscillation per second = 1 s −1 . (15-1) Period The period T is the time required for one complete oscil- lation, or cycle. It is related to the frequency by T = 1 f . (15-2) Simple Harmonic Motion In simple harmonic motion (SHM), the displacement x(t) of a particle from its equilibrium position is described by the equation x = x m cos(ωt + ϕ) (displacement), (15-3) in which x m is the amplitude of the displacement, ωt + ϕ is the phase of the motion, and ϕ is the phase constant. The angular fre- quency ω is related to the period and frequency of the motion by ω = 2 π T = 2 π f (angular frequency). (15-5) Differentiating Eq. 15-3 leads to equations for the particle’s SHM velocity and acceleration as functions of time: v = −ωx m sin(ωt + ϕ) (velocity) (15-6) and a = −ω 2 x m cos(ωt + ϕ) (acceleration). (15-7) In Eq. 15-6, the positive quantity ωx m is the velocity amplitude v m of the motion. In Eq. 15-7, the positive quantity ω 2 x m is the accel- eration amplitude a m of the motion. The Linear Oscillator A particle with mass m that moves under the influence of a Hooke’s law restoring force given by F = −kx exhibits simple harmonic motion with ω = √ k m (angular frequency) (15-12) and T = 2π √ m k (period). (15-13) Such a system is called a linear simple harmonic oscillator. Energy A particle in simple harmonic motion has, at any time, kinetic energy K = 1 2 mv 2 and potential energy U = 1 2 kx 2 . If no fric- tion is present, the mechanical energy E = K + U remains con- stant even though K and U change. Review & Summary Pendulums Examples of devices that undergo simple harmonic motion are the torsion pendulum of Fig. 15-9, the simple pendulum of Fig. 15-11, and the physical pendulum of Fig.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
