Mathematics
Harmonic Motion
Harmonic motion refers to a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium. This results in a sinusoidal motion, such as the back-and-forth movement of a pendulum or a mass on a spring. The motion can be described using trigonometric functions and is characterized by its amplitude, frequency, and phase.
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11 Key excerpts on "Harmonic Motion"
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 - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
UNIT 14: SIMPLE Harmonic Motion 467 2 4 6 8 10 y(m) Time (s) 0.20 –0.20 0.00 2 4 6 8 10 v y (m/s) Time (s) 0.00 b. To check your prediction, insert a new graph in your data collection soft- ware and plot the velocity as a function of time (using the data you already collected). How does the actual velocity-time graph compare to your prediction? How does the shape of the velocity-time graph com- pare to that of the position-time graph? Do you have any ideas how these two graphs might be related? Note: Once again, don’t clear your data as we will be using it again. You may be surprised by the similarity between the position-time graph and the velocity-time graph. The shapes of the two graphs look almost identical, except that one is shifted with respect to the other. To understand this similarity, we need to take a closer look at how to describe such graphs mathematically. 14.4 THE MATHEMATICS OF SIMPLE Harmonic Motion Simple Harmonic Motion is defined as any periodic motion in which the dis- placement from an equilibrium position varies sinusoidally in time. In other words, the displacement as a function of time can be described using either a 468 WORKSHOP PHYSICS ACTIVITY GUIDE sine or cosine function. 2 If the y-axis is chosen to be along the line of motion, a general sinusoidal equation describing the displacement y(t) can be written in the form 3 y(t) = Y cos ( t + 0 ) (14.2) where Y is the amplitude of the oscillation (the maximum displacement from equilibrium), its rotational frequency, and 0 its initial phase. Definition: Rotational frequency It is convenient to introduce a rotational frequency, , which is related to the (normal) frequency f via: = 2f = 2 T ( rad∕s ) (14.3) where has units of rad∕s, as compared to f , which has units of 1∕s, or Hz. Because the argument of a cosine (or sine) function goes through 2 radians in one complete cycle, it is the rotational frequency = 2f that appears in the argument of the function.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
(15-2) 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 as x(t) = x m cos(ωt + ϕ) (displacement), (15-3) in which x m , ω, and ϕ are quantities that we shall define. Freeze-Frames. Let’s take some freeze-frames of the motion and then arrange them one after another down the page (Fig. 15-2a). Our first freeze-frame is at t = 0 when the particle is at its rightmost position on the x axis. We label that coordi- nate as x m (the subscript means maximum); it is the symbol in front of the cosine +x m –x m x 0 Figure 15-1 A particle repeatedly oscillates left and right along an x axis, between extreme points x m and −x m . 415 15-1 SIMPLE Harmonic Motion 0 +x m –x m t = 0 t = T/4 t = T/2 t = 3T/4 t = T 0 +x m –x m (a) (c) x x m 0 Displacement Time (t) (d) –x m 0 +x m –x m t = 0 t = T/4 t = T/2 t = 3T/4 t = T T 0 +x m –x m (b) v v v v 0 x m –x m 0 T/2 T x x m 0 Displacement Time (t) (e) –x m A particle oscillates left and right in simple Harmonic Motion. Rotating the figure reveals that the motion forms a cosine function. This is a graph of the motion, with the period T indicated. The speed is zero at extreme points. The speed is greatest at x = 0. The speed is zero at the extreme points. The speed is greatest at the midpoint. Figure 15-2 (a) A sequence of “freeze-frames” (taken at equal time intervals) showing the position of a particle as it oscillates back and forth about the origin of an x axis, between the limits +x m and −x m . (b) The vector arrows are scaled to indicate the speed of the particle. The speed is maximum when the particle is at the origin and zero when it is at ± x m .
eBook - PDFTheoretical Mechanics for Sixth Forms
In Two Volumes
- C. Plumpton, W. A. Tomkys(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
Motion which is defined by this equation is called Simple Har-monic Motion (SHM). The position defined by z = 0 is called the centre of the motion. For a particle moving in a straight line the equation of SHM is x = -co 2 x, (14.1a) where x is the displacement of the particle from a fixed point in the line. From this equation we have mx = —τηω 2 χ 9 where m is the mass of the particle, and hence we obtain an alternative form of the definition of SHM for a particle moving in a straight line. The motion of a particle which moves in a straight line under the action of aforce which is directly proportional to the displacement of the particle from a fixed point in the line and which is directed towards that point is defined as Simple Harmonic Motion. EXERCISES 14.1. In each of the following questions obtain the equation of motion for the body and show that the motion is SHM. 1. A particle of mass m is attached to one end of a light elastic string of natural length / and modulus 2mg. The string is fixed at the other end and the particle hangs in equilibrium. The particle is pulled down so that the extended length of the string is less than 2/ and then released. 2. A smooth hollow cylinder, of internal radius r, is fixed with its axis horizontal. A smooth particle of mass m can slide on the inside of the cylinder and makes small oscillations in a vertical plane about the lowest generator. SIMPLE Harmonic Motion 337 3. Figure 14.4 shows a smooth ring P of mass m which is free to slide on a wire AB. This ring is attached by an elastic string, of natural length |/and modulus mg y to a fixed point O in the same horizontal plane as the wire and distant / from it and slightly dis-placed from its equilibrium position N. 4. A particle of mass m rests on a smooth plane inclined at an angle a to the horizon-tal. The particle is attached by means of a light inextensible string of length / which lies along a line of greatest slope to a fixed point of the plane.
- 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- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
C H A P T E R 6 Oscillatory Motion 6.1 Introduction In Chapter 3 the fundamental laws of mechanics were described in mathematical form. We now seek to apply these laws to an elementary, but quite important, physical problem—the harmonic oscillator. Problems involving harmonic oscillations occur repeatedly in classical physics and in quantum mechanics. The results presented in this chapter therefore find wide application in many fields of physics and engineering. For simplicity we shall first consider in some detail the motion of a particle that is constrained to move in one dimension. We take the origin to be located at the position of equilibrium of the particle, and we assume that if the particle is displaced from this position, then there exists some force which acts to restore the particle to its equilibrium position. Now, we shall assume that the function which describes this force possesses continuous derivatives of all orders so that the function may be expanded in a Taylor series : 128 6.1 INTRODUCTION 129 The term F 0 is the value of the force at the equilibrium position, which, by the definition of equilibrium, must vanish. Then, if we confine our attention to displacements of the particle that are sufficiently small, we may neglect all terms involving x 2 and higher powers of x. We have, therefore, the approximate relation F(x) = -kx (6.2) where we have substituted k = —(dF/dx) 0 . Since the restoring force is always directed toward the equilibrium position (the origin), the derivative (dF/dx) 0 is negative and therefore k is a positive constant. Physical systems which can be described in terms of Eq. (6.2) are said to obey Hooke's law.* A large class of physical processes which can be treated by applying Hooke's law are those involving elastic deformations. As long as the displacements are small and the elastic limits are not exceeded, a linear restoring force can be used for problems of stretched springs, elastic strings, bending beams, etc.
- Christof M. Aegerter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
In the case of idealized pendulums, the differential equation can also be deduced directly from the equations of motion, which we shall discuss in Section 6.2. The total energy of a harmonic oscillating system is constant, since the solutions found here describe idealized systems in which the masses swing on the springs without stopping or where the pendulum continues forever. In reality, all oscillations will stop at some point if they are not actively kept in motion. What happens in this case we will look at later in Section 3.8.1. 3.3 Describing Any Oscillation in Terms of Harmonic Ones: Fourier Series The oscillations we initially saw in Section 3.2 were only characterized by their periodicity. However, we have seen that harmonic vibrations are easier to describe, since we need much 74 Motions and Oscillations fewer parameters for a complete description of the motion. Therefore, it would be nice if we could capture all periodic movements in the same style, namely, as a sum of harmonic (sine and cosine) functions. This is actually possible and has been mathematically formulated by Joseph Fourier. Thus mathematically, it can be said that every periodic function can be interpreted as the sum of harmonics or, more specifically and precisely, if u(t) is a periodic function with a period T, i.e., u(t + T) = u(t), then we find the following: u(t) = ∞ n=0 (A n cos(ω n t) + B n sin(ω n t)) where ω n = 2π n T If we know the Fourier coefficients A n and B n , of which in general there can of course be infinitely many, we directly know u(t). Practically, however, only a small number of Fourier coefficients are necessary to give a reasonable description for most functions, one encounters.
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 - 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 - PDF- Patrick Hamill(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
Part IV Oscillations and Waves 11 Harmonic Motion This chapter is an in-depth study of Harmonic Motion. You are familiar with two examples of simple harmonic oscillators, namely, a pendulum and a mass on a spring. Now you will study several different kinds of Harmonic Motion, including underdamped, overdamped, and critically damped oscillatory motion, as well as forced Harmonic Motion and the motion of coupled harmonic oscillators. The techniques you will learn here are used in nearly every branch of physics. 11.1 Springs and Pendulums A very simple oscillatory system consists of a mass m connected to a spring of constant k. To avoid complications due to forces such as gravity or friction, assume the mass is on a perfectly frictionless horizontal surface and can slide back and forth, as shown in Figure 11.1. The kinetic energy of the mass is T = 1 2 m ˙ x 2 and the potential energy of the spring is V = 1 2 kx 2 . Therefore, as described in Section 4.3 the Lagrangian for the system is L = T − V = 1 2 m ˙ x 2 − 1 2 kx 2 . The equation of motion is obtained from Lagrange’s equation: d dt ∂L ∂ ˙ x − ∂L ∂x = 0, which yields ¨ x + k m x = 0. (11.1) Another simple oscillatory system is a pendulum consisting of a particle of mass m (the bob) on an inextensible, massless string of length l . See Figure 11.2. The kinetic energy is Figure 11.1 A mass on a frictionless surface connected to a spring. The distance x is measured from the equilibrium position to the center of mass of the block. k x m 284 11 Harmonic Motion q l Figure 11.2 A simple pendulum. T = 1 2 m( ˙ x 2 + ˙ y 2 ), where x = l sin θ and y = −l cos θ (assuming the origin is located at the point where the string is attached to the support). T = 1 2 m(l 2 ˙ θ 2 cos 2 θ + l 2 ˙ θ 2 sin 2 θ) = 1 2 ml 2 ˙ θ 2 . The potential energy is V = −mgl cos θ . Consequently, see Equation (4.4), L = T − V = 1 2 ml 2 ˙ θ 2 + mgl cos θ . The Lagrange equation of motion is: d dt ∂L ∂ ˙ θ − ∂L ∂θ = 0.
- H. D. Ram, A. K. Chauhan(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
9 MECHANICAL VIBRATION 9.1 Introduction Vibration is a very common phenomenon in machines. In certain situations, the control of vibration is very essential. To control the vibration, proper understanding of the phenomenon is desired. The essence of vibration is discussed below. A mechanical system is said to vibrate if it executes a simple Harmonic Motion or oscillation. A simple Harmonic Motion is given below: x A t = + ( ) sin ω α (9.1) where, x is the displacement from mean position, A is amplitude, ω is frequency and ω α t + ( ) is phase, ϕ . 9.2 Characteristics of simple Harmonic Motion (a) Amplitude: Magnitude of maximum displacement is called amplitude of displacement. In equation 9.1, the amplitude of displacement is A . Displacement varies from +A to –A . (b) Velocity: Velocity of a point in simple Harmonic Motion (SHM) is give by: dx dt A t = + ( ) ω ω α cos For the sake of simplicity of writing, dx dt will be written as dotnosp x . So, dotnosp x A t = + ( ) ω ω α cos 486 | Foundations and Applications of Engineering Mechanics The amplitude of velocity is A ω and it varies from A ω to -A ω . (c) Acceleration: Acceleration of a point in SHM is given by: d x dt A t 2 2 2 = -+ ( ) ω ω α sin For the sake of simplicity of writing, d x dt 2 2 will be written as dotnospdotnosp x . So, dotnospdotnosp x A t = -+ ( ) ω ω α 2 sin The amplitude of acceleration is A ω 2 and it varies from A ω 2 to -A ω 2 . (d) Phase angle ( ϕ ): Phase angle is given by: ϕ ω α = + t or, d dt ϕ ω = So, the Phase velocity is ω rad/s. Initial phase = α Following table shows the displacement, velocity and acceleration of particle executing SHM for special values of phase angle. ϕ 0 π/ 2 π 3 π/ 2 2 π x 0 A 0 -A 0 dotnosp x A ω 0 -A ω 0 A ω dotnospdotnosp x 0 -A ω 2 0 A ω 2 0 9.3 Differential equation governing the simple Harmonic Motion Differentiating the equation 9.1 twice with respect to t , we get d x dt A t 2 2 2 = -+ ( ) ω ω α sin d x dt x 2 2 2 = -ω
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