Physics
Simple Harmonic Motion
Simple Harmonic Motion refers to the repetitive back-and-forth movement exhibited by certain systems, such as a mass on a spring or a pendulum. It is characterized by a restoring force that is directly proportional to the displacement from the equilibrium position, resulting in a sinusoidal motion. The period and frequency of the motion are determined by the system's mass and the stiffness of the spring.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "Simple Harmonic Motion"
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- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
173 © 2010 Taylor & Francis Group, LLC Simple Harmonic Motion Two kinds of motion, linear and rotational, have been studied so far. Another type, known as the Simple Harmonic Motion (SHM), where the acceleration of the object is not constant is of special interest. The acceleration of an object executing SHM is proportional and opposite in direction to the displacement of the object from its equilibrium position. This is because the force acting on the moving object is proportional, but opposite to its displacement. The simple pendulum and the oscil-lation of a block, attached to an ideal spring, once displaced slightly from the equilibrium position are examples of SHM. 9.1 HOOKE’S LAW Hooke’s law demonstrates the response of linear elastic media when acted upon by a force. Springs are a good example. The situation is demonstrated in the following where a spring of a natural length ℓ (Figure 9.1a), attached to a rigid support at one end, has a mass m attached to its other end. The mass is pulled a distance x by a force F EXT (Figure 9.1b). As long as the displacement is within the spring’s elastic limit, the spring exhibits a self-created force F SP , equal in magnitude but opposite in direction to the applied force, F EXT . Once F EXT is removed, the spring force becomes the only force acting on m, pulling it back toward its original position, O. That is why F SP is called the restor-ing force . In all cases of a mass-spring system, the spring is assumed to be of a negligible mass. Hooke’s law established that the external force F EXT needed to stretch a spring an amount x is F EXT = kx (9.1) and the spring force is F SP = − kx, (9.2) 9 F EXT F SP (b ) x + x m Equilibrium position O O (a) + x Equilibrium position m uni2113 FIGURE 9.1 (a) Illustration of a horizontal spring of natural length ℓ attached to a rigid support on one end and mass m attached to its other end. (b) The mass is pulled a distance x from O by a force F EXT .
- Joseph C. Amato, Enrique J. Galvez(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
A body undergoing simple SHM, or sinusoidal oscillation , experiences a linear restoring force proportional to its displacement x from its stable equilibrium position: F x ( t ) = − kx ( t ). In either case, the motion of the body is periodic and can be described by trigonometric functions. For SHM, x ( t ) = A cos( ω t + θ 0 ) or equivalently x t A t ( ) sin( ) = + ′ ω θ 0 , where A is the amplitude of the oscillation and ω is the angular frequency. The initial phase angle θ θ π 0 0 2 ( / = ′ + ) is determined by the position x 0 and velocity v 0 of the body at t = 0. Periodic systems that are not inherently sinusoidal (such as the pendulum) behave like simple harmonic oscillators in the limit of small displace -ments from equilibrium. The mathematics of the mass–spring system or the pendulum is applicable to all sinusoidal oscilla -tors, on all time and length scales, throughout the universe. Whenever a system’s differential equation of motion takes the form a x = d 2 x / dt 2 = − C 2 x , the system undergoes SHM, x ( t ) = A cos( ω t + θ 0 ) with angular frequency ω = C . For the one-body mass–spring system, ω = k m / , while for the pendulum ω = g l / . For the two-body mass–spring oscillator, the equations of motion can be combined to form a new differential equation that predicts SHM for the reduced mass μ = m 1 m 2 /( m 1 + m 2 ) with frequency ω µ = k / . This pro -cedure can be extended to systems with much greater complexity. The sensation of weightlessness led Einstein to propose his Equivalence Principle, and from it, the General Theory of Relativity. Using the principle, we found that a gravitational field deflects light, and also that it causes clocks (time) at different altitudes to advance at different rates, an effect clearly seen in the orbiting atomic clocks of the Global Positional System. In the Principia , Isaac Newton took great pains to define his scientific terms precisely—with several notable exceptions.
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- 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 .
- 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 - ePub- H. John Pain(Author)
- 2013(Publication Date)
- Wiley(Publisher)
1
Simple Harmonic Motion
At first sight the eight physical systems in Figure 1.1 appear to have little in common.1.1(a) is a simple pendulum, a mass m swinging at the end of a light rigid rod of length l . 1.1(b) is a flat disc supported by a rigid wire through its centre and oscillating through small angles in the plane of its circumference. 1.1(c) is a mass fixed to a wall via a spring of stiffness s sliding to and fro in the x direction on a frictionless plane. 1.1(d) is a mass m at the centre of a light string of length 2l fixed at both ends under a constant tension T . The mass vibrates in the plane of the paper. 1.1(e) is a frictionless U-tube of constant cross-sectional area containing a length l of liquid, density ρ , oscillating about its equilibrium position of equal levels in each limb. 1.1(f) is an open flask of volume V and a neck of length l and constant cross-sectional area A in which the air of density ρ vibrates as sound passes across the neck. 1.1(g) is a hydrometer, a body of mass m floating in a liquid of density ρ with a neck of constant cross-sectional area cutting the liquid surface. When depressed slightly from its equilibrium position it performs small vertical oscillations. 1.1(h) is an electrical circuit, an inductance L connected across a capacitance C carrying a charge q . All of these systems are simple harmonic oscillators which, when slightly disturbed from their equilibrium or rest postion, will oscillate with Simple Harmonic Motion. This is the most fundamental vibration of a single particle or one-dimensional system. A small displacement x from its equilibrium position sets up a restoring force which is proportional to x acting in a direction towards the equilibrium position.Thus, this restoring force F may be writtenwhere s , the constant of proportionality, is called the stiffness and the negative sign shows that the force is acting against the direction of increasing displacement and back towards the equilibrium position. A constant value of the stiffness restricts the displacement x
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
What causes Simple Harmonic Motion? We start by considering the velocity and acceleration of an object in Simple Harmonic Motion, which is illustrated in Animated Figure 11.1.4. Start the animation and watch carefully. Animated Figure 11.1.4 The green arrow represents the velocity and the purple arrow represents the acceleration. The acceleration is always directed toward the origin, where x 0 = . I N T E R A C T I V E F E A T U R E I N T E R A C T I V E F E A T U R E Velocity and Acceleration in Simple Harmonic Motion | 285 When the object (the green dot) is at x A = − , its instantaneous velocity is zero and it is accelerating in the positive x direction, reaching its highest speed at x 0 = , at which point the acceleration is momentarily zero. When the object is at x A = + , its instantaneous velo- city is zero and it is accelerating in the negative x direction, once again reaching its highest speed at x 0 = . The acceleration (and therefore the net force) is always pointed toward the origin. At the position x 0 = , the net force is zero. This position is called the equilibrium position, and a force that always acts to push an object back toward an equilibrium pos- ition is called a restoring force. Simple Harmonic Motion can be defined in terms of the restoring force that causes it: Simple Harmonic Motion Let the net force on an object be proportional to the magnitude of its displacement from an equilibrium position. If the direction of this force is always towards the equilibrium position, then the resulting motion will be Simple Harmonic Motion. If the motion is parallel to the x axis with the equilibrium position at x 0 = , then the force can be written F Cx = − (11.1.4) where C is a proportionality constant. The proportionality constant C has SI units of newtons per meter (N m / ) and it depends on what causes the force. There are many forces in nature that are approximated well by Equation 11.1.4.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
(c) Express A and 0 in terms ofx 0 and V 0 . (d) Find the time dependence of the force acting on the body and relate it to the displacement. (e) Show that a necessary and sufficient condition for SHM is that the force and the displacement are related by F = -Cx y where C is a positive constant {such a force is called a restoring force), (f) Express co and T in terms of C and m. Chapter 8 Harmonic Motion 245 Solution (a) These are given by (8.2), (8.4) and (8.5). (b) We substitute t = 0 into (8.3) and (8.4): x 0 = Acos(<9), V 0 = coAsin(9). (c) Dividing V 0 by x 0 we get 6 = arctan(Vo/o)x 0 ) and A 2 = x 0 2 + (VVco) 2 . (d) By the second law and (8.5) we have F(i) = ma(t) = -m co 2 Acos(cot + 0) = -mco 2 x(t), so that F(x) = -ma> 2 x. (e) This is clear from part d, where the constant C - mco 2 > 0. Thus SHM is characterized by a restoring force acting on the body which is linear in its displacement and opposes it. (f) From part e and (8.2), 0) = y /C/m a nd T = iK^mlC (8.6) Example 3 Differential equation for SHM. A vast array of physical phenomena are governed by differential equations. Deriving the differential equation for SHM will provide us with added insight and a clue to the origin of the harmonic functions which characterize the time dependence of the displacement and its derivatives. The prototype for deriving the differential equation is the simple system we have met before, namely, a block of mass m attached to a spring of force constant k and having displacement x(t) from the equilibrium position. There are two ways to derive the differential equation.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The back-and-forth motion illus- trated in the drawing then repeats itself, continuing forever, since no friction acts on the object or the spring. *As we will see in Section 10.8, Equation 10.2 is similar to a relationship first discovered by Robert Hooke (1635–1703). Unstrained length of the spring x x A B x C F x F x F x Figure 10.4 The restoring force F x (see blue arrows) produced by an ideal spring always points opposite to the displacement x (see black arrows) of the spring and leads to a back-and-forth motion of the object. 10.1 | The Ideal Spring and Simple Harmonic Motion 225 When the restoring force has the mathematical form given by F x 5 2kx, the type of friction-free motion illustrated in Figure 10.4 is designated as “Simple Harmonic Motion.” By attaching a pen to the object and moving a strip of paper past it at a steady rate, we can record the position of the vibrating object as time passes. Figure 10.5 illustrates the resulting graphical record of Simple Harmonic Motion. The maximum excursion from equi- librium is the amplitude A of the motion. The shape of this graph is characteristic of Simple Harmonic Motion and is called “sinusoidal,” because it has the shape of a trigonometric sine or cosine function. The restoring force also leads to Simple Harmonic Motion when the object is attached to a vertical spring, just as it does when the spring is horizontal. When the spring is vertical, however, the weight of the object causes the spring to stretch, and the motion occurs with respect to the equilibrium position of the object on the stretched spring, as Figure 10.6 indicates. The amount of initial stretching d 0 due to the weight can be calculated by equat- ing the weight to the magnitude of the restoring force that supports it; thus, mg 5 kd 0 , which gives d 0 5 mg/k. Check Your Understanding (The answers are given at the end of the book.) 1.
eBook - ePub- Robert H. Randall(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
CHAPTER 1FUNDAMENTAL PARTICLE VIBRATION THEORY
The production of sound always involves some vibrating source. Such a source is often of irregular shape, and rarely do all parts of the vibrating surface move as a unit. It is the very complexity of the vibration of a sound source that makes it necessary to consider first the simplest vibrating body, the particle . The motion of actual sources may approximate that of a particle, particularly at low frequencies. Whenever this approximation may not be made, the vibrating surface may be broken up into smaller areas, infinitesimal if desired, the sum effect of which is equivalent to that of the total surface area of the actual source. The mathematics of this summation may be extremely complicated, but approximations will often lead to useful results.1-1 Simple Harmonic Motion of a particle.
Simple Harmonic Motion originates, in mechanics, because of the existence of some kind of unbalanced elastic force. With such a force, Newton’s second law becomes, for a particle of mass m, free to move along the x -axis,(1–1)In the expression on the right for the force, K is called the elastic constant, and the negative sign indicates that the restoring force always acts towards the origin. Equation (1-1) may also be written(1–2)where ω 2 = K/m. This differential equation completely defines the type of motion and from it all other properties of Simple Harmonic Motion may be obtained. By integrating Eq. (1-2) twice, the displacement equation may be shown to be of the form(1–3)where x m . is the amplitude of the motion and α is called the phase angle. The quantities x m and α are essentially constants of integration, whose values depend upon the mathematical boundary conditions. They may easily be determined, for instance, if one knows the value of x and of the velocity, , at either the time t = 0, or at any other specific value of the time. Whether the cosine or the sine function appears in Eq. (1-3) is dependent upon these boundary conditions. If, for instance, α turns out to be ±π/ 2, Eq. (1-3) may be written in the sine form. The angular frequency, ω , is equal to 2πƒ, where f
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.
