Simple Harmonic Motion Energy

10 Key excerpts on "Simple Harmonic Motion Energy"

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  Essential Physics

  • 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 .

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  Elementary Plane Rigid Dynamics

  • H. W. Harkness(Author)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER FOUR Undamped Simple Harmonic Motion A very common type of motion is one in which a body vibrates about some mean position. The motion of a mass on a spring, the vibrations of a structure such as a bridge, the motion of the pistons in a reciprocating engine, the motion of the electric charges in an oscillating circuit are a few examples. These vibra-tory motions may follow a variety of laws, but there is one type of vibratory motion which is very common in the physical world for a reason which will become clear as examples are discussed. This particular type of vibratory motion is called Simple Har-monic Motion (hereinafter referred to as SHM) and will now be defined. Definition of Simple Harmonic Motion Consider the motion of a point P in Figure IV-I. The point vibrates between the extreme positions B and C, having the point O at the middle of its path. If we measure time from the instant P passes through O, the displacement of P from O as time goes on will first increase until its maximum value OC is reached when it will decrease to zero again and become negative, reaching a maximum negative value at B when it will decrease to zero again. The motion will be repeated. If this type of motion is to be realized it is clear that when the point P is to the right B · -O FIG. IV-1 95 96 IV. UNDAMPED SIMPLE HARMONIC MOTION of O its acceleration must be toward the left since it stops at C and returns. Similarly when the displacement is to the left of O its acceleration must be toward the right. In other words its acceleration must always be toward the midpoint of its path. Further, since the acceleration changes sign at O, being negative until it reaches the point O from the right and positive immedi-ately after passing the point O, it must be zero just at O. So much is fixed simply because the point vibrates with O as the midpoint of the motion. If we now fix the magnitude of the acceleration at all times the motion will be completely defined.

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  Fundamentals 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.

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  An Introduction to Mathematics for Engineers

  Mechanics

  • 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 .

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  Introduction to Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  222 Chapter 10 | Simple Harmonic Motion and Elasticity 10 | Simple Harmonic Motion and Elasticity Sarah Reinertsen is a professional athlete who holds numerous world records in her disability division. Her athletic performance is made possible by a high-tech prosthetic leg made of carbon fiber, which flexes and stores elastic potential energy, like a spring does. The elastic potential energy stored by a spring is one of the topics in this chapter. Chapter | 10 LEARNING OBJECTIVES After reading this module, you should be able to... 10.1 | Apply Hooke’s law to simple harmonic motion. 10.2 | Apply simple harmonic motion relations to the reference circle. 10.3 | Apply conservation-of-energy principles to solve simple harmonic motion problems involving springs. 10.4 | Analyze pendulum motion. 10.5 | Define damped harmonic motion. 10.6 | Define driven harmonic motion. 10.7 | Apply elastic deformations to define stress and strain. 10.8 | Relate Hooke’s law to stress and strain. ©Patrik Giardino/Corbis 10.1 | The Ideal Spring and Simple Harmonic Motion Springs are familiar objects that have many applications, ranging from push-button switches on electronic components, to automobile suspension systems, to mattresses. In use, they can be stretched or compressed. For example, the top drawing in Figure 10.1 shows a spring being stretched. Here a hand applies a pulling force F x Applied to the spring. The subscript x reminds us that F x Applied lies along the x axis (not shown in the drawing), which is parallel to the length of the spring. In response, the spring stretches and undergoes a displacement of x from its original, or “unstrained,” length. The bottom drawing in Figure 10.1 illustrates the spring being compressed. Now the hand applies a pushing force to the spring, and it again undergoes a displacement from its unstrained length.

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  The Physics of Vibrations and Waves

  • H. John Pain(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Use the method of example (3) (p. 28) to show that

  Problem 1.19

  If we represent the sum of the series by the complex exponential form show that Summary of Important Results

  Simple Harmonic Oscillator (mass m, stiffness s, amplitude a )

  Equation of motion + ω 2 x = 0 where ω 2 = s /m

  Displacement x = a sin (ωt + ϕ )

  Energy = = constant

  Superposition (Amplitude and Phase) of two SHMs One-dimensional

  Equal ω , different amplitudes, phase difference δ , resultant R where

  Different ω , equal amplitude,

  Two-dimensional: perpendicular axes

  Equal ω , different amplitude—giving general conic section

  (basis of optical polarization)

  Superposition of n SHM Vectors (equal amplitude a , constant successive phase difference δ ) The resultant is R cos (ωt + a ), where

  and Important in optical diffraction and wave groups of many components.

  * This section may be omitted at a first reading.

  *

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  A B C x f = 0 cm x f x 0 v 0 = 0 m/s INTERACTIVE FIGURE 10.17 The total mechanical energy of this system is entirely elastic potential energy (A), partly elastic potential energy and partly kinetic energy (B), and entirely kinetic energy (C). Conceptual Example 8 takes advantage of energy conservation to illustrate what happens to the maximum speed, amplitude, and angular frequency of a simple harmonic oscillator when its mass is changed suddenly at a certain point in the motion. CONCEPTUAL EXAMPLE 8 Changing the Mass of a Simple Harmonic Oscillator Figure 10.18a shows a box of mass m attached to a spring that has a force constant k. The box rests on a horizontal, frictionless surface. The spring is initially stretched to x = A and then released from rest. The box executes simple harmonic motion that is characterized by a maximum speed  x max , an amplitude A, and an angular frequency . When the box is passing through the point where the spring is unstrained (x = 0 m), a second box of the same mass m and speed  x max is attached to it, as in part b of the drawing. Discuss what happens to (a) the maximum speed, (b) the amplitude, and (c) the angular frequency of the subsequent simple harmonic motion. Reasoning and Solution (a) The maximum speed of an object in simple harmonic motion occurs when the object is passing through the point where the spring is unstrained (x = 0 m), as in Figure 10.18b. Since the second box is attached at this point with the same speed, the maximum speed of the two-box system remains the same as that of the one-box system. 10.3 Energy and Simple Harmonic Motion 269 In the previous two examples, gravitational potential energy plays no role because the spring is horizontal. The next example illustrates that gravitational potential energy must be taken into account when a spring is oriented vertically. (b) At the same speed, the maximum kinetic energy of the two boxes is twice that of a single box, since the mass is twice as much.

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  Classical and Relativistic Mechanics

  • 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.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  The total mechanical energy E is the sum of these two energies: E 5 0.414 J 1 0.138 J 5 0.552 J. Because the total mechanical energy remains constant during the motion, this value equals the initial total mechanical energy when the object is stationary and the energy is entirely elastic potential energy (E 0 5 1 2 kx 2 0 5 0.552 J). (b) When x 0 5 0.0450 m and x f 5 0 m, we have v f 5 B k m (x 0 2 2 x f 2 ) 5 B 545 N/m 0.200 kg [(0.0450 m) 2 2 (0 m) 2 ] 5 2.35 m/s Now the total mechanical energy is due entirely to the translational kinetic energy ( 1 2 mv 2 f 5 0.552 J), since the elastic potential energy is zero (see part C of Figure 10.17). Note that the total mechanical energy is the same as it is in Solution part (a). In the absence of fric- tion, the simple harmonic motion of a spring converts the different types of energy between one form and another, the total always remaining the same. m m (a) (b) m x = 0 m  x max x = A 0 = 0 m/s   x max  x max x = 0 m Figure 10.18 (a) A box of mass m, starting from rest at x 5 A, undergoes simple har- monic motion about x 5 0 m. (b) When x 5 0 m, a second box, with the same mass and speed, is attached. Conceptual Example 8 takes advantage of energy conservation to illustrate what hap- pens to the maximum speed, amplitude, and angular frequency of a simple harmonic oscil- lator when its mass is changed suddenly at a certain point in the motion. CONCEPTUAL EXAMPLE 8 | Changing the Mass of a Simple Harmonic Oscillator Figure 10.18a shows a box of mass m attached to a spring that has a force constant k. The box rests on a horizontal, frictionless surface. The spring is initially stretched to x 5 A and then released from rest. The box executes simple harmonic motion that is characterized by a 262 Chapter 10 | Simple Harmonic Motion and Elasticity maximum speed v x max , an amplitude A, and an angular frequency v.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  The total mechanical energy E is the sum of these two energies: E= 0.414 J + 0.138 J = 0.552 J. Because the total mechanical energy remains constant during the motion, this value equals the initial total mechanical energy when the object is stationary and the energy is entirely elastic potential energy ( E 0 = 1 _ 2 k x 0 2 = 0.552 J). (b) When x 0 = 0.0450 m and x f = 0 m, we have υ f = √ ___________ k  __ m  (x 0 2 − x f 2 ) = √ ___________________________ 545 N/m ________ 0.200 kg [(0.0450 m) 2 − (0 m) 2 ] = 2.35 m/s Now the total mechanical energy is due entirely to the transla- tional kinetic energy ( 1 _ 2 mυ f 2 = 0.552 J), since the elastic potential energy is zero (see part C of Interactive Figure 10.17). Note that the total mechanical energy is the same as it is in Solution part (a). In the absence of friction, the simple harmonic motion of a spring converts the different types of energy between one form and another, the total always remaining the same. INTERACTIVE FIGURE 10.17 The total mechanical energy of this system is entirely elastic potential energy (A), partly elastic potential energy and partly kinetic energy (B), and entirely kinetic energy (C). A B C x f = 0 cm x f x 0 v 0 = 0 m/s CONCEPTUAL EX AMPLE 8 Changing the Mass of a Simple Harmonic Oscillator Figure 10.18a shows a box of mass m attached to a spring that has a force constant k. The box rests on a horizontal, friction- less surface. The spring is initially stretched to x = A and then released from rest. The box executes simple harmonic motion that is characterized by a maximum speed υ x max , an amplitude A, and an angular frequency . When the box is passing through the point where the spring is unstrained (x = 0 m), a second box of the same mass m and speed υ x max , is attached to it, as in part b of the drawing. Discuss what happens to (a) the maximum speed, (b) the amplitude, and (c) the angular frequency of the subse- quent simple harmonic motion.

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