RI Chap 9 Oscillations Lecture Notes

9 Oscillations H2 Physics 9478 Content Page 9.1 Introduction to Oscillations 2 9.2 Simple Harmonic Motion 4 9.3 Energy in Simple Harmonic Motion 16 9.4 Damped Oscillations 19 9.5 Forced Oscillations and Resonance 23 9.6 Summary of S.H.M Equations 27 9.7 Appendix 28 Learning Outcomes Candidates should be able to: (a) describe simple examples of free oscillations, where particles periodically return to an equilibrium position without gaining energy from or losing energy to the environment. (b) investigate the motion of an oscillator using experimental and graphical methods. (c) show an understanding of and use the terms amplitude, period, frequency, angular frequency, phase and phase difference, and express the period in terms of both frequency and angular frequency. (d) show an understanding that 2ax =− is the defining equation of simple harmonic motion, where acceleration is (directly) proportional to displacement from an equilibrium position and acceleration is always directed towards the equilibrium position. (e) recognise and use 0 sinx x t = as a solution to the equation 2ax =− . (f) recognise and use the equations 0 cosv v t = and ( ) 22 0v x x =  − . (g) describe, with graphical illustrations, the relationships between displacement, velocity, and acceleration during simple harmonic motion. (h) describe the interchange between kinetic and potential energy during simple harmonic motion. (i) describe practical examples of damped oscillations, with particular reference to the effects of the degree of damping (light/under, critical, heavy/over), and to the importance of critical damping in applications such as a car suspension system. (j) describe graphically how the amplitude of a forced oscillation changes with driving frequency, resulting in maximum amplitude at resonance when the driving frequency is close to or at the natural frequency of the system (natural frequency refers to the frequency when the system is in free oscillation). (k) show a qualitative understanding of the effects of damping on the frequency response and sharpness of the resonance. (l) describe practical examples of forced oscillations and resonance, and show an appreciation that there are some circumstances in which resonance is useful, and other circumstances in which resonance should be avoided.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT P a g e | 2 9.1 Introduction to Oscillations A periodic motion is one in which a body continually retraces its path at equal time intervals. Many systems exhibit periodic motion. The molecules in a solid oscillate about their equilibrium positions; electromagnetic waves are characterised by oscillating electric and magnetic field vectors; and in alternating -current electrical c ircuits, voltage and current vary periodically with time. An oscillation is a special periodic motion in which the oscillator moves to and fro about an equilibrium position. This is also called harmonic motion. Simple harmonic motion is a type of such a motion. Consider a body attached to the end of a fixed spring in Fig. 9.1(a), or a pendulum in Fig. 9.1(b). In both cases, the body and the pendulum are at their equilibrium positions. The equilibrium position is the position where the resultant force on the body is zero. If the body or pendulum is displaced from its equilibrium position (i.e. pulled either to the left or right), it will experience a resultant force that tries to restore it to its equilibrium position. This is called the restoring force. The mass-spring system and pendulum systems described above are examples of oscillators undergoing harmonic motion. We will examine these two systems in greater detail later in this chapter. ❖ Free Oscillation If a body is displaced from its equilibrium position and then released, it oscillates at its natural frequency about the equilibrium position. A free oscillation occurs when a body oscillates with no driving and/or resistive forces acting on it. Since the oscillating system does not gain energy from or lose energy to the surroundings, the total energy and amplitude remain constant with time. Fig. 9.1(a) Fig. 9.1(b)

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT P a g e | 3 ❖ Definitions of Physical Quantities T and f are related by the equation = 1T f S.I. unit for T: s S.I. unit for f: Hz (Hertz) or −1s A complete cycle has a phase of 2 . In one period T, the body undergoing a complete oscillation through a phase angle of 2 . Hence the angular frequency of the oscillation ==2 2 fT S.I. unit of angular frequency : −1rad s Angular frequency is not the same as angular velocity, even though both have the same units of −1rad s and are represented by the same symbol . In oscillations,  represents angular frequency while in circular motion, it represents angular velocity (rate of change of angular displacement). The phase difference is a measure of how “out of step ” two oscillating bodies are with each other. Phase and phase difference are usually measured in degrees or radians. Phase (Angle) The phase (angle)  of an oscillation that a body is in, is the fraction of an oscillation cycle it is at relative to its starting displacement. Period and Frequency Period T is the time taken for the body to complete one oscillation. Frequency f is the number of oscillations per unit time. Amplitude Amplitude is the magnitude of the maximum displacement of the particle from its equilibrium position. Angular Frequency Angular frequency  is the rate of change of phase angle of the oscillation and is equal to the product of 2 and its frequency f. Phase Difference Phase difference  is the difference in the positions of two oscillating bodies in their cycles.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT P a g e | 4 9.2 Simple Harmonic Motion When a particle experiences a force F that is directly proportional to its displacement x from a fixed point and that it is always directed towards that fixed point, = = −F ma kx The acceleration of the particle = − = − 2ka x x m where  is the angular frequency of oscillation. This is the defining equation for simple harmonic motion (S.H.M.). A particle whose motion satisfies this equation is said to be in S.H.M. The “fixed point ” in S.H.M. refers to the equilibrium position of the particle. The negative sign indicates that the force is directed opposite to its displacement. Fig. 9.2 The displacement x of the particle is measured from the fixed point and is directed away from the fixed point while the a cceleration a of the particle is directed towards the fixed point. This means that x and a are always opposite in direction as implied by the negative sign in the equation. Since the acceleration of the particle is directly proportional to its displacement from equilibrium position, the acceleration is not constant. Therefore, kinematics equations cannot be used to analyse the motion of particles in S.H.M. By convention, vector quantities directed to the right are positive. OB positive a negative x O A positive x negative a Simple Harmonic Motion (S.H.M.) Simple harmonic motion is the motion of a particle about a fixed point such that its acceleration is proportional to its displacement from the fixed point and is always directed towards the point. [Text]

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT P a g e | 5 ❖ Characteristics of S.H.M. Consider a particle N undergoing S.H.M. between two points A and B, about an equilibrium position O. Throughout the motion, N experiences a restoring force and an acceleration towards O. Fig. 9.3 illustrates the motion of N at various instances of the motion. The displacement x from point O, the velocity v and acceleration a of