HCI 08 Oscillations Lecture Notes

Hwa Chong Institution (College) H2 Physics C1 2023 1 Chapter 8 Oscillations A metronome, which is basically an upside down pendulum, is a device that produces regular ticks (beats). It dates back to the early 19th century. A metronome is used by some performing musicians for practice in maintaining a consistent tempo; it gives the composer an approximate way of specifying the tempo. From its inception, however, the metronome has been a highly controversial tool, and there are musicians who reject its use altogether. - http://en.wikipedia.org/wiki/Metronome

Hwa Chong Institution (College) H2 Physics C1 2023 2 Oscillations Candidates should be able to: (a) describe simple examples of free oscillations. (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 and phase difference and express the period in terms of both frequency and angular frequency. (d) recall and use the equation 2ax  as the defining equation of simple harmonic motion. (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 changes in displacement, velocity and acceleration during simple harmonic motion. (h) describe the interchange between kinetic energy 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 and to the importance of critical damping in applications such as a car suspension system. (j) describe practical examples of forced oscillations and resonance. (k) describe graphically how the amplitude of a forced oscillation changes with driving frequency near to the natural frequency of the system, and understand qualitatively the factors which determine the frequency response and sharpness of the resonance. (l) show an appreciation that there are some circumstances in which resonance is us eful and other circumstances in which resonance should be avoided. RELEVANT E-LEARNING WEBSITES Applets 1) Spring-Mass SHM http://ngsir.netfirms.com/j/Eng/springSHM/springSHM_js.htm 2) Mass Spring Lab https://phet.colorado.edu/en/simulation/mass-spring-lab Online lessons 3) University of Salford Online Lessons: http://www.acoustics.salford.ac.uk/feschools/waves/shm.htm#motion 4) PHYSCLIP Online Lessons a. Mechanics: SHM (http://www.animations.physics.unsw.edu.au/mechanics/chapter4_simpleharmonicmotion.html) b. Waves and Sound: Oscillations (http://www.animations.physics.unsw.edu.au/waves-sound/oscillations/) 5) Schoolphysics: http://www.schoolphysics.co.uk/age16-19/Mechanics/Simple%20harmonic%20motion/ 6) How Radio works http://electronics.howstuffworks.com/radio8.htm

Hwa Chong Institution (College) H2 Physics C1 2023 3 Table of Contents 8.0 Introduction ................................ ................................ ................................ ......................... 4 8.1 Simple Harmonic Motion ................................ ................................ ................................ .... 4 8.1.1 Terms used to describe SHM ................................ ................................ ...................... 5 8.1.2 SHM - Displacement-time (x-t) relationship ................................ ................................ .. 6 8.1.3 SHM - Velocity-time (v-t) relationship and acceleration-time (a-t) relationship ............. 6 8.1.4 Definition of SHM – relationship between acceleration and displacement ................... 9 8.1.5 Simple Harmonic Motion: velocity-displacement (v-x) relationship ............................ 12 8.2 Relationship between Simple Harmonic Motion and Uniform Circular Motion ............ 14 8.3 Energy in SHM ................................ ................................ ................................ ................... 17 8.3.1 Energy-displacement Graphs of SHM ................................ ................................ ....... 17 8.3.2 Energy-time Graphs of SHM ................................ ................................ ..................... 18 8.4 Damped Oscillations, Forced Oscillations and Resonance ................................ ........... 19 8.4.1 Damped oscillations ................................ ................................ ................................ .. 19 8.4.2 Forced oscillations and resonance ................................ ................................ ............ 21 8.4.3 Examples of Destructive and Useful Resonance ................................ ....................... 26 Appendix 1 Case study: Oscillations of A Simple Pendulum ................................ ............... 27 Appendix 2 Case study: Oscillations of a Mass on Vertical Spring ................................ ..... 28 Appendix 3 Equations of motion for damped simple harmonic oscillations ....................... 29 Appendix 4 Demonstration of Forced Oscillation and Resonance - Barton’s pendulums . 30 Tutorial 8 Oscillations ................................ ................................ ................................ .................. 31 Playlist of teaching videos and lecture examples: https://youtube.com/playlist?list=PL_b5cjrUKDlYadbhfFX4Z8AwqHX1vdl9l

Hwa Chong Institution (College) H2 Physics C1 2023 4 8.0 Introduction Oscillations are very common in everyday life such as the motion of a clock pendulum , the vibrations of strings in musical instruments, the vertical oscillations of a car after passing over a bump on the road or even the vibrations of atoms in a lattice. Oscillations can occur when displacing an object that is in a state of stable equilibrium from its equilibrium position results in a restoring force acting on the object directed towards the equilibrium position. The motion is said to be periodic if the displacement of a body in oscillatory motion repeats itself at equal intervals. The full range of its motion that is repeated is referred to as a cycle and the time taken for the body to go through a cycle is the period of the oscillation. Example 8.1: Periodic Motion Which of the following cases are periodic oscillations? 8.1 Simple Harmonic Motion The free oscillation of a mass attached to a spring is a very special example of a periodic oscillation that we call simple harmonic motion (SHM). It is special because the displacement of the mass about its equilibrium position varies sinusoidally. We will prove this analytically but first, le t us look at an experiment that shows the sinusoidal variation of its displacement with time. (*) See Appendix 2 for the derivation of this relationship. m Spring-Mass System (*) where T = period of oscillation m = mass of the oscillating object k = spring constant Spring-Mass system in vertical SHM Spring-Mass system in horizontal SHM

Hwa Chong Institution (College) H2 Physics C1 2023 5 We can obtain a graph of the variation in the displacement of the mass hanging from a spring via the following experiment. We attach a pen to the mass such that its motion is traced onto a vertical sheet of graph paper that is pulled to the left at a constant rate. 8.1.1 Terms used to describe SHM (with reference to Fig 8.1) Terms Definitions Displacement, x The displacement x is the linear distance of the mass from its equilibrium position (x = 0) in a specified direction. It is a vector; the + / – sign tells the direction of displacement from equilibrium. If you set the direction for displacement above equilibrium to be positive, then displacement below the equilibrium would be negative and vice versa. Equilibrium Position (x = 0) When the oscillating mass is at equili