NJC Oscillations notes 2025

National Junior College Science Department | Physics 1 9. Oscillations Content Page 9.1 Simple harmonic motion ................................ ................................ ................................ ............ 3 9.1.1 Free Oscillation and Simple Harmonic Motion ................................ ................................ . 3 Examples of oscillations ................................ ................................ ................................ .............. 4 9.1.2 The Meaning of Some Terms used in SHM ................................ ................................ ..... 5 9.1.3 Simple Harmonic Motion (SHM)................................ ................................ ....................... 6 9.1.4 Equations of Motion in SHM ................................ ................................ ............................ 8 Exercise 1: Free Oscillations ................................ ................................ ................................ ..... 11 9.1.5 Graphing the equations of motion ................................ ................................ .................. 13 9.1.6 Relation between uniform circular motion and simple harmonic motion ..................... 15 9.2 Energy in oscillations ................................ ................................ ................................ .............. 17 9.2.1 Energy in Simple Harmonic Motion ................................ ................................ ................... 17 9.2.2 Expressions for Energy ................................ ................................ ................................ .. 18 9.3 Damped Oscillations ................................ ................................ ................................ ............... 23 Light Damping (Underdamped) ................................ ................................ .............................. 23 Critical Damping................................ ................................ ................................ ..................... 25 Heavy Damping (Overdamped) ................................ ................................ .............................. 25 Examples of Damping ................................ ................................ ................................ ............ 26 9.4 Forced Oscillations and Resonance ................................ ................................ .................. 27 9.4.1 Forced Oscillation ................................ ................................ ................................ .......... 27 9.4.2 Frequency Response................................ ................................ ................................ ..... 28 9.4.3 Barton's Pendulums ................................ ................................ ................................ ....... 31 9.4.4 Other Examples of Resonance ................................ ................................ ......................... 32 Exercise 2: Application ................................ ................................ ................................ .............. 33 Appendix A: Mathematical Intuition (optional but good to have) ................................ .................... 35

National Junior College Science Department | Physics 2 Learning Objectives Note: this entire chapter is H2 only. 9.1 Simple harmonic motion (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 𝑎 = −𝜔2𝑥 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 𝑥 = 𝑥0sin 𝜔𝑡 as a solution to the equation 𝑎 = −𝜔2𝑥. (f) Recognise and use the equations 𝑣 = 𝑣0cos 𝜔𝑡 and 𝑣 = ±𝜔√𝑥02 − 𝑥2. (g) Describe, with graphical illustrations, the relationships between displacement, velocity, and acceleration during simple harmonic motion. 9.2 Energy in oscillations (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 1of the system. (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. Important Note: For this topic, your calculator needs to be in radian mode. Otherwise, there will be calculation errors. 1 Natural frequency refers to the frequency when the system is in free oscillation.

National Junior College Science Department | Physics 3 9.1 SIMPLE HARMONIC MOTION 9.1.1 Free Oscillation and Simple Harmonic Motion Self-study resources SLS lesson on SHM (1/5) https://for.edu.sg/shm1 A system is said to be undergoing a free oscillation when the only external force acting on it is a restoring force. A restoring force is a resultant force that acts in opposite direction to the displacement of the object from its equilibrium position (where the resultant force is zero). A free oscillation is usually set into motion by displacing the system from its equilibrium position and then releasing it. Simple harmonic motion is the simplest type of free oscillation.

National Junior College Science Department | Physics 4 Examples of oscillations Self-study resources Other examples https://youtu.be/VKtEzKcg6_s

National Junior College Science Department | Physics 5 9.1.2 The Meaning of Some Terms used in SHM Term Symbol Meaning SI unit Vector / Scalar Relevant Equation Period 𝑇 time taken for one complete oscillation s scalar Frequency 𝑓 number of oscillations per unit time 1 Hz = 1 cycle (or oscillation) per second Hz (hertz) scalar 𝑓 = 1 𝑇 Angular Frequency 𝜔 product of frequency and 2π rad can also be understood as the rate of change of phase rad s−1 scalar 𝜔 = 2π𝑓 𝜔 = 2π 𝑇 Equilibrium Position position at which no net force acts on an oscillating object Displacement 𝑥 distance from a fixed point in a specified direction measured with respect from the equilibrium position in the context of oscillatory motion m vector Amplitude 𝑥0 magnitude of the maximum displacement of an oscillation m scalar 𝑥 = 𝑥o sin(𝜔𝑡 + ) Phase 𝜃 denotes the state of the oscillatory cycle at a particular time 𝑡 rad scalar 𝜃 = 𝜔𝑡 +  Initial Phase  state of the oscillatory cycle at 𝑡 = 0 rad scalar Phase Difference Δ𝜃 used to compare the difference in phase between two oscillators or two instances in time for one oscillator a measure of how much an oscillation is out of step with another rad scalar in phase Two oscillations that are exactly in step with one another are said to be in phase with one another The phase difference these two oscillations is any even integer multiples of π rad, including 0 rad. in antiphase Two oscillations that are always moving in opposite dire