JPJC 2026 Oscillations Lecture Notes Tutor

1 JURONG PIONEER JUNIOR COLLEGE 9478 H2 PHYSICS OSCILLATIONS Content • Simple harmonic motion • Energy in oscillations 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 xa 2−= 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 xa 2−= (f) recognise and use the equations 0 cosv v t = and 22 0()v 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 energ y 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 frequency6 of 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. ____________________________ 6 Natural frequency refers to the frequency when the system is in free oscillation.

2 Introduction In the previous lectures, we discussed two types of motion: motion under uniform acceleration and motion in a circle. In both types of motion , we had seen the importance of forces acting on a moving body. In this topic , we shall inv estigate the periodic motion of a body. We can gain a deep understanding of periodic motion by studying simple harmonic motion, which is essentially a projection of uniform circular motion. Naturally, we will revisit concepts in kinematics, dynamics, forces and energy in trying to understand simple harmonic motion. 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 1.1 Free oscillations ▪ An oscillation is a type of motion in which an object moves back and forth over the same path repeatedly in a regular manner. Such a motion is also known as a periodic motion or a harmonic motion. ▪ In a free oscillation, it is assumed that damping forces (e.g. dissipative forces like air resistance and friction) and driving forces (from energy sources) are absent. ▪ The body vibrates at the natural frequency of the system about its equilibrium position with no external forces acting on it. ▪ Figure 1 below shows some examples of oscillatory motion. Figure 1 Some examples of oscillatory objects and their waveforms. Oscillating object Waveform (a) tuning fork prong (b) watch balance wheel (d) body of guitar (e) cone of loudspeaker (c) piston in engine

3 (b) investigate the motion of an oscillator using experimental and graphical methods 1.2 Oscillations of a simple pendulum ▪ The variation of displacement of a simple pendulum with time can be investigated with the set-up as shown in Figure 2. Figure 2 Simple harmonic motion ▪ The paper is pulled at constant speed, while the funnel is oscillating back-and-forth. The sand falling out from the funnel traces out a sinusoidal curve on the paper , which can be represented by the displacement-time graph in Figure 3. Figure 3 A sinusoidal curve ▪ An equation for a sinusoidal oscillation is of the form: 0 sinx x t = where x is the displacement, 0x is the amplitude, t is the time and  is the angular frequency. simple pendulum paper sand in a funnel oscillating paper pulled at a constant speed curve traced out by sand dropping from funnel displacement x time t xo - xo T 2T

4 ▪ The mass does not actually move along the sinusoidal curve. It always moves back and forth over the same path (periodic). ▪ Graph need not always be a sine curve. It can also be a cosine curve (oscillator at maximum displacement at t = 0 as shown in Figure 4 ) or any other sinusoidal curve, depending on the initial conditions. Figure 4 A displacement-time graph for an oscillator at maximum displacement at t = 0 ▪ If the oscillati ng object starts at the maximum amplitude position at 0t = , the sinusoidal equation can be expressed as: 0 cosx x t = (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 1.3 Some terms describing the harmonic motion Some common terms used to describe the harmonic motion of an object are as follows. ▪ Period T : The period of a motion is the length of time taken for one complete oscillation. ▪ Frequency f : The frequency of a periodic motion is the number of oscillations per unit time. Tf 1= The SI unit is hertz (Hz). 1 Hz is one cycle per second. ▪ Displacement x : Displacement is the distance an object has moved from its equilibrium position in a stated direction. It is a vector quantity. The equilibrium position of an oscillatory motion is the position where the net force acting on the object is zero. ▪ Amplitude 0x : The amplitude is the magnitude of the maximum displace ment of the object from the equilibrium point. displacement x time t x0 - x0 T 2T

5 ▪ Angular frequency,  : The angular frequency is a scalar measure of rotation rate, expressed in radians per second (rad s−1). It describes how fast the phase of the oscillation changes, with higher angular frequency corresponding to faster oscillations. 2 2 fT== since 1f T= Hence, the equation for a sinusoidal oscillation can also be written as 0 sin2x x ft = (d) show an understanding that xa 2−= 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 xa 2−= 1.4 Definition of simple harmonic motion Spring-mass system: Motion of a mass attached to a spring in the horizontal direction Figure 5 Spring-mass system undergoing simple harmonic motion When an object of mass m is displaced from the equilibrium position as shown in Figure 5 and in the diagram on the right , the stretched spring exerts a restoring force F on m, towards the equilibrium position O. When the mass is released , the restoring force causes the mass to accelerate towards the equilibrium position O. At equilibrium point, the restoring force is zero. Taking the right as positive, w hen the spring is compressed, x is negative (to the left ) and th