TJC 9 Oscillations notes

i Unit 9: 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 a =  2x 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 = xo sint as a solution to the equation a =  2x (f) recognise and use the equations v = vo cost and 2 2 ov 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 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 (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 *Natural frequency refers to the frequency when the system is in free oscillation. Temasek Junior College 2025 Name : ________________________________ Class : _________

2 1 Introduction An oscillation is a motion in which a system moves to-and-fro along the same path repeatedly. The oscillation of a pendulum of a clock is apparent, but the oscillation of atoms within a solid is hidden. The oscillation associated with wave motion can be appreciated when a body floating in a liquid rises and falls as a wave travels past it, but the oscillation associated with light waves cannot be perceived by our senses. In fact, all communication by sight and by hearing makes use of wave oscillation. 1.1 Free oscillations LO (a) In a free oscillation, the particle periodically returns to an equilibrium position without gaining energy from or losing energy to the environment. The only external force acting on it is the restoring force and it oscillates at its natural frequency with constant amplitude and constant total energy. Simple harmonic oscillations are free oscillations. 1.2 Motion of an oscillator LO (b) Fig (a) shows an oscillating mass-spring system. The motion is easily monitored with a motion sensor connected to a data logger to record the variation with time of the position of the mass. Fig (b) shows a typical displacement–time graph generated by a motion sensor. The curve is sinusoidal and the motion is said to be sinusoidal. An oscillation that follows a sinusoidal function of time is called a simple harmonic motion. In practice, dissipative forces exists and the energy and hence amplitude of the oscillations diminish over time. Fig (a) Fig (b)

3 1.3 Important terms LO (c) In the context of an oscillating system, 1. Displacement x is the distance of an oscillating particle from its equilibrium position in a specific direction. 2. Amplitude xo is the maximum magnitude of displacement of the oscillating particle from the equilibrium position. 3. Period T is the time for one complete oscillation. 4. Frequency f is the number of complete oscillations per unit time. SI unit is hertz (Hz) where 1 hertz = 1 cycle per second = 1 s-1. And 1f T 5. Angular frequency  is defined as the product of 2 and the frequency. i.e. 22 f T    SI unit is rad s-1. As  is a constant, T is a constant and is independent of the amplitude xo of the oscillation. This is an important characteristic of simple harmonic motion. 6. Phase  = t refers to the stage or position reached within a cycle of an oscillation with respect to a reference (zero) position, usually expressed as a fraction of a cycle or a period or as an angle. i.e. phase 2t t T         . Phase difference  is an angular measure of the fraction of a cycle that two oscillations having the same frequency are out of step. Consider an oscillation of displacement x1 = xo sint and a second oscillation of displacement x2 = xo sin(t +) as shown. Both oscillations have the same amplitude xo, same angular frequency  and hence same frequency f and period T. But they have a phase difference of orad = 902   . Note: x2 = xo sin(t +) reaches its peak first. We say that x2 = xo sin(t +) leads x1 = xo sint by orad or 902   . displacement x time t x2 = xo sin(t +) x1 = xo sint 0 T xo

4 Example 1: The figure below shows four different oscillations A, B, C and D on the same displacement- time graph. Complete the sentences below with regards to their phase difference. Phase difference between A and B = _________, they are ___________________. Phase difference between A and C = _________, they are ___________________. Phase difference between A and D = _________, they are ___________________. 1.4 Defining equation of SHM LO (d) Simple harmonic motion is an oscillatory motion of a particle whose acceleration is (directly) proportional to displacement from an equilibrium position and acceleration is always directed towards the equilibrium position Mathematically, the defining equation of SHM is a =  2x where a is acceleration, x is displacement from equilibrium position and  is the angular frequency. The negative sign implies that the acceleration points in the opposite direction to the displacement vector. From Newton’s second Law, force F = ma =  m2x. Thus, F   x Graphically, a motion is said to be simple harmonic only if a straight line of negative slope passing through the origin is observed for the acceleration versus displacement graph as shown. x a 0 D

5 To prove that a motion is simple harmonic, it is sufficient to establish that a   x. For example, consider a mass-spring system. In Fig (a), a body of mass m attached to a light spring of force constant k rests on a frictionless table. In Fig (b), the body is displaced by an amount x from its equilibrium position (x = 0). The spring is extended and exerts a restoring force F on the mass in the direction opposite to displacement x. If the spring obeys Hooke’s law, F = – kx where k is the force constant. The negative sign indicates that the restoring force on the body due to the spring is opposite to displacement. From Newton’s second law, F = ma = – kx and hence ka x a xm     Thus the motion is simple harmonic. Compare with the defining equation of SHM a =  2 x, 2 1 and 2 2 k k f fm m       Practice of science: How do astronauts weigh themselves in space? Example 2: Values of the acceleration a of a particle moving in simple harmonic motion as a function of its displacement x are given in the table below: a/cm s-2 16 8 0  8  16 x/c