DHS 08 Oscillations (Lecture Notes & Tutorial)

Dunman High School (Senior High Physics) - Oscillations DHS Y5 Physics H2 (2023) For Internal Use Only Page 1 of 34 Topic 10 Oscillations Guiding Questions • What are the characteristics of periodic motion? How can we study and describe such motion? • How can circular motion be related to simple harmonic motion? • How do we analyse simple harmonic motion? Content • Simple harmonic motion • Energy in simple harmonic motion • Damped and forced oscillations, resonance Learning Outcomes 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 as the defining equation of simple harmonic motion (e) recognise and use as a solution to the equation (f) recognise and use the equations and (g) describe, with graphical illustrations, the changes in 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 cases 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 the sharpness of the resonance (l) show an appreciation that there are some circumstances in which resonance is useful and other circumstances in which resonance should be avoided. 2ax =− 0 sinx x t = 2ax =− 0 cosv v t = ( ) 22 0v x x= −

Dunman High School (Senior High Physics) - Oscillations DHS Y5 Physics H2 (2023) For Internal Use Only Page 2 of 34 Introduction An oscillation is a periodic to-and-fro motion of an object between two limits. Periodic means the oscillations repeat themselves. There are three types of oscillations: (1) free oscillation, (2) damped oscillation, & (3) forced oscillation (a) describe simple examples of free oscillations Free Oscillations (oscillating system is isolated) When an object undergoes free oscillation, it oscillates with no energy gain or loss . The object oscillates with constant amplitude, as there are no external force acting on it. Examples of free oscillations include, when friction and air resistance are negligible, (1) a simple vertical (vibrating) spring-mass system, (2) a simple horizontal spring-mass system, and (3) a simple (swinging) pendulum It can be shown (in the Annex) that the natural frequency of p, q and r are 1 2 p kf m=  , 12 2 q kf m=  and 1 2 r gf = l respectively where the spring constant of the spring is k, the length of the pendulum is l and the mass of the oscillator is m. (b) investigate the motion of an oscillator using experimental and graphical methods Experimental Investigation We can plot the displacement-time graphs for oscillators. One possible experimental method is illustrated below. (PLAN view) O mass oscillating horizontally, with pen attached to draw on paper pen tracks on paper paper pulled at constant velocity in direction shown t / s x / m The displacement of the oscillating body varies with time sinusoidally. equilibrium position

Dunman High School (Senior High Physics) - Oscillations DHS Y5 Physics H2 (2023) For Internal Use Only Page 3 of 34 Note: • In the absence of air resistance and friction, a displacement-time graph is sinusoidal. This is a characteristic of an important type of oscillation called simple harmonic motion. • The mass does not actually move along the sinusoidal curve. It always moves back and forth over the same path (periodic). • Graph is sinusoidal, but need not always be a sine curve (i.e. oscillator which is at equilibrium position at t = 0). It can also be a cosine curve (i.e. oscillator at maximum displacement at t = 0) or any other sinusoidal curve, depending on the initial conditions. (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 Since an oscillatory motion can be graphically represented by a sinusoidal representation, we can mathematically describe its displacement (x) with respect to time (t) as the general equation 0 sin( )x x t =+ , and usually we are concerned with the particular solutions in which o0 = or o90 = , depending on where the object starts oscillation at t = 0: For x = x 0 sin (t), at t = 0 (starting time), the value of x = 0. i.e. the object passes the equilibrium position at t = 0. For x = x0 cos (t), at t = 0 (starting time), the value of x = x0. i.e. the object is at the extreme position at t = 0. In the table that follows, we will define the terms that is important in the topic of “Oscillations”: Physical Quantity Definition SI Unit (symbol) Type amplitude, x0 The maximum displacement of the oscillating object from the equilibrium position. metre (m) scalar displacement, x The distance of the oscillating object from its equilibrium position in a stated direction. metre (m) vector period, T The time taken for one complete oscillation. second (s) scalar frequency, f The number of complete to-and-fro cycles per unit time made by the oscillating object. 1f T= hertz (Hz) scalar T T 2T t / s x / m x = x0 cos (t) t / s x / m x = x0 sin (t) 0 0 x0 − x0 x0 − x0

Dunman High School (Senior High Physics) - Oscillations DHS Y5 Physics H2 (2023) For Internal Use Only Page 4 of 34 angular frequency,  Refers to the constant which characterizes the particular simple harmonic oscillator and is related to its natural frequency given by 2πf Note that the displacement x must return to its original value after one period T of the motion, i.e. ( ) ( ) ( ) ( ) ( ) 0 0 0cos cos cos cos cos x t x t T x t x t T x t T t t T        =+ = + = + =+ Since ( )cos cos 2  =+ , we have 2 2 fT == radian per second (rad s−1) scalar phase An angle, in either degrees or radians, which gives a measure of the fraction of a cycle that has been completed by an oscillating particle or by a wave. e.g. at the stage when ¼ cycle completed, phase = /2 rad at the stage when ½ cycle completed, phase =  rad at the stage when ¾ cycle completed, phase = 3/2 rad radian (rad) scalar For instance, in a simple pendulum, ONE complete oscillation occurs when it moves from position A to O to B, then back to O and A. When the pendulum reaches position O from A, we say it has completed ¼ of a cycle. Hence, phase = ¼ × 2 = /2 rad at this stage of the cycle. phase difference between two oscillations Phase difference between two oscillations is a measure of how much one oscillation is out of step with another. If two oscillations are in step with one another, they are said to be in phase with one another OR phase difference = zero. Oscillations are said to be in anti-phase OR phase difference =  rad when the displacement of one oscillation reaches a positive maximum value at the same instant as the other reaches a negative maximum (i.e. the displacements are opposite to each other). radian (rad) scalar

Dunman High School (Senior High Physics) - Oscillations DHS Y5 Physics H2 (2023) For Internal Use Only