NJC 2023 Oscillations Notes

NATIONAL JUNIOR COLLEGE 1 2023SH1 | Science | Physics © 2023 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy, recording or any other information storage an d retrieval system,without prior permission in writing from the copyright owner. Oscillations (Topic 10) Content 1. Mathematical Pre-requisite 2. Free Oscillation and Simple Harmonic Motion 3. Energy in Simple Harmonic Motion 4. Damped Oscillation 5. Forced Oscillation and 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 both frequency and angular frequency. (d) recall and use the equation a = −2x as the defining equation of simple harmonic motion. (e) recognise and use x = x0 sin t as a solution to the equation a = −2x. (f) recognise and use the equations v = v0 cos t and 22 0()v x x = − (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 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 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 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.

NATIONAL JUNIOR COLLEGE 2 2023SH1 | Science | Physics © 2023 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy, recording or any other information storage an d retrieval system,without prior permission in writing from the copyright owner. 1. Mathematical Pre-requisite In this chapter, you will use the sine (sinusoidal) graph frequently and hence, you need to know how to sketch the sine graph given an equation of the form 𝑥 = 𝑥0 sin 𝜃 as shown in Fig. 1A below: Notice that the sine graph repeats itself after every 2 radians or 360 degrees. The repeated pattern is shown in dotted line in Fig. 1a above. Also, the displacement (𝑥), of an object executing simple harmonic motion (for example, a pendulum) can be represented by an equation of the form: 𝑥 = 𝑥0 sin(𝜔𝑡). The term (𝜔𝑡) has units of radians (i.e. it is dimensionless). The graph of 𝑥 against 𝑡 is shown in Fig. 1B below: Notice that the graph in Fig. 1 b repeats itself after every 𝑇 seconds where 𝑇 represents the period of oscillation. We can use this fact to convert 𝜃 in radian to 𝑡 in second, or vice versa, using direct proportion as follows: 2𝜋 → 𝑇 𝜃 → 𝑡 By proportion: 𝜃 2𝜋 = 𝑡 𝑇 hence, 𝜃 = ( 2𝜋 𝑇 ) 𝑡 = 𝜔𝑡 where 𝜔 = 2𝜋 𝑇 Replacing 𝜃 with (𝜔𝑡) in the equation 𝑥 = 𝑥0 sin 𝜃, we get 𝑥 = 𝑥0 sin(𝜔𝑡) 𝑥 / m 𝜃 / rad 𝑥0 −𝑥0 0 𝜋 2 𝜋 3𝜋 2 2𝜋 Fig. 1a 4𝜋 𝑥 / metres 𝑡 / seconds 𝑥0 −𝑥0 0 𝑇 Fig. 1b 2𝑇

NATIONAL JUNIOR COLLEGE 3 2023SH1 | Science | Physics © 2023 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy, recording or any other information storage an d retrieval system,without prior permission in writing from the copyright owner. The equation 𝑥 = 𝑥0 sin(𝜃 −) = 𝑥0 sin(𝜔𝑡 −) represents a translation (or shifting) of the sine graph along the positive 𝜃 (or positive 𝑡) direction by  radians as shown in Fig. 1c below: Hence if  = 90, then you will end up with a negative cosine function: 𝑥 = 𝑥0 sin (𝜃 − 𝜋 2) = 𝑥0 [sin 𝜃 cos ( 𝜋 2) − cos 𝜃 sin ( 𝜋 2)] = −𝑥0 cos 𝜃 since sin ( 𝜋 2) = 1 and cos ( 𝜋 2) = 0 You also need to know the following relations: 𝑑 𝑑𝑡 [sin(𝜔𝑡)] = 𝜔 cos(𝜔𝑡) i.e. differentiating a sine function gives a positive cosine function 𝑑 𝑑𝑡 [cos(𝜔𝑡)] = −𝜔 sin(𝜔𝑡) i.e. differentiating a cosine function gives a negative sine function 𝑥 / metres 𝜃 / radians 𝑥0 −𝑥0 0 𝜋 2 𝜋 3𝜋 2 2𝜋 Fig. 1c  𝑥 = 𝑥0 sin(𝜃 −)

NATIONAL JUNIOR COLLEGE 4 2023SH1 | Science | Physics © 2023 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy, recording or any other information storage an d retrieval system,without prior permission in writing from the copyright owner. 2. Free Oscillation and Simple Harmonic Motion An object is said to be undergoing free oscillations when the only external force acting on it is the restoring force. A restoring force is a resultant force that acts in opposite direction to the displacement of the object from an equilibrium position (where the resultant force is zero). Simple harmonic motion is the simplest type of free oscillation. 2.1 Examples of oscillations Other examples : https://www.youtube.com/watch?v=VKtEzKcg6_s

NATIONAL JUNIOR COLLEGE 5 2023SH1 | Science | Physics © 2023 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy, recording or any other information storage an d retrieval system,without prior permission in writing from the copyright owner. 2.2 The Meaning of Some Terms used in SHM - Period (𝑻): The period 𝑻 is the time taken for one complete oscillation. - Frequency (𝒇): The number of oscillations per unit time is called frequency, 𝑓. Frequency is equal to the inverse of the period, 𝑓 = 1 𝑇. The SI units of frequency is Hz (hertz) where 1 Hz = 1 cycle (or oscillation) per second. - Angular frequency (𝝎): Angular frequency,  is defined as the product of the frequency and 2. 𝜔 = 2𝜋𝑓 = 2𝜋 𝑇 . Units of angular frequency is rad s-1. - Equilibrium position: The position at which no net force acts on the oscillating object is called the equilibrium position. - Displacement (𝒙): Displacement is the distance the object has moved from its equilibrium position in a stated direction. It is a vector quantity and it measured with respect from the equilibrium position. - Amplitude (𝒙𝐨): Amplitude is the magnitude of the maximum displacement from equilibrium position. It is a scalar quantity. For example, the displacement, 𝑥 of an object in simple har monic motion can be represented by the equation: 𝑥 = 𝑥o sin(𝜔𝑡 +) = 5 sin (𝜔𝑡 + 𝜋 2) The amplitude is 𝑥o = 5 units - Phase: The term (𝜔𝑡 ±) is called the phase. The phase denotes the state of oscillation at a particular time 𝑡 and it has units of radian. In general, the state of motion/oscillation at time 𝑡 means the displacement, velocity and acceleration of the object at time 𝑡. The term ± inside the bracket represents the initial phase of the oscillating object which is the state of oscillation at 𝑡 = 0. The initial phase (ie the value of