ASRJC Oscillations Notes

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 10-1 Additional Notes Topic 10: Oscillations 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 a = -ω2x as the defining equation of simple harmonic motion. (e) recognise and use x = xo sin ωt as a solution to the equation a = -ω2x. (f) recognise and use v = vo cos ωt and v = ±ω ( ) 22 oxx − (g) describe, with graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion. (h) describe the interchange between kinetic energy 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 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 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.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 10-2 Additional Notes A.1 Periodic Motion • Any motion that repeats itself in equal time intervals. Examples: (a) mass oscillating at end of a spring (b) oscillation of a simple pendulum (c) planetary motion A.2 Oscillatory or Vibratory Motion • A particle in oscillatory or vibratory motion moves back and forth over the same path. Examples: (a) mass oscillating at end of a spring (b) oscillation of a simple pendulum (c) vibrations of strings in musical instrument Oscillation Free Oscillation Forced oscillation (dealt with later in Section E) the to-and-fro movement of a body about a fixed point in which no external driving force acts on it. Occurs when the body is acted upon by an external periodic driving force, causing the body to oscillate at the frequency of the periodic force , rather than the natural frequency of the body. Example: A child on a swing given a n initial push and swings freely. Example: A child on a swing kept in motion by pushing the child at equal intervals of time. Nature of Science We can gain a deep understanding of periodic motion by analysing the mathematically simplest case of free oscillations, known as simple harmonic motion (SHM). Such sinusoidally varying motion is essentially a projection of uniform circular motion, and provi des a mathematical basis upon which to describe more complicated oscillations. Relating Science and Society Oscillations play an important role in engineering and nature. The study and control of oscillation is needed to achieve important goals in engineering, e.g. to prevent the collapse of a building due to waves created by an earthquake. A Introduction

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 10-3 Additional Notes • An example of oscillatory motion is the spring-mass system. If we attach a pen to the mass (above diagram) and the mass is set into vertical oscillations, the p en will trace a fine line on the paper placed behind the mass . If the paper is pulled steadily (at a constant speed) at right angles to the oscillations, then the pen ought to trace out a sinusoidal wave on the paper. The motion of the mass is about an equilibrium point O and limited between C and C'. The path marked by the pointer on the paper represents the displacement-time plot of the mass. Since the displacement-time plot of the mass having such a motion is sinusoidal (can be expressed as sine or cosine functions), this periodic, oscillatory or vibratory motion is commonly referred to as simple harmonic motion. In a mechanical oscillation there is a continual interchange of potential and kin etic energies - during parts of the cycle, potential energy is changed to kinetic energy of the moving body, and during other parts of the cycle the kinetic energy is converted to potential energy. The motion can continue indefinitely if there is no energy loss, although energy is lost in real situations. (This is dealt with later in Section D.) • In this topic we will discuss the following quantities of an oscillating body: o Its motion (i.e. displacement, velocity, acceleration) – note that these quantities are related to each other, as well as related to time. o The forces acting on it, and in particular, the net force (i.e. restoring force) which results in the above motion. o The variation of its mechanical energy (i.e. potential energy and kinetic energy) as a result of its motion. C C’ O Motion of paper Equilibrium line Restoring force is the net force that acts on the object to bring it back to its equilibrium position.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 10-4 Additional Notes A.3 Key terms specific to oscillations • Equilibrium position O is the position at which no net force acts on an oscillating body. • Displacement x of the object is the l inear distance of the oscillating body from its equilibrium position in a specified direction. It is a vector quantity and can be positive or negative. • Amplitude of the oscillation, x0 is the maximum displacement of an oscillating body from its equilibrium position. It is a scalar quantity, always positive. • Period T of the motion is the time taken to complete one oscillation. • Frequency f is the number of complete oscillations per unit time (Unit: Hertz (Hz)). For a system in free oscillation, this frequency is known as the natural frequency. • Angular frequency ω is a constant of a given oscillator and is related to its natural frequency f by ω = 2f. 1f T= displacement x O Motion of paper Equilibrium line T x0 There is a difference in the meaning of frequency (f) and angular frequency (ω). Although the symbols (ω) used are the same. Angular frequency is different from angular velocity.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 10-5 Additional Notes Check Your Understanding 1 The displacement of the mass described on page 8-3 varies with time as shown. Determine for this oscillation, its (a) amplitude, (b) period and (c) frequency. Worked Example 1 A bungee jumper undergoes an oscillation between a height of 3.0 m and 8.0 m above sea level. It takes 2.0 s for him to move from the lowest point to the highest point. Determine the (a) amplitude and (b) period of oscillation. displacement /m time /s 0.12 0 −0.12 1.0 2.0 3.0 4.0 Solution: (a) amplitude = ½ (distance between the highest and lowest points) = ½ (5.0) = 2.5 m (b) period = 2 x (time taken to travel between the highest and lowest points) = 2 x 2.0 = 4.0 s

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 10-6 Additional Notes B.1 Example of simple harmonic motion: Vertical Spring-Mass Oscillation A mass m attached to the end of a long spring (with force constant k) can be made to oscillate in a straight line about O, between points A and B. At Position 1 Spring is uns