Oscillation Lecture Notes

Dunman High School (Senior High Physics Department) 9646 Physics Topic 9: Oscillations 9-1 H2 Topic 9 Oscillations Do you know why that most grandfather clocks, invented in 1656, will have a pendulum of length ≈1m while smaller versions have a pendulum of length ≈25cm? Do you know that these clocks will ‘slow down’ on top of Mount Everest? These clocks were the most accurate time keepers until quartz was invented in 1927! What is the period of these clocks (1.0m and 0.25m)? A 0.5 s B 1.0 s C 2.0 s D 1 minute

Dunman High School (Senior High Physics Department) 9646 Physics Topic 9: Oscillations 9-2 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) understand 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) recognise and use the equation xa 2ω−= as the defining equation of simple harmonic motion. (e) recall and use txx ωsin0= as a solution to the equation xa 2ω−= . (f) recognise and use )( cos 22 0 0 xxv tvv −±= = ω ω (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 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 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. References: 1 Advanced Level Physics by Loo Kwok Wai 2 College Physics by Young and Geller 3 Physics for Scientist and Engineers (5th Edition) by Serway

Dunman High School (Senior High Physics Department) 9646 Physics Topic 9: Oscillations 9-3 1 Introduction (a) describe simple examples of free oscillations. An oscillation is a periodic motion of an object about a certain mean equilibrium position with a continuous interchange of kinetic energy and potential energy. Periodic (or harmonic) motion refers to any motion that repeats itself in equal intervals of time. Examples: vibration of tuning fork, boat bobbing at anchor, a child playing on a swing, bells, pendulum of a grandfather clock, diaphragms in telephones and speaker systems and surging pistons in engines of cars. When an object is displaced from its fixed equilibrium position and is free to oscillate, it will oscillate about this position with the natural frequency of the system. (We will go into more details of this natural frequency with a simple pendulum case study later.) The object will oscillate forever under free oscillations as energy is conserved and quantities such as amplitude and period remain constant. In real life, dissipative forces like air resistance and friction will cause the oscillations to f ade with time, as we observed in our many practical exercises on oscillations. These oscillations are known as damped oscillations. Damped oscillations occur when there is a continuous transfer of energy to the surroundings such that the energy in the syst em decreases with time, hence the amplitude of the motion progressively decreases with time. Forced oscillations are caused by continual input of energy by external source to an oscillating system to compensate the loss due to damping in order to maintain the amplitude of the oscillation.

Dunman High School (Senior High Physics Department) 9646 Physics Topic 9: Oscillations 9-4 1.1 Describing Oscillations (c) understand and use the terms amplitude, period, frequency, angular frequency and phase difference and express the period in terms of both frequency and angular frequency. Consider displacement with time of a spring-mass oscillating in frictionless environment: a. Equilibrium position (or neutral position) is the position at which no net force acts on the oscillating mass. b. Displacement (x) is the distance of the oscillating mass from its equilibrium position at any instant in a stated direction. c. Amplitude (xo) is the maximum displacement of the oscillating mass from the equilibrium position in either direction. d. Period (T) is the time taken for one complete oscillation. Unit: second. e. Frequency (f) is the number of complete to- and-fro cycles per unit time made by the oscillating object. Unit: Hertz. 1 Hz is equal to one cycle per second. f. Angular frequency (ω) of an oscillation is frequency × 2π. It is the angle in radians by which the phase of the motion changes per unit time. Unit: radian per second (rad s−1). g. Phase is 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. h. Phase difference is a measure of how much one wave is out of step with another. It is measured in either degrees or radians. i. In-phrase, Out of Phase, Antiphase In phase: phase difference is zero. Out of phase: phase difference is not zero. Antiphase: phase difference is 180 degrees or π radians. Time PERIOD T x x = 0 x > 0 x < 0 Amplitude Xo Equilibrium position Amplitude Xo

Dunman High School (Senior High Physics Department) 9646 Physics Topic 9: Oscillations 9-5 Examples of Phase Difference Phase difference of π/2, (∆t = ¼T) If v = v0 cos kt, then v’ = v0’cos (kt − π/2) = −v0’sin kt Phase difference of π, completely out of phase, (∆t = ½T) If v = v0 sin kt, then v’ = v0 ’ sin (kt − π) = − v0 ’ sin kt Example 1 A man standing at the dock was observing the bobbing motion of a speedboat on the seawater just below the dock. He estimated that the boat could reach two extreme positions which were 3.0 m and 4.0 m below the dock surface respecti vely. He also counted that the boat would bob on average 30 times in 15 seconds. Assume that the speedboat was executing free oscillation, determine its (a) amplitude, (b) period, (c) frequency, (d) vertical distance between the equilibrium position and the dock surface and (e) the angular frequency. Solution (a) x0 = ½ (4.0 – 3.0) = ½ (1.0) = 0.50 m (b) T = 30 15 = 0.50 s (c) f = == 50.0 11 T 2.0 Hz (d) vertical distance = ½ (4.0 + 3.0) = 3.5 m (e) ω = 2 π f = 2 π f = 2 π (2.00) = 4.00 π = 12.6 rads-1 v v’ v’ v

Dunman High School (Senior High Physics Department) 9646 Physics Topic 9: Oscillations 9-6 Example 2 A pendulum takes 31.7 seconds to complete 25 oscillations, if another similar pendulum is 1.70 seconds behind the first, what is the phase difference between the pendulums? Solution 31.7 ÷ 25 = 1.268 = 1.27 s The period is 1.27 s Since the period of the pendulum is 1.27s, the actual time difference in the oscillation is ∆t= 1.70 − 1.268 = 0.432 s φ = 2π∆t/T = 2.14 rad Example 3 The velocity time graph of the oscillating cart is shown below. Determine the (a) Period (b) Frequency (c) Angular Frequency Solution (a) 2.5 waves in 1.6 seconds, therefore period is 0.64 s (b) f = 1/T = 1.56 Hz (c) ω = 2