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1 UCT PHY1025F: Vibrations & Waves Physics 1025F Vibrations & Waves Dr. Steve Peterson Steve.peterson@uct.ac.za OSCILLATIONS

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2 UCT PHY1025F: Vibrations & Waves Chapter 11: Vibrations and Waves Periodic motion occurs when an object vibrates or oscillates back and forth over the same path

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3 UCT PHY1025F: Vibrations & Waves Periodic motion, processes that repeat, is one of the important kinds of behaviours in Physics Periodic Motion

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4 UCT PHY1025F: Vibrations & Waves Equilibrium position – position where net force is zero Restoring force – force acting to restore equilibrium Oscillation – periodic motion governed by a restoring force Equilibrium and Oscillation

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5 UCT PHY1025F: Vibrations & Waves A graph or motion that has the form of a sine or cosine function is called sinusoidal. A sinusoidal oscillation is called simple harmonic motion (SHM) Equilibrium and Oscillation

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6 UCT PHY1025F: Vibrations & Waves SHM is characterised by… Amplitude A: maximum distance of object from equilibrium position Period T: time it takes for object to complete one complete cycle of motion; e.g. from x = A to x = −A and back to x = A Frequency ƒ: number of complete cycles or vibrations per unit time Displacement x: is the distance measured from the equilibrium point Simple Harmonic Motion

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7 UCT PHY1025F: Vibrations & Waves SHM occurs whenever the net force along direction of 1D motion obeys Hooke’s Law -(i.e. force proportional to displacement and always directed towards equilibrium position) Not all periodic motion over the same path can be classified as SHM Initially, we will look at the horizontal mass-spring system as a representative example of SHM Simple Harmonic Motion

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8 UCT PHY1025F: Vibrations & Waves x is the displacement of the mass m from its equilibrium position (x = 0 at the equilibrium position) The negative sign indicates that the force is always directed opposite to displacement (i.e. restoring force towards equilibrium) Hooke’s Law Review k is the spring constant spring force

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9 UCT PHY1025F: Vibrations & Waves A prosthetic leg contains a spring to absorb shock as the person is walking. If an 80 kg man compresses the spring by 5 mm when standing with his full weight on the prosthetic, what is the spring constant (k)? How far would the spring compress for a 100 kg man? Example: Hooke’s Law

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10 UCT PHY1025F: Vibrations & Waves From Newton II, for a mass-spring system: For a horizontal mass-spring system & all other cases of SHM, acceleration depends on position Since acceleration is not constant in SHM standard “equations of motion” cannot be applied Horizontal Mass on a Spring

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11 UCT PHY1025F: Vibrations & Waves V&S Example 13.2: A 0.350-kg object attached to a spring of force constant 1.30 x 10 2 N/m is free to move on a frictionless horizontal surface. If the object is released from rest at x = 0.10 m, find the force on it and its acceleration at x = 0.10 m, x = 0.05 m, x = 0 m, x = -0.05 m, and x = -0.10 m. Example: SHM

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12 UCT PHY1025F: Vibrations & Waves SHM occurs whenever the net force along direction of 1D motion obeys Hooke’s Law For a pendulum, the restoring force is Does this motion qualify as simple harmonic motion? A. Yes B. No The Simple Pendulum

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13 UCT PHY1025F: Vibrations & Waves A pendulum only exhibits SHM if it is restricted to small-angle oscillations (< 10°). For such small angles (in radians), we get the small-angle approximation, where The Simple Pendulum

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14 UCT PHY1025F: Vibrations & Waves Using the small-angle approximation, the restoring force becomes The pendulum displacement (the arclength s) is proportional to the angle giving The Simple Pendulum Linear restoring force

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15 UCT PHY1025F: Vibrations & Waves The potential energy of a spring (Section 6-4): The kinetic energy of the mass (Section 6-3): Therefore the total energy of the spring-mass system is: This total energy is conserved (assuming no friction, etc…) Energy in a Mass-Spring System

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16 UCT PHY1025F: Vibrations & Waves Energy in Simple Harmonic Motion

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17 UCT PHY1025F: Vibrations & Waves A 4.0 kg mass attached to a horizontal spring with stiffness 400 N/m is executing simple harmonic motion. When the object is 0.1 m from equilibrium position it moves with 2.0 m/s. Calculate the amplitude of the oscillation Calculate the maximum velocity of the oscillation Example: Energy of Spring

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18 UCT PHY1025F: Vibrations & Waves Conservation of energy allows the calculation of the velocity of an object attached to a spring at any position in its motion: Energy in Simple Harmonic Motion

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19 UCT PHY1025F: Vibrations & Waves SHM and Uniform Circular Motion

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20 UCT PHY1025F: Vibrations & Waves Simple Harmonic Motion The position, velocity and acceleration are all sinusoidal The frequency does not depend on the amplitude The object’s motion can be written as

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21 UCT PHY1025F: Vibrations & Waves Example: SHM

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22 UCT PHY1025F: Vibrations & Waves The Simple Pendulum (Review)

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23 UCT PHY1025F: Vibrations & Waves Frequency of Simple Pendulum

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24 UCT PHY1025F: Vibrations & Waves Frequency and Period

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25 UCT PHY1025F: Vibrations & Waves Damped harmonic motion happens when energy is removed (by friction, or design) from the oscillating system. Resonance occurs when energy is added to an oscillator at the natural frequency of the oscillator. Damping & Resonance

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26 UCT PHY1025F: Vibrations & Waves All systems have a natural frequency, the frequency at which a system will oscillate if left by itself. Natural Frequency

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27 UCT PHY1025F: Vibrations & Waves Resonance occurs when energy is added to an oscillator at the natural frequency of the oscillator. If an external force of this frequency is applied, the resulting SHM has huge amplitude! Resonance

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28 UCT PHY1025F: Vibrations & Waves The basic properties of waves (the wave model) cover aspects of wave behaviour common to all waves. A wave is the motion of a disturbance. The Wave Model Waves carry energy & momentum without the physical transfer of material. A traveling wave is an organized disturbance with a well- defined wave speed.

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29 UCT PHY1025F: Vibrations & Waves Mechanical Waves … require some source of disturbance and a medium that can be disturbed with some physical connection or mechanism through which adjacent portions can influence each other (e.g. waves on a string, sound, water waves) Two Types of Waves: Mechanical

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30 UCT PHY1025F: Vibrations & Waves Electromagnetic Waves... don’t require a medium and can travel in a vacuum (e.g. visible light, x-rays etc) Two Types of Waves: Electromagnetic

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31 UCT PHY1025F: Vibrations & Waves A wave pulse can be created with a single ‘snap’ on a rope Energy is transmitted from one point on the rope to the next A periodic (continuous) wave can be created by wiggling the rope up and down continuously Energy is continuously being transmitted along the rope Making a wave

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32 UCT PHY1025F: Vibrations & Waves Types of Mechanical Travelling Waves Transverse waves: Longitudinal waves: In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion. In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave. A longitudinal wave is also called a compression wave.

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33 UCT PHY1025F: Vibrations & Waves Some definitions…

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34 UCT PHY1025F: Vibrations & Waves Waves on a string (transverse waves) are propagated by the difference in directions of the tensions. Sounds waves (longitudinal waves) are pressure waves. Waves on a String and in Air

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35 UCT PHY1025F: Vibrations & Waves Wave Speed: String

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36 UCT PHY1025F: Vibrations & Waves Two travelling waves can meet and pass through each other without being destroyed or even altered. The Principle of Superposition Principle of Superposition -when two waves pass through the same point, the displacement is the sum of the individual displacements Pulses are unchanged after the interference.

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37 UCT PHY1025F: Vibrations & Waves Constructive: Two waves, 1 and 2, have the same frequency and amplitude and are “in phase.” The combined wave, 3, has the same frequency but a greater amplitude. Constructive Interference

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38 UCT PHY1025F: Vibrations & Waves Destructive: Two waves, 1 and 2, have the same amplitude and frequency but one is inverted relative to the other (i.e. they are 180° “out of phase”) When they combine, the waveforms cancel. Destructive Interference

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39 UCT PHY1025F: Vibrations & Waves Just like light reflects off water or an echo bounces off a cliff, a wave pulse on a string will reflect at a boundary. Whenever a traveling pulse reaches a boundary, some or all of the pulse is reflected. There are two types of boundaries: -Fixed end -Loose end Wave Pulse Reflection

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40 UCT PHY1025F: Vibrations & Waves When a pulse is reflected from a fixed end, the pulse is inverted, but the shape and amplitude remains the same. Reflection of Pulses – Fixed End Think about Newton’s 3 rd law at the boundary point.

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41 UCT PHY1025F: Vibrations & Waves When reflected from a free end, the pulse is not inverted, again the shape and amplitude remains the same. Reflection of Pulses – Free End Think about Newton’s 3 rd law at the boundary point.

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42 UCT PHY1025F: Vibrations & Waves Pulse Refection at a Discontinuity A discontinuity can act like a fixed or a free end depending on how the medium changes. Low to high linear mass density acts like fixed end High to low linear mass density acts like free end

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43 UCT PHY1025F: Vibrations & Waves When a travelling wave reflects back on itself, it creates travelling waves in both directions. The wave and its reflection interfere according to the Principle of Superposition. The wave appears to stand still, producing a standing wave. Standing Waves

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44 UCT PHY1025F: Vibrations & Waves A simple example of a standing wave is a wave on a string, like you will see in Vibrating String practical. The mechanical oscillator creates a traveling wave that is reflected off the fixed end and interferes with itself. The result is a series of nodes and antinodes, with the exact number depending on the oscillating frequency. Standing Waves on a String

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45 UCT PHY1025F: Vibrations & Waves Nodes are points where the amplitude is 0. (destructive interference) Anti-nodes are points where the amplitude is maximum. (constructive interference) Distance between two successive nodes is ½ λ. Standing Waves on a String

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46 UCT PHY1025F: Vibrations & Waves The figure shows the “n = 2” standing wave mode. The red arrows indicate the direction of motion of the parts of the string. All points on the string oscillate together vertically with the same frequency, but different points have different amplitudes of motion. Standing Waves on a String

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47 UCT PHY1025F: Vibrations & Waves Standing Wave on a String

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48 UCT PHY1025F: Vibrations & Waves Each mode has a specific wavelength. Standing Wave on a String

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49 UCT PHY1025F: Vibrations & Waves Standing Wave on a String

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50 UCT PHY1025F: Vibrations & Waves Standing Wave on a String

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