Physics 1025 F Vibrations Waves OSCILLATIONS Dr Steve
Physics 1025 F Vibrations & Waves OSCILLATIONS Dr. Steve Peterson Steve. peterson@uct. ac. za UCT PHY 1025 F: Vibrations & Waves 1
Chapter 11: Vibrations and Waves Periodic motion occurs when an object vibrates or oscillates back and forth over the same path UCT PHY 1025 F: Vibrations & Waves 2
Periodic Motion Periodic motion, processes that repeat, is one of the important kinds of behaviours in Physics UCT PHY 1025 F: Vibrations & Waves 3
Equilibrium and Oscillation Equilibrium position – position where net force is zero Restoring force – force acting to restore equilibrium Oscillation – periodic motion governed by a restoring force UCT PHY 1025 F: Vibrations & Waves 4
Equilibrium and Oscillation 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) UCT PHY 1025 F: Vibrations & Waves 5
Simple Harmonic Motion 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 UCT PHY 1025 F: Vibrations & Waves 6
Simple Harmonic Motion SHM occurs whenever the net force along direction of 1 D 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 UCT PHY 1025 F: Vibrations & Waves 7
Hooke’s Law Review spring force k is the spring constant 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) UCT PHY 1025 F: Vibrations & Waves 8
Example: Hooke’s Law 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? UCT PHY 1025 F: Vibrations & Waves 9
Horizontal Mass on a Spring 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 UCT PHY 1025 F: Vibrations & Waves 10
Example: SHM V&S Example 13. 2: A 0. 350 -kg object attached to a spring of force constant 1. 30 x 102 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. UCT PHY 1025 F: Vibrations & Waves 11
The Simple Pendulum SHM occurs whenever the net force along direction of 1 D motion obeys Hooke’s Law For a pendulum, the restoring force is Does this motion qualify as simple harmonic motion? A. Yes B. No UCT PHY 1025 F: Vibrations & Waves 12
The Simple Pendulum 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 UCT PHY 1025 F: Vibrations & Waves 13
The Simple Pendulum Using the small-angle approximation, the restoring force becomes The pendulum displacement (the arclength s) is proportional to the angle giving Linear restoring force UCT PHY 1025 F: Vibrations & Waves 14
Energy in a Mass-Spring System 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…) UCT PHY 1025 F: Vibrations & Waves 15
Energy in Simple Harmonic Motion • UCT PHY 1025 F: Vibrations & Waves 16
Example: Energy of Spring 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 UCT PHY 1025 F: Vibrations & Waves 17
Energy in Simple Harmonic Motion Conservation of energy allows the calculation of the velocity of an object attached to a spring at any position in its motion: UCT PHY 1025 F: Vibrations & Waves 18
SHM and Uniform Circular Motion • UCT PHY 1025 F: Vibrations & Waves 19
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 UCT PHY 1025 F: Vibrations & Waves 20
Example: SHM • UCT PHY 1025 F: Vibrations & Waves 21
The Simple Pendulum (Review) • UCT PHY 1025 F: Vibrations & Waves 22
Frequency of Simple Pendulum • UCT PHY 1025 F: Vibrations & Waves 23
Frequency and Period • UCT PHY 1025 F: Vibrations & Waves 24
Damping & Resonance • 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. UCT PHY 1025 F: Vibrations & Waves 25
Natural Frequency All systems have a natural frequency, the frequency at which a system will oscillate if left by itself. UCT PHY 1025 F: Vibrations & Waves 26
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! UCT PHY 1025 F: Vibrations & Waves 27
The Wave Model 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. Waves carry energy & momentum without the physical transfer of material. A traveling wave is an organized disturbance with a welldefined wave speed. UCT PHY 1025 F: Vibrations & Waves 28
Two Types of 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) UCT PHY 1025 F: Vibrations & Waves 29
Two Types of Waves: Electromagnetic Waves. . . don’t require a medium and can travel in a vacuum (e. g. visible light, x-rays etc) UCT PHY 1025 F: Vibrations & Waves 30
Making a wave 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 UCT PHY 1025 F: Vibrations & Waves 31
Types of Mechanical Travelling Waves Transverse waves: In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion. Longitudinal waves: 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. UCT PHY 1025 F: Vibrations & Waves 32
Some definitions… • UCT PHY 1025 F: Vibrations & Waves 33
Waves on a String and in Air Waves on a string (transverse waves) are propagated by the difference in directions of the tensions. Sounds waves (longitudinal waves) are pressure waves. UCT PHY 1025 F: Vibrations & Waves 34
Wave Speed: String • UCT PHY 1025 F: Vibrations & Waves 35
The Principle of Superposition Two travelling waves can meet and pass through each other without being destroyed or even altered. 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. UCT PHY 1025 F: Vibrations & Waves 36
Constructive Interference 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. UCT PHY 1025 F: Vibrations & Waves 37
Destructive Interference 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. UCT PHY 1025 F: Vibrations & Waves 38
Wave Pulse Reflection 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 UCT PHY 1025 F: Vibrations & Waves 39
Reflection of Pulses – Fixed End When a pulse is reflected from a fixed end, the pulse is inverted, but the shape and amplitude remains the same. Think about Newton’s 3 rd law at the boundary point. UCT PHY 1025 F: Vibrations & Waves 40
Reflection of Pulses – Free End When reflected from a free end, the pulse is not inverted, again the shape and amplitude remains the same. Think about Newton’s 3 rd law at the boundary point. UCT PHY 1025 F: Vibrations & Waves 41
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 UCT PHY 1025 F: Vibrations & Waves High to low linear mass density acts like free end 42
Standing 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. UCT PHY 1025 F: Vibrations & Waves 43
Standing Waves on a String 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. UCT PHY 1025 F: Vibrations & Waves 44
Standing Waves on a String 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. UCT PHY 1025 F: Vibrations & Waves 45
Standing Waves on a String 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. UCT PHY 1025 F: Vibrations & Waves 46
Standing Wave on a String • UCT PHY 1025 F: Vibrations & Waves 47
Standing Wave on a String Each mode has a specific wavelength. UCT PHY 1025 F: Vibrations & Waves 48
Standing Wave on a String • UCT PHY 1025 F: Vibrations & Waves 49
Standing Wave on a String • UCT PHY 1025 F: Vibrations & Waves 50
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