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This 13 page Class Notes was uploaded by Milton O'Hara on Sunday October 11, 2015. The Class Notes belongs to PHYS170 at Duquesne University taught by Staff in Fall. Since its upload, it has received 42 views. For similar materials see /class/221281/phys170-duquesne-university in Physics 2 at Duquesne University.
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Date Created: 10/11/15
Review Chapter 2 0 Simple Harmonic Motion SHlVI X XMAXcoswt X is the particle s displacement also called amplitude at any given time XMAX is the maximum displacement of the particle wis known as the particle s angular frequency and w is what s known as a phase constant In most cases but not all a 0 and the equation looks a bit simpler X X W coswt The time between repetitions of the same value is called the period of the motion is represented by the symbol T and has units of seconds 0 Linear frequency is how many repetitions of the motion occur in one second is denoted by the symbol f and has units of Hertz HZ which are cyclessecond or vibrationssecond 0 Linear frequency and period are related by a simple mathematical formula namely 73 Angular frequency is given by the formula a 2rg has units of radianssecond and is purely needed because the sine and cosine functions model the behavior of something undergoing SHM One characteristic of SHM is that the period of motion is independent of the particle s amplitude or displacement from equilibrium Review Chapter 2 SHM is motion where the force or acceleration is proportional to the displacement of the particle and the motion is periodic in time o The springmass system Simple Pendulums 1 1 f fi 5 2 m 2 l M Spring ltlt Load For small amplitudes displacements of 15 or less from equilibrium Air Springs For air at standard temperature and pressure Y 14 Helmholtz resonator Review Chapter 2 From the Position vs Time graph of a springmass system we created a Velocity vs Time graph I Ill 1 I 14 I l L l We created PE and KB graphs from these two graphs and the respective formulae u lt lll 1 III III III J SPRING PE lkAx2 KE lmv2 2 2 Ignoring friction energy is changed from potential energy to kinetic energy back to potential energy back to kinetic energy However the total energy remains constant PE KE constant Kan 111 Tin I Review Chapter 2 If we consider friction energy will be transferred to types we can t put back into our system and oscillations die out There are two types of oscillation Longitudinal motion in the direction of propagation Iquot I l I llllimllllll Transverse motion is perpendicular to the direction of propagation I ll4 33 I ll 0 In systems of more than one spring and mass more modes of vibration are possible and very often each mode has its own frequency The larger frequency occurs when the masses are moving in opposite directions 1 7 WW Wp 0 If two independent vibrational modes happen to have the same frequency of vibration they are called degenerate modes fundamental mode of vibration the mode of vibration with the lowest frequency harmonic mode of vibration a higher mode of vibration that is a wholenumber multiple of the fundamental Tuning fork convenient standards of frequency maintain set frequency for a long time Chladni patterns way to study modes of vibration Nada circlc Nada diameter Q2394 adage Fourier analysis a mathematical technique used to identify each frequency present as well as its relative strength gac s 335 Amplitude Frequency Review Chapter 3 There are two characteristics common to all waves 1 A wave is a traveling disturbance 2 A wave carries energy Electromagnetic waves light waves radio waves are the only waves that can travel without a medium Transverse waves the disturbance of the particles is perpendicular to the disturbance s direction of travel I quotxquot Longitudinal waves the disturbance of the particles is parallel to the the disturbance s direction of travel Y Acoskxwt This equation is only for waves traveling in one dimension Notice the amplitude depends on position x and time I The wave number k has units of radiansmeter and is related to the wavelength k 27 Hal Wavelength is the distance between two repeated amplitudes on the wave The equations for nding the speed of a wave is really just the de nition of speed put in the lingo of waves V vtf H P Review Chapter 3 0 Graphs of traveling waves In a solid longitudinal waves travel with a speed given by V 7 while the transverse speed of a wave on a string is given by V Z P M What happens to the pulse or wave when it reaches the end of the rope or junction connecting a different rope Situation 1 The rope is attached to an immovable object such as a wall The pulse is ipped after it hits the end of the rope It returns to the sender with an inverted shape Mathematically this is called undergoing a 180 or 7 radians phase shift Situation 2 The rope is not attached to anything such as a whip The pulse is not ipped when it hits the end of the rope and returns to the sender no phase shift Situation 3a two ropes are attached to each other and pi gt pm There will be a re ected pulse at is in phase with the original pulse and a transmitted pulse in the less massive rope that is in phase with the original pulse g lt T Situation 3b two ropes are attached to each other and pm lt pm There will be a re ected pulse that is out of phase with the original pulse and a transmitted pulse in the more massive rope that is in phase with the original pulse WW Review Chapter 3 Principle of Linear Superposition the total displacement at a point on the string will be the summation of the pulses at that point Destructive Interference the superposition of waves gives an amplitude that is less than the amplitudes of the waves that are interfering Fully Destructive Interference when two waves have the same frequency and amplitude but are out of phase by 180 Constructive Interference the superposition of waves gives an amplitude that is greater than the amplitudes of the waves that are interfering Fully Constructive Interference when two waves have the same frequency and amplitude but are in phase The resulting amplitude will be twice the amplitudes of the original waves Standing Wave this is the result of interference between two identical waves same amplitude and frequency that are traveling in opposite directions Sound waves are longitudinal disturbances in the air and are regions of high and low pressure The speed of sound in air depends on temperature As a general rule sound travels slowest in gases faster in liquids and fastest in solids v331306t v VRT Doppler Effect An effect in which the observed frequency is different than the frequency being emitted by a source and is caused by a relative motion between the observer and source of the sound Larger wavelength Smaller wavelength Truck mowng Review Chapter 3 0 Doppler Effect An effect in which the observed frequency is different than the frequency being emitted by a source and is caused by a relative motion between the ob server and source of the sound flf Vivol LViV J S 0 Reflection waves bounce off of barriers Refraction when the speed of a wave changes for some reason Often but not always refraction will cause a wave to change direction Refraction can happen when waves go from one medium to another or if the properties of a medium change 0 Diffraction the ability of waves to bend around obstacles v OVER SHOA l o Interference same ideas as earlier in the chapter Review Chapter 4 Resonance an effect when a system is driven at or near a natural frequency The resulting amplitude of the Vibration dramatically increases A There are two types of resonances narrow and broad The Qfactor is the man way to categorize resonances f Q a The smaller the linewidth Af the sharper the resonance or the higher the Q factor Basically the higher the Q factor the closer you have to get to resonance to see an effect While resonance is an important acoustical phenomenon resonant effects are not always desired This is how the book arrives at the generalized formula for the frequencies of modes of Vibration for a string xed at both ends T fn2 L here n l 2 3 and fn will be the frequency of the nth harmonic the 1st harmonic is usually called the fundamental Review Chapter 4 Overtone higher modes of vibration than the fundamental but not restricted to a Whole number multiple of the fundamental 0 Partial all modes of vibration the fundamental pulse overtones everything The upper partial excludes the fundamental and is the same thing as the overtones 0 Pipe open at both ends the boundary conditions require an antinode at each open end The resonant frequencies are given by 71V ff n123 Note all harmonics are present in this system Review Chapter 4 0 Pipe open at one end the boundary conditions require an antinode at the open end and a node at the closed end The resonant frequencies are given by I lV f n135 Note only the odd harmonics are present or audible in this system 0 Metallic rods With both ends free the boundary conditions require antinodes at both ends The resonant frequencies are given by Review Chapter 4 Sympathetic Vibrations the Vibration of one object due to the Vibration of another object Examples of this include the tuning fork and speakers Review Chapter 5 Outer ear l Mlddle Inner ear I Semicircular canals Eardrum Round window Eustachian tube logyNx Logarithms y is the base x is the power or exponent and N is the number generated by taking the base to the power Review Chapter 5 Logarithms amp my expectations be able to find the logarithms of simple cases for any base I give you without using a calculator be able to give the correct range of a logarithm for any base I give you without using a calculator be able to find the logarithm in base 10 of any number I give you using the table amp identities below not a calculator X logX X logX a log AB logA logB 1 0000 6 0778 2 0301 7 0845 b log AB logA logB 3 0477 8 0903 c 10g An quotlog A 4 0602 9 0954 5 0699 10 1000 Many of you may ask what this has to do with music This is the foundation for decibels a decibel scale is a logarithmic scale
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