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Physics Coach Kelsoe Pages 366-395 Section 1-Simple Harmonic Motion Objectives: Identify the conditions of simple harmonic motion Explain how force, velocity, and acceleration change as an object vibrates with simple harmonic motion Calculate the spring force using Hooke’s Law Intro to Hooke’s Law At the equilibrium position, speed reaches a maximum At maximum displacement, spring force and acceleration reach a maximum In simple harmonic motion, restoring force is proportional to displacement Simple Harmonic Motion Simple harmonic motion-vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium Putting it in perspective… Think of a pendulum on a clock for example. As the pendulum swings outward it reaches its maximum displacement. The pendulum’s starting position is its equilibrium. Therefore as the pendulum swings it constantly moves in simple harmonic motion, constantly moving from its maximum displacement to its equilibrium. Hooke Hooke’s law is the relationship between the restoring force and the displacement of the mass. Easily written as… Felastic= -kx spring force= - (spring constant x displacement) Negative sign signifies that the direction of the spring force is always opposite the direction of the mass’s displacement from equilibrium. The Simple Pendulum The restoring force of a pendulum is a component of the bob’s weight For small angles, the pendulum’s motion is simple harmonic Gravitational potential increases as a pendulum’s displacement increases Section 2-Measuring Simple Harmonic Motion Objectives: Identify the amplitude of vibration Recognize the relationship between period and frequency Calculate the period and frequency of an object vibrating with simple harmonic motion Amplitude, Period, and Frequency Amplitude- the maximum displacement from equilibrium (radians or meters) Period (T)-the time that it takes a complete cycle to occur (seconds) Frequency (f)-the number of cycles or vibrations per unit of time (Hertz-Hz) Frequency is the inverse of period. The period of a simple pendulum depends on pendulum length and free-fall acceleration. T=2π √L/ag Mass does not affect acceleration so therefore the pendulums’ period is the same. Have you ever noticed that when you first watch a pendulum it moves its maximum displacement when it first takes off? This is because when the amplitude increases the restoring force increases proportionally. Force is proportional to acceleration, so the initial acceleration will be greater. Period of a mass-spring system depends on mass and spring constant. Period of a mass-spring system in simple harmonic motion: T=2π√m/k (period=2π/square root of (mass divided by spring constant) Section 3-Properties of Waves Objectives: Interpret waveforms of transverse and longitudinal waves. Apply the relationship among wave speed, frequency, and wavelength to solve problems. Transverse vs. Longitudinal There are two types of waves: transverse and longitudinal. Transverse waves are perpendicular to the wave motion. Longitudinal waves are parallel to the wave motion. Waves can be measured in terms of its displacement. Important Definitions: Transverse wave- a wave whose particles vibrate perpendicularly to the direction the wave is traveling. Crest-the highest point above the equilibrium position Trough- the lowest point below the equilibrium position Wavelength-distance between two adjacent similar points of a wave, such as from crest to crest or from trough to through Longitudinal wave- wave whose particles vibrate parallel to the direction the wave is traveling … Medium- a physical environment through which a disturbance can travel Mechanical wave- a wave that requires a medium through which to travel Finding speed of wave After deriving, the speed of the wave is equal to frequency times wavelength… V=f λ Section 4- Wave Interactions Constructive Interference- a superposition of two or more waves in which individual displacements on the same side of the equilibrium position are added together to form the resultant wave Destructive Interference- a superposition of two or more waves in which individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave. Standing Wave-a wave pattern that results when two waves are the same frequency, wavelength, and amplitude travel in opposite directions and interfere. Continues with… Node- a point in a standing wave that maintains zero displacement Antinode- a point in a standing wave, halfway between two nodes, at which the largest displacement occurs. Just a few things to know… If two or more waves are moving through a medium, the resultant wave is found by adding the individual displacements together point by point. Standing waves are formed when two waves that have the same frequency, amplitude, and wavelength travel in opposite directions and interfere. Chapter 12 Sound Sound Physics Compression is the region of a longitudinal wave in which the density and pressure are at a maximum. Rarefaction is the region of a longitudinal wave in which the density and pressure are at a minimum. Pitch is a measure of how high or low a sound is perceived to be, depending on the frequency of the sound wave. The Doppler Effect is an observed change in frequency when there is relative motion between the source of waves and an observer. Characteristics of Sound Waves Frequency determines pitch. The frequency of an audible sound wave determines how high or low we perceive the sound to be, which is known as pitch. As the frequency of a sound wave increases, the pitch rises. The speed of sound depends on the medium. Because waves consist of particle vibrations, the speed of a wave depends on how quickly one particle can transfer its motion to another particle. The speed also depends on the temperature of the medium. Sound waves propagate in three dimensions. Sound waves travel away from a vibrating source in all three dimensions. The wave fronts of sound waves spreading in three dimensions are approximately spherical. Rarefaction and Compression Example of Refraction and Compression The sound from a tuning fork is produced by the vibrations of each of its prongs. When a prong swings to the right, there is a region of high density and pressure. When the prong swings back to the left, a region of lower density and pressure exists. Doppler Effect! As a police car or ambulance passes you, with its siren wailing, the sound of the siren seems to drop to a lower note. This is due to something known as the Doppler Effect, named after the Austrian scientist Christian Johann Doppler, who discovered the reason for it in 1842. The Doppler Effect is defined as an observed change in frequency when there is relative motion between the source of waves and an observer. Doppler found that if a source of sound is moving, the sound waves are crowded together in front of it and spread apart behind it. Squeezing sound waves together increases their frequency, while spreading them lowers their frequency. As a result, to someone at rest the sound will seem higher when the source is approaching and lower when it is moving away. At the time when Doppler announced his findings, there was no easy means to test them. The fastest form of transport which carried a horn was the horse-drawn mail coach. This only moved at about ten miles an hour – too slow for the Doppler Effect to be noticeable. It was only in 1845, when a scientist took a trumpet aboard a train locomotive, that the effect was first demonstrated. Sound Waves Section Two Vocabulary Intensity is the rate at which energy flows through a unit area perpendicular to the direction of wave motion. A decibel (dB) is a dimensionless unit that describes the ratio of two intensities of sound; the threshold of hearing is commonly used as the reference intensity Resonance is a phenomenon that occurs when the frequency of a force applied to a system matches the natural frequency of vibration of the system, resulting in a large amplitude of vibration Sound Intensity! Intensity is the rate of energy flow through a given area. Sound waves traveling in air are longitudinal waves. As the sound waves travel outward from the source, energy is transferred from one air molecule to the next. The rate at which this energy is transferred through a unit area of the plane wave is called the intensity of the wave. Because power, P, is defined as the rate of energy transfer, intensity can also be described in terms of power. Intensity has units of watts per square meter (W/m₂) Intensity= ΔE/ Δt = area P area Intensity of a Spherical Wave In a spherical wave, energy propagates equally in all directions; no one direction is preferred over any other. The power emitted by the source (P) is distributed over a spherical surface (area=4∏r₂), assuming that there is no absorption in the medium. • Intensity = (power) . (4∏)(distance from the source)₂ Decibels and Sound Moderation Intensity and frequency determine which sounds are audible. The softest sound that can be heard by the average human ear occurs at a frequency of about 1000 Hz and an intensity of 1.0 x 10-12. Such a sound is said to be at the threshold of hearing. The loudest sounds that the human ear can tolerate have an intensity of about 1.0 W/m2. This is known as the threshold of pain. Relative intensity is measured in decibels. Relative intensity is the ratio of the intensity of a given sound wave to the intensity at the threshold of hearing. Because of the logarithmic dependence of perceived loudness on intensity, using a number equal to 10 times the logarithm of the relative intensity provides a good indicator for human perceptions of loudness. This is referred to as the decibel level. The decibel level is dimensionless because it is proportional the logarithm of a ratio. Forced Vibrations and Resonance Vibrations at the natural frequency produces resonance. Every pendulum vibrates at a certain frequency known as its natural frequency. The human ear transmits vibrations that cause nerve impulses. Sound waves travel through the three regions of the ear and are then transmitted to the brain as impulses through nerve endings on the basilar membrane. Sound waves of varying frequencies resonate at different spots along the basilar membrane, creating impulses in hair cells embedded in the membrane. These impulses are then sent to the brain, which interprets them as sounds of varying frequencies. Section Three Vocabulary Fundamental frequency is the lowest frequency of vibration of a standing wave. A harmonic series is a series of frequencies that includes the fundamental frequency and integral multiples of the fundamental frequency. Timbre is the musical quality of a tone resulting from the combination of harmonics present at different intensities. A beat is the periodic variation in the amplitude of a wave that is the superposition of two waves of slightly different frequencies. Standing Waves on a Vibrating String The vibrating strings of a violin produce standing waves whose frequencies depend on the string lengths. Wave Basics •High points of the wave are called Crests •Low points are troughs •Amplitude is the maximum displacement from the undisturbed state. •Can be positive or negative. •The Wavelength is the distance between two adjacent corresponding points on the wave. •A waves frequency refers to how many waves are made per time interval, usually in seconds. The Doppler effect (or Doppler shift), named after Austrian physicist Christian Doppler who proposed it in 1842 The Doppler effect is how wave properties, specifically frequencies, are influenced by the movement of the source and the listen. The most common examples of this would be a train passing or police sirens. The frequency of the sounds that the source emits does not actually change. Merely the wavelength and the received are affected. The sounds are being sent out at the same rate but the person moving towards or away from the source hear them at changing intervals. To calculate the change in frequency due to the Doppler effect, consider a situation where the motion is set between a Listener L and the source S with the direction from listener to the source as the positive direction. fL = [(v + vL)/(v + vS)] fS The velocities vL and vS are the velocities of the listener and source relative to the wave medium. The speed of the sound wave, v, is always considered positive. After figuring all that out we get the frequency heard by the listener (fL) in terms of the frequency of the source (fS): If the listener is at rest vL = 0. If the source is at rest vS = 0. That means if both the listener and the source are at rest fL = fS. If the listener is moving toward the source, then vL > 0, though if it's moving away from the source then vL < 0. Alternately, if the source is moving toward the listener the motion is in the negative direction, so vS < 0, but if the source is moving away from the listener then vS > 0. The Doppler Effect in Light The major difference between light waves and sound waves is that light waves do not require a medium to travel through. Light waves from a moving source experience the Doppler effect to result in either as a red shift or blue shift in the light's frequency. A “Blue shift” occurs when the wavelength of light decreases and or the frequency increases. A “Red Shift” happens when a star or other object is moving away from the observer.