L 21 – Vibrations and Sound [1] Flow past an object • Resonance object vorticies • Tacoma Narrows Bridge Collapse wind • clocks – pendulum • springs November 7, 1940 • harmonic motion vortex street - exerts a • mechanical waves periodic force on the object • sound waves • musical instruments an example of resonance in mechanical systems Vortex street behind Selkirk Island The earth is shaking in the South Pacific S waves P waves Earthquakes http://www.geo.mtu.edu/UPSeis/waves.html http://www.classzone.com/books/earth_science/terc/content/visualizations/es1005/es1005page01.cfm?chapter_no=visualization Keeping time Î Clocks Earthquake and Tsunami hourglass sundial Length candle burns 1800 Clocks Digital clock with mainspring 1 water clock- Greeks length of a shadow As the water leaks out, level goes down. Non-repetitive Clocks Repetitive Clocks • measures a single interval of time • based on an object whose motion repeats • hourglass itself at regular intervals • shadows-sundials • does not require frequent operator • candles intervention • water clocks • Î pendulum clock • poorly suited to subdividing a day • first used by Galileo to measure time • requires frequent operator • based on harmonic oscillators – objects intervention that vibrate back and forth • not portable The pendulum- a closer look The “restoring” force • The pendulum is driven by gravity – the mass is falling • To start the pendulum, you L L from point A to point B then displace it from point B to rises from B to C point A and let it go! T T • the tension in the string T • point B is the equilibrium T forces it to move in a circle position of the pendulum C A C A • one component of mg Í • on either side of B the blue B T mg B mg is along the circular arc – T force always act to bring mg always pointing toward T (restore) the pendulum point B on either side. At back to equilibrium, point B point B this blue force mg mg vanishes. • this is a force mg “restoring” 2 the role of the restoring force Repeating motions • the restoring force is the key to understanding all systems that oscillate or repeat a motion • if there are no forces (friction or air over and over. resistance) to interfere with the motion, the • the restoring force always points in the motion repeats itself forever Æ it is a direction to bring the object back to harmonic oscillator equilibrium (for a pendulum at the bottom) • harmonic – repeats at the same intervals • from A to B the restoring force accelerates • notice that at the very bottom of the the pendulum down pendulum’s swing (at B ) the restoring • from B to C it slows the pendulum down so force is ZERO, so what keeps it going? that at point C it can turn around it’s the INERTIA ! let’s look at energy • to start the pendulum, we move it from B to A. A t • even though the restoring force is zero at point A it has only gravitational potential energy the bottom of the pendulum swing, the ball (GPE) due to gravity is moving and since it has inertia it keeps • from A to B, its GPE is converted to kinetic moving to the left. energy, which is maximum at B (its speed is • as it moves from B to C, gravity slows it maximum at B too) down (as it would any object that is • from B to C, it uses its kinetic energy to climb moving up), until at C it momentarily up the hill, converting its KE back to GPE comes to rest. • at C it has just as much GPE as it did at A • ÎÎÎlarge pendulum demoÍÍÍ Some terminology period and frequency A A The mass/spring oscillator is the simplest example • The period T and frequency f are related 0 to each other. • the maximum displacement of an object • if it takes ½ second for an oscillator to go from equilibrium is called the AMPLITUDE through one cycle, its period is T = 0.5 s. • the time that it takes to complete one full • in one second, then the oscillator would cycle (A Æ B Æ C Æ B Æ A ) is called complete exactly 2 cycles ( f = 2 per the PERIOD of the motion second or 2 Hertz, Hz) • if we count the number of full cycles the • 1 Hz = 1 cycle per second. oscillator completes in a given time, that is • thus the frequency is: f = 1/T and, T = 1/f called the FREQUENCY of the oscillator 3 Mass hanging from a spring springs Æ amazing devices! • a mass hanging from a spring also executes harmonic motion up and down. stretching the harder I pull on a spring, • to understand this the harder it pulls back motion we have to first understand how the harder I push on springs work. a spring, the harder it pushes back compression Springs obey Hooke’s Law springs are useful ! elastic limit of the spring 1 2 1 spring force (N) 1 2 amount of stretching or compressing in meters • the strength of a spring is measured by how much force it provides for a given amount of stretch • we call this quantity k, the spring constant in N/m • magnitude of spring force = k × amount of stretch springs help you sleep more comfortably! the mass/spring oscillator the mass spring oscillator does not need gravity • as the mass falls down it stretches the spring, which spring that can makes the spring force be stretched or bigger, thus slowing the compressed frictionless mass down surface • after the mass has come momentarily to rest at the bottom, the spring pulls it back up • at the top, the mass starts falling again and the process the time to complete an oscillation does not continues – oscillation! depend on where the mass starts! 4 T = period, time for one complete cycle Pendulum Mass-spring L m T = 2π T = 2π pendulum g mass− spring k • L = length (m) • m = mass in kg • g = 10 m/s2 • k = spring constant • does not depend in N/m on mass 5.