Harmonic Motion

[Music] there are many things in the physical world that rest in balance and are called stable systems they are in a state of equilibrium in such a state the chords of this piano for example do not produce any sounds I must first be struck or disturbed before they will produce their individual notes a property of stable systems is that once disturbed from equilibrium the force that acts on them tends to push them back and that they tend to overshoot the equilibrium point when they return and wind up moving back and forth or oscillating even complex ecological systems demonstrate this fluctuation back and forth past the point of stability in this situation for example the hawk population is tied to that of the rabbits with the numbers of both prey and predators swelling and shrinking rather than remaining constant in physics this fundamental principle is called harmonic motion or sometimes it is referred to as periodic motion the accuracy of any timekeeping device whether it is mechanical electronic or even nuclear depends on harmonic motion harmonic motion is a mechanical phenomena it refers to motion that repeats itself over equal time intervals [Music] when an object moves back and forth over the same path in harmonic motion it is said to be oscillating any bit of matter even an atom that pasa lates would uniform frequency is demonstrating harmonic motion the time needed to complete one oscillation is called the period or frequency of motion the pendulum in this grandfather's clock has a period of about one second at the heart of this electronic watch is a quartz crystal it's molecules exhibit harmonic motion vibrating at several hundred million beats per second all harmonic sounds have this periodic motion as their underlying principle whether it is the vibration of a piano chord or the column of air in a wind instrument pleasing resonance is a result of all the air molecules vibrating in harmonic unison every object has its own natural frequency or resonance when this frequency is intensified or amplified some pretty spectacular things start to happen but harmonics and residents are best put into perspective when harmonic motion is clearly understood and for science that understanding grew out of a need for more accurate timepieces a need that began long ago for thousands of years civilization struggled with a problem of keeping accurate time some of the earliest devices depended on the progression of the Sun across the sky these Sun dials were able to segment the day into rather large undefined intervals early sand clocks marked the passage of time with somewhat greater refinement the ancient Chinese water clocks were also an improvement however they were for the most part in practical since freezing temperatures literally made time standstill the beautiful mechanical clocks of 13th and 14th century Europe had no uniformity what was desperately needed was something that provided uniform intervals to segment time something the very nature of which would allow it to become the heart to the clock itself the famous Italian scientist Galileo discovered just what was needed and as legend has it in a rather unorthodox manner as the story goes Galileo was sitting in church one day soon his attention wandered over to a lamp swinging back and forth from a bell tower room at this point he made some startling observations using his pulse to mark the period between sweeps Galileo discovered that no matter how large or how small the arc of the swing the period always remained the same in other words while the swinging lamp gradually decreased its sweep the interval between each cycle remained constant Galileo quickly grasped the significance of his discovery and soon pendulums were built into all clocks providing the uniform intervals so necessary to accurate timekeeping any object mounted so that it can swing back and forth about an axis is called a pendulum Sir Isaac Newton one of the most brilliant thinkers of all time provided some very significant insights into the nature of pendulum motion Newton demonstrated that the period of a pendulum is independent of the size of the bob that is the quantity of mass at the tip of the pendulum this is true for the same reason that gravity attracts all bodies equally regardless of mass in space away from the Earth's gravity a pendulum could not swing he proved that the period is directly proportional to the square root of the pendulums life [Music] for example a pendulum four times the length of another pendulum will take twice the time to complete one oscillation Newton also proved that the period of a pendulum is independent of its amplitude the amplitude is the maximum movement on either side of the pendulums equilibrium point Newton showed that both large and small sweeps of a given pendulum occur within identical time cage a pendulum is a specific example of harmonic motion you can explain the physics behind the movement of a pendulum to demonstrate in part just how fundamental to all of nature were his laws of motion Newton believed that virtually every phenomenon in the physical world to be explained through his three laws of motion and harmonic motion is a keystone to understanding much of the mechanics of the physical world an object undergoing harmonic motion is called a harmonic oscillator the two rods joined to this weight are used to demonstrate a very important property of harmonic oscillators and that is the potential for the total transfer of energy and momentum a force on the spring gives the weight its initial oscillation and the mass Bob's up and down however the spirals of the spring also give a twisting component to the motion so the vertical oscillations gradually yield to the angular oscillations which in turn give way to the vertical motion as one builds the other waves until the energy transfer is complete and the motion briefly stops in one plane only to quickly renew again [Laughter] [Music] here two pendulums are joined together with a floppy spring creating something called a coupled oscillator [Music] yet just one pendulum is set into motion something very interesting happens with profound implications the first pendulum gradually slows while the second slowly picks up momentum soon the first pendulum has completely stopped while only the second one moves [Music] before long the entire process has reversed itself [Music] what is happening is that energy is being transferred from one oscillating object to another and then back again these pendulums are known as loosely coupled oscillators there are also examples of oscillators with a stiff or even momentary coupling the sudden impact transfers the energy from one object or group of objects to another with the cycling of each oscillation the energy moves from potential to kinetic back to potential and so on friction or the loss of energy through heat and air resistance prevent this exchange from going on forever but the deeper significance of these demonstrations is that the fundamental principle of all wave motion is revealed [Music] an elastic medium like water or air is in reality an infinite number of molecules behaving like coupled oscillators when the calm surface tension of a pool of water is disturbed it bounces back setting up oscillations but something else happens ripples spread out across the surface this spreading out is much like the oscillations passing between the coupled pendulums in disturbing the surface of the water a wave is created and the water behaves like many oscillators strongly coupled together always are the propagation of a disturbance from one oscillator to another every object or system has a natural frequency at which it vibrates in quality musical instruments the components are chosen for the tones they can produce prints and finishes affect the quality of the tones the types of wood how they are handled and joined together all contribute to the quality of the sound in the final stages of assembling a piano for example close attention is paid to matching components so that the richest most resonant tones are produced in sound waves the source of the vibration in this case a tuning fork vibrates the air around it at a specific frequency each oscillation of the tuning fork gives a tiny push against the air molecules surrounding air molecules like those of water also behave like strongly coupled oscillators propagating the vibration throughout the medium an identical tuning fork placed adjacent to the first would begin vibrating just from the tiny pushes of airs frightening a child's swing is a type of pendulum everyone knows that if you want the swing to go higher you must push it in step or in phase with the motion of the swing [Music] in fact if you push at the right point each time you don't have to push harder to make the swing go higher in physics this is described as forced oscillations and is similar to what the tiny pushes of air were doing to the second tuning fork the repeated application of a small force can create large amplitude vibrations if the force is applied in phase four the oscillating system under these conditions resonance occurs sometimes resonance can cause an oscillating system to fall apart shattering glass with one's voice for example is not a real life possibility however it can be done in this experiment a microphone picks up the sound of a glass beaker as it is gently tapped it's frequency is analyzed and a frequency generator is used to match the natural frequency of the beaker a loudspeaker reproduces the exact frequency in effect adding energy in phase to the already oscillating system too much energy added to the system and even though the driving force may be small the results of residents can be spectacularly lost as with a child swing small amounts of energy are added to the system exactly in step four the existing motion the resulting amplitudes are considerably increased many things in everyday life exhibit resonance most cars produce a rattle at certain motor speeds this means that somewhere there is a natural harmonic oscillator whose frequency is matched by that motor speed fortunately there are dentling devices that reduce the effective resonance these weights are oscillating up and down at a specific frequency [Music] connected to the weights is a motor-driven disc of variable frequency [Music] when its frequency is matched to that of the oscillating waves the system's resonant frequency is approached and it begins to fly apart here a beaker of water provides enough dampening to prevent destruction of the systems even at resonant frequencies [Music] sometimes the effects of resonance can be quite threatening in an earthquake for example seismic waves are generated from the epicenter in a broad range of frequencies these frequencies are mostly low compared to audible sound buildings between five and ten storeys tall are typically resonant at earthquake frequencies these cans were made to match the seismic patterns of an earthquake that occurred in Long Beach California in the 1930s each cam represents ground movement in a separate plane in an earthquake the real-life counterparts of these model buildings and literally break apart due to resonance Architects create model buildings to study their response to a variety of resonant frequencies [Music] damage can be minimized by increased dampening such as increasing the friction in joints telephone wires singing in the wind are another interesting example of residence this effect is created by the flow of wind around the wire in high winds the normally smooth flow of air around the wire becomes increasingly unstable complicated flow patterns develop and vortices are created under the right wind speeds the vortices rapidly peel off at a rate matching the resonant frequency of the wire at that point the wire starts to vibrate or sing the ancient Greeks notice the eerie sounds produced by their harps due to this effect which we now call an A or iam car you are about to witness what has become known as the world's greatest Aeolian park the Tacoma Narrows Bridge opened in July 1940 and closed and took four months later what is happening here is a classic example of engineering vs. aerodynamics [Music] the high winds of the Olympic Peninsula proved too much for the bridges design and on November 7th 1940 the bridge reached its resonant frequency [Music] [Applause] [Music] since then every new suspension bridge built has been scaled tested in wind tunnels [Music] from the tiny vibrating molecules of a quartz crystal in an electronic watch to the resonant frequency of a suspension breaks harmonic motion is present in some you [Music] in nature an endless flow of energy begins on an inconceivably small scale and builds to distances to vast for the mind to comprehend and throughout this broad display of creativity our monic motion brings rhythm to the universe every bit of matter vibrates with its own natural frequency and in some form contributes to the harmonies of life