"Annenberg Media" "¶" "NEWTON'S SECOND LA+F = ma," "COMBINEDWITH ARESTORINGFORCE," "LEADSTOAMOTION THATREPEATSITSELF" "OVEREQUALPERIODSOFTIME ," "PRODUCINGWHATIS KNOWNAS HARMONICMOTION." "I HAVE A STORY." "I'M NOT SURE IT'S TRUE, BUT IT'S GOOD," "SO I'LL TELL IT TO YOU ANYWAY." "IT'S ABOUT OUR OLD FRIEND GALILEO." "ONE DAY, WHEN GALILEO WAS A YOUNG MAN IN PISA," "HE ATTENDED MASS AT THE DUOMO." "THAT'S THE CATHEDRAL IN PISA" "WHOSE BELL TOWER IS THE LEANING TOWER OF PISA." "IN THE DUOMO, THERE'S A LAMP THAT HANGS FROM THE CEILING." "ON THIS PARTICULAR DAY," "THE LAMP WAS SWINGING BACK AND FORTH." "GALILEO NOTICED SOMETHING ABOUT THE SWINGING LAMP" "THAT NO ONE HAD NOTICED BEFORE." "HE NOTICED THAT EACH COMPLETE SWING" "TOOK EXACTLY THE SAME AMOUNT OF TIME." "AT FIRST, THE LAMP MOVED QUICKLY THROUGH LARGE ARCS." "AS THE SWINGING DIED DOWN," "IT SWUNG MORE SLOWLY THROUGH SMALLER ARCS." "BUT THE AMOUNT OF TIME FOR EACH SWING" "WAS ALWAYS THE SAME." "HOW DID HE DETERMINE THAT?" "HE TIMED THE SWINGS BY COUNTING HIS PULSE." "I'VE WONDERED ABOUT THAT" "HAVING REALIZED HE WAS MAKING A GREAT DISCOVERY," "WHY HIS PULSE DIDN'T START RACING" "AND RUIN HIS MEASUREMENT." "ANYWAY, HE TIMED HIS OWN PULSE." "HE HAD TO WATCH IT COMPLETE MANY SWINGS," "COUNTING HIS PULSE EACH TIME" "TO MAKE SURE EACH SWING TOOK THE SAME AMOUNT OF TIME." "IT MUST HAVE BEEN A VERY DULL SERMON." "THE LAMP IS STILL THERE." "IT'S CALLED GALILEO'S LAMP." "YOU CAN BUY A PICTURE OF IT" "FROM THE STREET VENDERS OUTSIDE ON THE PIAZZA." "THE TROUBLE WITH THE STORY IS," "ACCORDING TO CHURCH RECORDS," "THE LAMP WASN'T INSTALLED UNTIL AFTER GALILEO DIED." "BUT GALILEO DID MAKE THAT DISCOVERY," "AND IT WAS EXTREMELY IMPORTANT," "BECAUSE HE HAD DISCOVERED" "THAT A PENDULUM COULD BE USED AS A TIMING ELEMENT." "HIS DISCOVERY LED DIRECTLY TO THE INVENTION" "OF THE FIRST ACCURATE CLOCKS." "THE KIND OF MOTION HE WAS STUDYING" "IS CALLED SIMPLE HARMONIC MOTION." "BOBBING UP AND DOWN ON AN ORDINARY SPRING," "THIS SIMPLE WEIGHT REPEATS ITSELF ONCE EVERY SECOND." "AS TIME PASSES," "ITS MOTION DIES DOWN, ITS EXCURSIONS BECOME SMALLER." "BUT IT STILL TAKES EXACTLY ONE SECOND" "TO MAKE EACH CYCLE." "IT IS A PERFECT TIMEKEEPING DEVICE." "AND ITS MOTION IS CALLED SIMPLE HARMONIC MOTION." "KEEPING TIME HAS A MUSICAL CONNOTATION," "AS DOES THE WORD "HARMONIC."" "THAT'S NO ACCIDENT OF LANGUAGE." "INSTRUMENTS SHARE A SPECIAL PROPERTY" "WITH BOBBING WEIGHTS." "BOTH GENERATE VIBRATION AT A CERTAIN FREQUENCY," "WHICH PRODUCES A CERTAIN PITCH" "THAT DOESN'T CHANGE AS THE MOTION DECREASES." "MANY FACTORS ARE AT PLAY" "IN THE PRODUCTION OF MUSICAL TONE" "THE LENGTH OF AN INSTRUMENT'S STRING," "THE SIZE AND THE SHAPE OF AN INSTRUMENT," "THE TECHNIQUE WITH WHICH AN INSTRUMENT'S PLAYED," "AND CERTAINLY THE MUSICIAN'S ABILITY TO PLAY IT." "HOWEVER, IN THE PHYSICS OF MUSIC," "ONE FACTOR NEVER VARIES." "ONCE THE TONE HAS BEEN PRODUCED, THE PITCH REMAINS THE SAME," "EVEN THOUGH THE VIBRATIONS DIMINISH." "WHY DOES THE PITCH REMAIN CONSTANT?" "IN THE LANGUAGE OF CLASSICAL MECHANICS," "FORCE," "THE "F" IN ISAAC NEWTON'S" "F EQUALS MASS TIMES ACCELERATION." "INA CERTAINPOSITION, ALLFORCESAREBALANCED." "HOWEVER,WHENTHESPRING ISSTRETCHED," "ITATTEMPTSTO PULLTHE MASS BACKTOITSORIGINALPOSITION." "THEFARTHERTHEMASSMOVES," "THEGREATERTHEFORCE TOPULLIT BACK." "THESAMEPRINCIPLE WORKSCONVERSELY." "WHENTHESPRINGISCOMPRESSED, IT TRIES TO PUSH THE MASS BACK." "WHICHEVERDIRECTION THEMASSMOVES," "AFORCEEMERGES TO COUNTER THE DISPLACEMENT." "THE COMBINATION OF THIS FORCE ANDTHEMASS'SINERTIA" "WILLBETHEKEY  TOKEEPINGTIME." "AT EACH POINT IN ITS MOTION," "THE NET FORCE IS PROPORTIONAL" "TO THE DISTANCE x OF THE MASS FROM ITS EQUILIBRIUM POSITION" "ANDINTHEOPPOSITEDIRECTION." "THEEQUATIONIS F= - kx." "THEVALUEOF k" "DEPENDSONHOWSTIFF THESPRINGIS ." "IN WESTERN CULTURE," "FEW THINGS GOVERN HUMAN LIFE AS STEADILY" "AS THE MECHANISTIC CONCERN WITH TIME." "ALTHOUGH ADVANCING TECHNOLOGY" "GIVES MORE PEOPLE MORE TIME ON THEIR HANDS," "THERE'S SEEMS TO BE A NEED" "FOR INCREASINGLY ACCURATE TIMEKEEPING DEVICES." "PREOCCUPATION WITH TIME" "ISN'T A PHENOMENON OF THE FAST-PACED 20th CENTURY." "IT DEVELOPED GRADUALLY," "HOUR BY HOUR, DAY IN AND DAY OUT," "FOR THOUSANDS OF YEARS." "ACCURATE TIME, HOWEVER, CAME RELATIVELY LATE IN HISTORY." "IT ARRIVED WITH THE PHYSICAL PRINCIPLE" "OF HARMONIC MOTION." "AND ALONG THE WAY, OVER TIME," "HARMONIC MOTION BECAME THE ONLY MEANS" "OF TIMEKEEPING ACCURACY." "BUT IT'S TIME TO GO BACK TO THE BEGINNING." "HOW DID THE CONCEPT OF TIME DEVELOP?" "PERHAPS FROM WITNESSING THE CYCLES IN NATURE," "THE CHANGING SEASONS." "PERHAPS FROM NOTING" "THE ANNUAL SOLAR OR MONTHLY LUNAR CYCLES." "MANY EARLY CIVILIZATIONS BUILT MONUMENTS" "THAT SUGGEST INSIGHT INTO THE CYCLICAL MOVEMENTS" "OF THE SUN, MOON, AND EARTH." "THESE STRUCTURES" "MARKED THE PASSAGE OF TIME ON A BROAD SCALE." "EARLY CALENDARS REPRESENTED EFFORTS" "TO SEGMENT ANNUAL CYCLES INTO SMALLER INCREMENTS." "SUNDIALS DIVIDED A DAY, UNLESS IT WAS CLOUDY." "ABOUT 500 YEARS AFTER THE ADVENT OF THE SUNDIAL," "THE EGYPTIANS DEVELOPED THE WATER CLOCK." "THE DRIPPING WATER" "REVEALED SUCCESSIVE LINES WITHIN AN ALABASTER BOWL." "THIS BASIC DESIGN REMAINED UNCHANGED FOR CENTURIES," "UNTIL CHINESE ARTISANS" "IMPROVED IT BY ADDING SMALL, WHEEL-MOUNTED CUPS" "AND A TRIP LEVER." "AFTER MARCO POLO SPREAD THE WORD FROM THE ORIENT," "SEMIMECHANICAL CLOCKS ENJOYED CONSIDERABLE POPULARITY" "IN 13th-CENTURY EUROPE." "UNFORTUNATELY, EUROPEAN WINTERS," "FREEZING THE WATER CLOCKS IN PLACE," "OFTEN MADE TIME STAND STILL." "WHEN INTRODUCED IN THE 14th CENTURY," "THE SANDS OF TIME DIDN'T FREEZE." "HOWEVER, BECAUSE OF THE BULK OF THE SAND," "SUCH CLOCKS WERE LIMITED TO MEASURING SHORT INTERVALS." "AROUND THIS TIME," "THE FIRST MECHANICAL CLOCKS WERE INTRODUCED." "THEY HAD MOVING PARTS," "INCLUDING PULLEYS, WEIGHTS WITH ESCAPEMENTS," "GEARS, AND TRIP LEVERS." "OFTEN THEY WERE ORNATE, DELICATELY CRAFTED TIMEPIECES." "THEIR ACCURACY VARIED ACCORDING TO THE CRAFTSMAN'S SKILL" "AND THE QUALITY OF THE MATERIALS AT HAND." "EARLY MECHANICAL CLOCKS WERE FURTHER IMPROVEMENTS," "BUT TIME STILL WASN'T SOMETHING PEOPLE COULD COUNT ON." "THE LENGTH OF AN HOUR" "DEPENDED ON WHOSE CLOCK KEPT TIME." "DAY FOLLOWS NIGHT," "SEASON FOLLOWS SEASON," "AND YEAR FOLLOWS YEAR." "THE EARTH'S ENDLESS MOTIONS," "FOLLOWING THE INEXORABLE LAWS OF THE MECHANICAL UNIVERSE," "MARK PERFECT TIME." "EVERY HUMAN-MADE CLOCK" "IS AN ATTEMPT TO IMITATE NATURE'S PRECISION." "NO MATTER HOW INGENIOUS, ANCIENT WATER CLOCKS," "SAND CLOCKS, AND MECHANICAL CLOCKS" "ULTIMATELY FAED TO KEEP ACCURATE TIME" "FOR ANY LENGTH OF TIME." "WHY?" "BECAUSE THEY NEGLECTED TO IMITATE" "ONE ESSENTIAL ASPECT OF NATURE'S CLOCK," "THE CYCLE." "TIME AND NATURE ARE CYCLIC." "DAY AND NIGHT, WINTER AND SUMMER," "THE CYCLES REPEAT THEMSELVES ENDLESSLY." "RANGING FROM A MICROSCOPIC CELL TO THE ROTATIONS OF GALAXIES," "NATURE REPEATS HERSELF IN OSCILLATIONS." "THERE ARE NATURAL CYCLES." "[GUNSHOT]" "THE BEATING OF THE HUMAN HEART." "THE FROZEN PACE OF THE ADVANCING AND RETREATING GLACIER." "AND THERE ARE PERHAPS LESS NATURAL CYCLES," "SUCH AS THE BOOM AND BUST OF THE ECONOMY" "AND OTHER THINGS THAT TEND TO RISE AND FALL." "HOWEVER, THESE CYCLES DON'T OPERATE" "UNDER THE SAME LAWS THAT GOVERN NATURE'S CLOCK." "THEY DON'T OSCILLATE WITH NATURAL PRECISION." "THEY LACK PRECISE INTERVALS," "REPEATING FOREVER ACCORDING TO AN INEXORABLE LAW OF NATURE." "HUMAN IMITATIONS OF THE CLOCK OF THE HEAVENS" "ARE BASED ON A DIFFERENT BUT EQUALLY INEXORABLE LAW." "THEMOSTINEXORABLELAW  INMECHANICSIS F= ma." "aISTHESECONDDERIVATIVE OFx ." "INSIMPLEHARMONICMOTION," "FORCECOMESFROM THEDISPLACEMENTxITSELF," "F= - kx." "TOGETHER,THESETWOEQUATIONS GIVETHEDIFFERENTIALEQUATION" "THATDESCRIBES SIMPLEHARMONICMOTION." "THAT DIFFERENTIAL EQUATION" "REFERS NOT ONLY TO A MASS ON A SPRING," "BUT TO ANY PHYSICAL SYSTEM THAT, WHEN DISTURBED," "IS RESTORED TOWARD EQUILIBRIUM" "WITH A FORCE PROPORTIONAL TO THE DISTURBANCE." "FOR EXAMPLE, THE AIR PRESSURE IN AN ORGAN PIPE," "THE ANGLE OF A PENDULUM, THE BENDING OF A TUNING FORK," "OR THE ROTATION OF A CLOCK MOVEMENT." "THESE SYSTEMS UNDERGO HARMONIC OSCILLATIONS." "THESE OSCILLATIONS CAN BE TOO FAST TO SEE," "AT LEAST UNDER NORMAL CIRCUMSTANCES," "OR TOO SLOW TO WAIT FOR." "THEY CAN BE ANY OTHER FREQUENCY, HIGH OR LOW." "NO MATTER THE FREQUENCY, HOWEVER," "EACH CAN BE DESCRIBED" "BY THE SAME DIFFERENTIAL EQUATION." "THE DIFFERENTIAL EQUATION" "IS SOLVED BY THE TIME-HONORED TECHNIQUE OF GUESSING." "A LONG TIME AGO," "SOMEONE GUESSED" "THAT THE SOLUTION MAY BE A SINE FUNCTION." "IT WAS AN EDUCATED GUESS," "BECAUSE THE MOTION OF A MASS OSCILLATING ON A SPRING" "RESEMBLES THE MOTION OF THE SHADOW OF A PARTICLE" "MOVING IN UNIFORM CIRCULAR MOTION." "SO THE IDEA" "OF GUESSING A SINE FUNCTION" "SEEMED QUITE NATURAL." "IN THIS CASE," "A SINE FUNCTION WITH AMPLITUDE A" "AND ANGULAR FREQUENCY OMEGA." "BUT IF THAT'S x," "IS ITS SECOND DERIVATIVE EQUAL TO -k/m TIMES x?" "THE ANSWER IS YES, IF OMEGA IS CHOSEN CORRECTLY." "FORTHEMASSONA SPRING," "THEANGULARFREQUENCY" "ISEQUALTO  THESQUAREROOTOFk /m ." "THESTIFFERTHESPRING, THEHIGHERTHEFREQUENCY." "THEGREATERTHEMASS, THELOWERTHEFREQUENCY." "BUTWHATEVERTHEFREQUENCY," "ITDEPENDSON THEMECHANICAL PROPERTIES OF THE OSCILLATOR," "SUCHASTHEMASS ANDTHESPRINGCONSTANT." "THETIMENEEDED TOMAKEACOMPLETECYCLE" "DOESNOTDEPEND ONHOWBIGTHE SWINGSARE," "REPRESENTEDBY"A " INTHEEQUATIONFOR x." "EVEN AS THE OSCILLATIONS DIE DOWN," "HARMONIC OSCILLATORS CONTINUE TO KEEP PERFECT TIME." "THE FREQUENCY IS MEASURED IN RADIANS PER SECOND." "THERE ARE 2 PI RADIANS IN EACH COMPLETE CYCLE." "LIKEWISE, IN THE GREAT CLOCK IN THE SKY," "EACH CYCLE IS DIVIDED INTO RADIANS," "AS IF THE UNIVERSE WERE AN ENORMOUS CIRCLE." "TO AN EXTENT, IT IS." "THE NATURES OF UNIFORM CIRCULAR MOTION" "AND SIMPLE HARMONIC MOTION" "ARE CLOSELY RELATED." "EACH CLOCK," "IN ITS ATTEMPT TO DUPLICATE THE CYCLES OF NATURE," "IS A MERE SHADOW OF THE COSMIC CLOCK." "JUST AS UNIFORM CIRCULAR MOTION AND SIMPLE HARMONIC MOTION" "SHAPE UP MUCH THE SAME IN NATURE," "THE POTENTIAL ENERGY OF HARMONIC MOTION" "CAN BE VISUALIZED AS A CURVE SHAPED LIKE A BOWL." "WHY THIS SHAPE?" "BECAUSE OF THE FORCE LAW" "AND THE DEFINITION" "OF POTENTIAL ENERGY." "THE BALL STARTS OUT" "WITH SOME POTENTIAL ENERGY." "AS IT ROLLS TOWARD THE BOTTOM," "THE POTENTIAL ENERGY" "CHANGES TO KINETIC ENERGY." "AS IT ROLLS UP THE OTHER SIDE," "THE KINETIC ENERGY CHANGES BACK" "INTO POTENTIAL ENERGY." "OVER TIME," "THERE'S A CONTINUAL INTERCHANGE" "BETWEEN KINETIC" "AND POTENTIAL ENERGY." "BUT EVEN THOUGH" "BOTH THE POTENTIAL ENERGY" "AND KINETIC ENERGY" "ARE ALWAYS CHANGING," "THE TOTAL ENERGY" "IS ALWAYS CONSTANT." "WHEN ANY HARMONIC OSCILLATOR" "IS DISTURBED," "THE DISTURBANCE PRODUCES A FORCE" "THAT PUSHES IT BACK TO WHERE IT WAS." "THERE, THE FORCE IS ZERO," "BUT INERTIA KEEPS IT GOING" "UNTIL THE REVERSE FORCE STOPS IT AND PUSHES IT BACK ONCE AGAIN." "THIS IS THE ESSENCE OF HARMONIC MOTION." "A SIMPLE HARMONIC OSCILLATOR." "IF IT WERE REALLY SIMPLE, IT WOULD GO ON FOREVER," "A PERPETUAL MOTION MACHINE." "BUT REAL OSCILLATORS ARE NOT SIMPLE." "THEY ALWAYS HAVE OTHER FORCES" "FRICTION, AIR RESISTANCE, AND SO ON" "THAT TEND TO MAKE THEM SLOW DOWN." "THAT'S WHY, FROM TIME TO TIME," "CLOCKS MUST BE WOUND OR HAVE THEIR BATTERIES CHANGED." "FRICTION TURNS ENERGY INTO HEAT." "THAT ENERGY MUST BE RESTORED" "IN ORDER TO KEEP THE CLOCK GOING." "BUT EVEN AS THE CLOCK WINDS DOWN," "THE PERIOD OF TIME FOR EACH OF ITS CYCLES REMAINS CONSTANT." "THAT FACT WAS FIRST DISCOVERED BY GALILEO GALILEI." "HIS TIMELY DISCOVERY," "WHICH STEPPED UP THE PACE" "FOR IMPROVED TIMEKEEPING IN THE RENAISSANCE," "WAS SUMMARIZED IN TWO NEW SCIENCES." "IN THAT WORK, GALILEO OBSERVED THAT A PENDULUM" "TAKES THE SAME TIME TO COMPLETE EACH SWING," "EVEN THOUGH ITS MOTION DIES DOWN." "GALILEO ALSO OBSERVED" "THAT ALL PENDULUMS OF THE SAME LENGTH" "OSCILLATE AT THE SAME RATE," "REGARDLESS OF THEIR MASS." "OF COURSE," "GALILEO, WHO DISCOVERED THE LAW OF FALLING BODIES," "REALIZED THAT A PENDULUM ITSELF IS A FALLING BODY." "IF ALL BODIES FALL AT THE SAME RATE," "REGARDLESS OF THEIR MASS," "THEN ALL PENDULUMS OF THE SAME LENGTH" "SHOULD OSCILLATE AT THE SAME RATE, REGARDLESS OF MASS." "WHY DOESN'T THE TIME OF THE PENDULUM'S SWING" "DEPEND ON ITS MASS?" "ISAAC NEWTON ANSWERED THAT QUESTION." "THANKS TO GALILEO'S LAW OF FALLING BODIES," "NEWTON REALIZED ALL OBJECTS FALL TO THE EARTH'S SURFACE" "WITH THE SAME CONSTANT ACCELERATION." "THEN, FROM A CONCEPTUAL POINT OF VIEW," "HE SAW A CONNECTION BETWEEN, FOR INSTANCE," "PENDULUMS OF DIFFERENT MASSES AND FREELY FALLING BODIES." "THE FORCE OF GRAVITY," "WHICH DEPENDS ON THE MASS OF THE PENDULUM," "CAN CANCEL THE MASS IN THE EQUATION F=ma." "IN A SENSE, THEN," "BY USING PENDULUMS OF DIFFERENT MASSES," "NEWTON TESTED THE LAW OF FALLING BODIES" "WITHOUT THE ADVANTAGE OF A VACUUM." "THANKS TO NEWTON," "NOT ONLY PHYSICS, BUT THE NATURE OF TIME ITSELF" "WAS NEVER PERCEIVED AS QUITE THE SAME." "WITH HARMONIC OSCILLATORS," "CLOCKMAKERS WERE ABLE TO INTRODUCE" "ACCURACY AND UNIFORMITY TO TIMEKEEPING." "AN HOUR IN CAMBRIDGE" "BECAME A PRECISELY AS LONG AS AN HOUR IN VENICE." "ACCURACY PROVED A BOON TO NAVIGATION." "IN DETERMINING LONGITUDE," "NAVIGATORS NEEDED TO BRING ALONG AN ACCURATE CLOCK" "THAT KEPT PRECISE TIME WITH GREENWICH, ENGLAND," "THE POINT OF ZERO LONGITUDE." "SAILORS COULD TELL LOCAL TIME FROM THE SUN'S POSITION," "AND KNOWING HOW THAT COMPARED WITH GREENWICH TIME," "THEY COULD DETERMINE THEIR LONGITUDE." "BUT AN ERROR OF ONLY A MINUTE" "COULD PUT THEM MILES OFF THEIR COURSE" "OR EVEN GET THEM LOST." "THE SAME HARMONIC MOTION" "THAT REGULATES THE ACCURACY OF A GRANDFATHER'S CLOCK" "IS THE PRINCIPLE BEHIND THE ACCURACY" "OF MODERN QUARTZ WATCHES." "WHERE MILLIONS OF OSCILLATIONS PER SECOND" "KEEP INCREDIBLY PRECISE TIME." "[PLAYING BEETHOVEN'S FIFTH SYMPHONY]" "HARMONIC MOTION CAN BE FOUND" "IN AN ENORMOUS VARIETY OF PHYSICAL PHENOMENA." "UNDERSTANDING IT HELPS TO EXPLAIN" "THE SUBTLE UNIFYING PRINCIPLES THAT GOVERN THE UNIVERSE." "DO YOU REMEMBER OUR DISCUSSION OF STABILITY?" "FOR AN OBJECT TO BE STABLE, IT MUST HAVE NO NET FORCE ON IT." "BUT IN ADDITION, IF I DISTURB IT," "THAT MUST PRODUCE A FORCE" "THAT PUSHES IT BACK TO WHERE IT WAS." "WHAT WE'VE STUDIED TODAY" "IS THE MOTION OF A STABLE OBJECT." "WHETHER IT'S A MASS ON A SPRING" "OR A PENDULUM OR A ROLLING BALL," "IF I DISTURB A STABLE OBJECT," "IT EXECUTES SIMPLE HARMONIC MOTION." "WHEN I WAS A STUDENT," "I STUDIED SIMPLE HARMONIC OSCILLATORS" "JUST AS YOU'RE DOING." "I WENT SLOGGING THROUGH ALL THE SINES AND COSINES." "BUT I COULDN'T UNDERSTAND WHY." "IT SEEMED LIKE MORE EXAMPLES OF PULLEYS AND INCLINED PLANES." "AS TIME WENT ON," "I BEGAN TO SEE" "MORE AND MORE EXAMPLES OF COMPLETELY DIFFERENT SYSTEMS," "EACH ONE EXECUTING THE SAME SIMPLE HARMONIC MOTION." "NOT JUST MASSES ON SPRINGS AND PENDULUMS," "BUT ORGAN PIPES AND ELECTRIC CIRCUITS" "AND EVEN ATOMS IN A CRYSTAL LATTICE." "THEN I BEGAN TO GET THE POINT." "SIMPLE HARMONIC MOTION" "IS NATURE'S RESPONSE TO TWANGING ANY STABLE SYSTEM." "THAT'S WHY IT'S IMPORTANT." "BUT THAT'S NOT THE END OF THE STORY." "IT'S JUST THE BEGINNING, AS WE'LL SEE NEXT TIME." "ALINEARRESTORINGFORCE, F= - kx," "LEADS TO SIMPLE HARMONIC MOTION" "WHENINSERTED INTONEWTON'SSECONDLAW ." "THESOLUTION TO THE DIFFERENTIAL EQUATION" "ISTHEDISTANCE FROM THE EQUILIBRIUM POSITION" "EXPRESSED BY THIS EQUATION..." "WHEREA ISTHEAMPLITUDE OFTHEMOTION." "CAPTIONING PERFORMED BY THE NATIONAL CAPTIONING INSTITUTE, INC." "CAPTIONS COPYRIGHT 1985 CALIFORNIA INSTITUTE OF TECHNOLOGY" "THE CORPORATION FOR COMMUNITY COLLEGE TELEVISION THE ANNENBERG/CPB PROJECT" "PUBLIC PERFORMANCE OF CAPTIONS PROIBITED WITHOUT PERMISSION OF NATIONAL CAPTIONING INSTITUTE" "Annenberg Media" "¶" "For information about this and other Annenberg Media programs call 1-800-LEARNER and visit us at"