NEWTON'S SECOND LA+F = ma,
 COMBINED WITH
 A RESTORING FORCE,
 LEADS TO A MOTION
 THAT REPEATS ITSELF
 OVER EQUAL PERIODS OF TIME,
 PRODUCING WHAT IS KNOWN AS
 HARMONIC MOTION.
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.
 IN A CERTAIN POSITION,
 ALL FORCES ARE BALANCED.
 HOWEVER, WHEN THE SPRING
 IS STRETCHED,
 IT ATTEMPTS TO PULL THE MASS
 BACK TO ITS ORIGINAL POSITION.
 THE FARTHER THE MASS MOVES,
 THE GREATER THE FORCE
 TO PULL IT BACK.
 THE SAME PRINCIPLE
 WORKS CONVERSELY.
 WHEN THE SPRING IS COMPRESSED,
IT TRIES TO PUSH THE MASS BACK.
 WHICHEVER DIRECTION
 THE MASS MOVES,
 A FORCE EMERGES
TO COUNTER THE DISPLACEMENT.
THE COMBINATION OF THIS FORCE
 AND THE MASS'S INERTIA
 WILL BE THE KEY
 TO KEEPING TIME.
AT EACH POINT IN ITS MOTION,
THE NET FORCE IS PROPORTIONAL
TO THE DISTANCE x OF THE MASS
FROM ITS EQUILIBRIUM POSITION
 AND IN THE OPPOSITE DIRECTION.
 THE EQUATION IS F = -kx.
 THE VALUE OF k
 DEPENDS ON HOW STIFF
 THE SPRING IS.
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.
 THE MOST INEXORABLE LAW
 IN MECHANICS IS F = ma.
 a IS THE SECOND DERIVATIVE
 OF x.
 IN SIMPLE HARMONIC MOTION,
 FORCE COMES FROM
 THE DISPLACEMENT x ITSELF,
 F = -kx.
 TOGETHER, THESE TWO EQUATIONS
 GIVE THE DIFFERENTIAL EQUATION
 THAT DESCRIBES
 SIMPLE HARMONIC MOTION.
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.
 FOR THE MASS ON A SPRING,
 THE ANGULAR FREQUENCY
 IS EQUAL TO
 THE SQUARE ROOT OF k/m.
 THE STIFFER THE SPRING,
 THE HIGHER THE FREQUENCY.
 THE GREATER THE MASS,
 THE LOWER THE FREQUENCY.
 BUT WHATEVER THE FREQUENCY,
 IT DEPENDS ON THE MECHANICAL
PROPERTIES OF THE OSCILLATOR,
 SUCH AS THE MASS
 AND THE SPRING CONSTANT.
 THE TIME NEEDED
 TO MAKE A COMPLETE CYCLE
 DOES NOT DEPEND
 ON HOW BIG THE SWINGS ARE,
 REPRESENTED BY "A"
 IN THE EQUATION FOR 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.
 A LINEAR RESTORING FORCE,
 F = -kx,
LEADS TO SIMPLE HARMONIC MOTION
 WHEN INSERTED
 INTO NEWTON'S SECOND LAW.
 THE SOLUTION
TO THE DIFFERENTIAL EQUATION
 IS THE DISTANCE
FROM THE EQUILIBRIUM POSITION
EXPRESSED BY THIS EQUATION...
 WHERE A IS THE AMPLITUDE
 OF THE MOTION.
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CAPTIONS COPYRIGHT 1985
CALIFORNIA INSTITUTE
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