9.1 clocks new ideas for today resonance harmonic oscillators timekeeping

9.1 Clocks9.1 Clocks

New ideas for todayNew ideas for today• Resonance• Harmonic oscillators• Timekeeping

•~3500 BC: sundials•~1500 BC: water clocks

History of timekeepingHistory of timekeeping

Earliest clocks: EgyptEarliest clocks: Egypt

Modern timekeepingModern timekeeping•1500-1510: spring powered clocks (Henlein / Germany)•1656, 1675: pendulum clocks, balance wheels (Hyugens, Netherlands)•1920s – : quartz clocks•1949 – : atomic clocks•1967: Cesium clock becomes official standard

Development of modernDevelopment of modern

timekeeping driven by navigationtimekeeping driven by navigation

Before mid-1700sBefore mid-1700sLatitude: use quadrant/sextant/octant to sight

North Star or sunLongitude: lunar time – very poor accuracy

1714:1714: British Parliament sets £20,000 prize ($10M in today’s dollars!) to make clock accurate to 2 minutes (0.5° longitude)

John Harrison – 1764John Harrison – 1764

H4 Linked balance mechanism

Today: GPS and LORAN-CToday: GPS and LORAN-C

We still use clocks to We still use clocks to navigatenavigate

Repetitive MotionsRepetitive Motions• An object with a stable equilibrium

tends to oscillate about that equilibrium• This oscillation involves at least two

types of energy: kinetic and a potential energy

• Once the motion has been started, it will repeat

When energy traded back and forth between When energy traded back and forth between kinetic and potential energy: “resonance”kinetic and potential energy: “resonance”

Mass on springMass on spring

Ball in bowlBall in bowl

Many objects in nature Many objects in nature have natural have natural resonances !resonances !

Repetitive motion characterized by a: period (or frequency) and amplitude

Resonance: energy can be stored in motion at a specific frequency

Period:Period: time of one full cycle

Frequency (1/Period):Frequency (1/Period):cycles completed per second

Amplitude:Amplitude:extent of repetitive motion

In an ideal clock, the period (and In an ideal clock, the period (and frequency) should not depend on frequency) should not depend on amplitudeamplitude

Properties of oscillationProperties of oscillation

Frequency depends on two propertiesFrequency depends on two properties

MassMass(beer gut)

StiffnessStiffness(diving board)

The Harmonic OscillatorThe Harmonic Oscillator

• Anything with a stable equilibrium and a restoring force (F) that’s proportional to the distortion away from equilibrium (x)(F = - k x, where k is a constant)

• Period is independent of amplitude• Examples:

1. Simple pendulum (small amplitude)2. Mass on a spring

A special example of something with a natural resonance

mg

Restoring force

F

PendulumPendulum

A harmonic oscillator!

That indicates simple harmonic motion

At low amplitude, the restoring force is proportional to the distance from equilibrium.

F = - k x (you have already seen this as Hooke’s Law!)

PendulumPendulum

• Period=

• Period only independent of amplitude for small amplitude

2length

g

Near earth’s surface, 1 m Near earth’s surface, 1 m pendulum has a 2 second pendulum has a 2 second periodperiod

Bowling, golf ball pendulaBowling, golf ball pendula

Variable length pendulaVariable length pendula

Chaotic pendulumChaotic pendulum

Clicker Clicker questionquestion

What happens to the period of a swing if you stand up?

A) The period gets longerB) The period gets shorterC) The period doesn’t change!

Balance ring clocksBalance ring clocks

A mass on a spring that is not sensitive to gravityA mass on a spring that is not sensitive to gravity

Balance Ring ClocksBalance Ring Clocks• A torsional coil spring causes a balanced ring to

twist back and forth as a harmonic oscillator• Gravity exerts no torque about the ring’s pivot

Torsion pendulaTorsion pendula

The Main Spring keeps tension on toothed escape wheel and needs to be re-wound.

•Coil Spring attached to Balance Ring and to the body of the watch•Coil Spring provides the restoring force for the Balance Ring•The restoring torque is proportional to the angle of rotation …simple harmonic motion ( = - k )!

Balance Ring ClocksBalance Ring Clocks

Quartz oscillatorsQuartz oscillators

Typical frequency in watch: 32,768 Hz (period is 31 s)

Most modern clocks use a quartz oscillator

Quartz OscillatorsQuartz Oscillators• Crystalline quartz is a harmonic oscillator• Oscillation decay is extremely slow (very pure

tone)• Quartz is piezoelectric

– Mechanically-electrically coupledmotion induced and measured electrically

PiezoelectricityPiezoelectricity

Can think of bonds between atoms in a crystal as springs. So, the restoring force is proportional to the distance from equilibrium. Simple harmonic motion! (F = -k x)

Quartz OscillatorsQuartz Oscillators

Quartz ClocksQuartz Clocks

• Vibration triggers electronic counter• Nearly insensitive to gravity,

temperature,pressure, and acceleration

• Slow vibration decayleads to precise period (loses/gains 0.1 sec in 1 year)

• Different shapes (bars, tuning fork) and cuts

Tuning forkTuning fork

Which clock should Neil Which clock should Neil Armstrong take to the moon?Armstrong take to the moon?

C. Quartz watchB. Balance ring clockA. Grandfather clock

Clicker questionClicker question

Time standardsTime standards

But every Cesium atom is exactly the same!

Courtesy of Mark Raizen’s group

What is one second?What is one second?

Problem: No two pendula or quartz oscillators are exactly the same

Atomic ClocksAtomic Clocks• Particles in an atom (neutrons, protons, electrons) can

have only a very specific amount of total energy.• Changing from one quantum state to another requires

or releases a fixed amount of energy• That energy can be converted into a frequency, so can

be the basis of a very accurate clock.

1 sec = 9,192,631,770 periods of the radiation corresponding to the ground state hyperfine transition in 133Cs

Spectral linesSpectral lines

NIST-F1 Cesium Fountain Atomic ClockThe Primary Time and Frequency

Standard for the United States

Atomic ClocksAtomic Clocks

Loses less than one second in 60 million years

NIST-7NIST-F1

F1 – the fountain clockF1 – the fountain clock

Every GPS satelliteEvery GPS satellite

contains an atomic clockcontains an atomic clock

Receivers: high quality quartz clock which is synchronized to atomic clock

See you next class!

For next class: Read Section 9.2

Bring a musical instrument!