T = 2π Lg

The pendulum clock would be useless on the pitching sea

Huygens improved the pendulum clock by eliminating the requirements for small swings.

For larger swings,The arc of the pendulum has to follow a special mathematical curve: cycloid (discovered by Galileo)

Huygens' clock could tell time to better than 1 minute/day

Pendulum period is only constant for small swings

A

B

C

D

Two golf balls are dropped from the top of a track in the shape of a cycloid as in panel A of the figure below. Which panel represents the result at the time when the first ball reaches the bottom?

Do this experiment when you visit the science center

But still the pendulum clock was not good enough to keep track of longitude

Temperature changes influence the value of L

g is not the same for everywhere on the planet! T = 2π Lg

Compute the magnitude of the centripetal acceleration of an object located on the earth's equator.

Radius of the earth at equator = 6378 kmThe period of rotation of earth = 24 hour = 86, 400 secCompute the tangential velocity, V

V = 2 π R

T

= 2 π x6378

86400

= 0.464 km

sec

= 464 m

sec

= 1670

km

hr

surprising 928 mph,

Compute the magnitude of the centripetal acceleration of an object on the equator.Radius of earth at equator = 6378 kmPeriod of rotation of earth

= 24 hour = 86,400 secCompute the tangential velocity, V

Note how small the result is compared to

Converted to mph, the rotation speed of the earth is a surprising 928 mph

More than twice as fast as an airplane!

Now we calculate the acceleration

V = 2πRT

a = V2

R = 4642

6378x 1000= 0.0338 m

sec2

g =9.8

m

sec2

= 464 m/sec

Rotation breaks the symmetry !

Axis of rotation

a = 4 π

2

R

T2

Radius of

latitude

circle = R

g changes over latitude

T = 2π Lg

Calculate the period of a pendulum with L = 1 meter. Give the closest answer.

A) 1 secondB) 2 secC) 3 secD) 0.5 sec

T = 2π Lg T = 2π L

g - 0.0338- 0.0338L = 1 meterL = 1 meter

= 2.0071 sec= 2.0071 sec =2.0106 =2.0106

0.0035 sec for one period0.0035 sec for one periodT = 2π 1

9.8T = 2π 1

9.8 - 0.0338

Pole Equator

Huygens had improved precision of the pendulum to tell time to better than one minute per day.Now his pendulum was sufficiently accurate to detect earth's rotation at the equator. Example: check time period between sunrise and sunrise around equinox, at the equator and at the north pole, difference 2.5 min

1 day = 86,400 secDifference = 151 sec = 2.5 minutes

A pendulum with a 2.5 minute daily error -> totally unsatisfactory for keeping track of longitude on a long ocean voyage across latitudes.

At the end of a month-long journey across the Atlantic ocean,

the corresponding time reading would be off by 75 minutes in time, and the longitude assessment by 19 degrees!

From the radius of the earth,

Each degree of longitude around the equator is about 112 km,so that the distance error of 19 degrees = 2128 km - a grievous miscalculation

Wanted!

A method to measure longitudeAccuracy better than 0.5 degree

(35 miles)

Reward offered20,000 British Pounds

($1,000,000 in today's currency)

Many weird ideas offered

Three main competitors•An accurate chronometer•Measure position of moon to stars and planets compare with predetermined accurate tables•Use the eclipses of the moons of Jupiter (Galileo) regular as a clock

What was the main problem with the lunar distance method to keep track of time with the accuracy necessary for longitude measurement?

A) The phases of the moon make it difficult to track of its position

B) There was no good data on star positions and times

C) The method required both good shipboard astronomical instruments as well as sophisticated analysis

D) All of the above.

What was the main problem with the regular eclipses of Jupiter's moons to keep track of time with the accuracy necessary for longitude measurement?

A) it was hard to keep a high power telescope focused on the heavenly bodies on a swaying ship

B) Jupiter is not always visible at night

C) Jupiter would not be visible on cloudy nights

D) All of the above

How to devise an accurate chronometer

The pendulum had too many problems

Hooke, Huygens... Harrisson

A spiral spring and flywheel oscillator Demo

The flywheel oscillates regularlyThe period is independent of amplitudePeriod is independent of g

Accuracy improved to 10 seconds per day

m

Equilibrium

position

m

m

Main problem: temperature varies length of springchanges period

Solution: make the spring out of a bi-metal copper/ironthey expand by different amounts, change in length is compensated by resulting curvature

Period now independent of temperature

Sidereal Day is shorter than solar dayBy 1/365 x 24 hours ≈ 4 minutes

You have already seen one connection between space and time

eg to measure your longitude position on earthyou could use an accurate time piece.

We use the daily rotation of the earth, with the sun as the fixed reference point

Now we are going to see deeper/more fundamental connection between space and time

Unity of Space and Time

A rotation mixes x and y co-ordinates to give new x’, y’ co-ordinates

A Lorentz transformation mixes space and time, mixing x' mixes x and t

There is an inseparable relation between space, time and velocity of light.

Minkowsky: "Space of itself, and time of itself will sink into mere shadow, and only a kind of union between them shall survive.

Sky and Water

Escher creates a metaphor for the inseparability of life from life-supporting elements, air and water.

There is no sharp division between dark and light.

Out of the darkness the black birds gain life. Bird forms melt into the dark water. Fish forms evaporate into white sky.

Fish and bird are also intertwined.

"My subjects are often also playful. I cannot keep them from having fun with our unassailable certainties. For instance, it is pleasure to deliberately mix up two and three dimensional reality, flat and spatial, and to poke fun at gravity. It is satisfying to note that quite a few people enjoy this kind of playfulness and that they are not afraid to revitalize their thought about rock-hard realities."