Measure the Earth Experiment

8/13/2019 Measure the Earth Experiment

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Measure the Earth Experiment

[22ndDecember, 2007]

How big is the Earth?

The impact of this question and the answer is always unclear toschool

children. Although they are taught in Geography class, all they get is someprinted data likeradius = 6400km. We also do not relate this importantquestion toapplied mathematics. Geometry is a whole branch that came outof measuring the Earth! To make this process apparent and easier forteachers, the following project has been undertaken by IUCAA and VigyanPrasar

Eratosthenes & his method

Eratosthenes, the Greekmathematician, poet, athlete, geographer and

astronomer, iscreditedfor the first estimation of the size of the Earth

around 240 BC. He was the librarian of the great libraryofAlexandria in

Egypt. Although he could read the latest scientific knowledge of his day, he

was always interested in learning new things. Being without muchequipment and technology, he used only tools like sticks, eyes, feet and

brains to do this experiment.It was not such a simple thing those days to believe that the Earth was

round! Most people thought it was flat. But Eratosthenes went ahead tomeasure the sphere he believed the Earth to be. But, he did not usemethods like walking around the Earth or digging a hole through its centreto the other side.

The Earth, if cut into two halves, would show a circular section. Theboundary of the circle would be its circumference, which we want to

measure. Suppose this circle was a Cake.Wecut out a slice that is one-sixth [1/6] ofit. The curved edge of this will then be1/6

thof the circumference. To put it the

other way, the circumference of the cakewill be six times the length of the curvededge of the slice. If we measure the anglemade by the other two sides, then wewould find that it is 60

o. It has to be so,

because 60ois 1/6

thof 360

owhich is the

whole circle. Thus if we know the distancebetween two points on a circumference


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and know the angle made by them at the centre, we can find out the lengthof the circle. This means that if we know the distance between two points

on the Earth and the angle they make with the Earths centre, we can findout the circumference of the Earth.

The distance between two places can be measured. The problem is to

measure the angle that they make. This wassolvedby Eratosthenes.

He had heard that in a place called Syene,southofAlexandria, the

walls of a well castnoshadow in

thewaterat noon on the

day of the middle ofSummer. In otherwords, there were noshadows of by verticalthings June 21

st, as the

Sun was exactlyoverhead there at noon.

But on the samedate he observed that a

vertical pillar inAlexandria cast a small but measurable length of shadow. Ifthe Earth was flat, then the sunrays coming perpendicular to the flatlandwould make all vertical objects cast similar shadows anywhere on theEarth. The difference in the length is possible only if the Sunlight comes atdifferent angles at different places. This means that the Earth is a curvedsurface.

If two sticks perpendicular to the ground in Syene and Alexandria wereto extend to the centre of

the Earth, they would thusmake an angle. Usingsimple geometry, hefound out thisangle fromthe lengthof the shadowatAlexandriaon 21

stJune.

It came out to be about7

owhich is 1/50

thof 360

o.

It is said that he got anarmy to walk to Syene

[now called Aswan] tomeasure the distance till


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there. It came out to be 800km. Fifty times that i.e. 40,000km was thecircumference of the Earth!

Aim

The aim of this activity is to get teachers and students to measure the dimensions ofthe Earth using the same simple method used by Eratosthenes over 2000 years ago.Participants are to make a simple setup and measure some shadows and note some

times. If s& are known in the figure below, we can find the circumference of theEarth.

To get the value of the Earth's circumference, Eratosthenes had tomeasure the distance between the two places that he did the experiment at.But now that we know the circumference very accurately, we can reversethe method and use it to find the latitude and longitude of the place where

our observations are taken.

More...

The scope of this project to teach applied geometry and trigonometry isgreat. The radius and other parameters of a sphere can be found out by thechildren using the circumference. This project, if done in a group across thecountry, can also help you verify your latitude and your longitude. Acomparison with other groups and some calculations can give us the easilygive us these values. If two places collaborate, they can also find out the

distance between them.


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Instead of doing it on just one day, this can also be repeatedperiodically for years to come so that a new group of students get to learnform this exercise every year.

How you and your group may participate in the project

With your group participating along with the others all over the country, wewill be able to repeat this fundamental measurement by Eratosthenes. Wegive here some basic instructions for you to follow. If you have understoodthe Eratosthenes method you may modify the method given here to suit

your local conditions. However, it is advisable that all of us to follow theseinstructions for the uniformity and to cross check the observations.

In this experiment we will be measuring a shadow of a tall pole due to the

sunlight, for about 4 hours from 10:00 a.m. to 2 p.m.

What you have to do

Requirements and Preparations

1. You should find a place where you can get a clear sunshine from 10am tillabout 2:30pm. Your cricket or football ground should be quite good.

2. We need a cylindrical pole of 25 mm in diameter and about 200 cm long. Thepole could be 220 to 240 cm too as long as only 200 cm of it used for casting

the shadow.

3. Measuring tape. This will be used to measure the length of the shadow of thepole.

4. An accurate clock.5. Big marbles or small white painted stones or big nails (~30 in number), chalk

etc.

6. Plumb-line

Prior to the actual experiment

o Three to five days before the actual date of the experimentlocate the place where you would like to do the experiment andfix the pole and watch the shadow of the pole. Check if it getsthe full sunshine from 10 am to 2 p.m. It can and might happenthat at a crucial time shadow of a nearby building might fall onthe pole spoiling your experiment. Shift the position of the poleif necessary.


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o The pole is to be rooted in the ground firmly. It is important, forthis group project, that it is exactly 200 cm above the ground.For this, measure 200cm length from one end of the pole andmark this point with coloured tape. Bury the pole such that theportion after this goes inside and the tape is just touching theground.

o Now if you are fixing the pole in soil then you should dig a deephole and put the base into it such that 200 cm stick out abovethe ground. Pack it with small stones and pebbles and then addloose soil. Make the pole exactly vertical with the help of aplumb-line. Put water to fix it.

o If you school has tiled floor or if you would like to do theexperiment on the terrace of your school then you may make astand to support the pole but remember that the place of thepole should not be disturbed during the entire experiment.

Again use the plumb-line to make the pole vertical.

o You have to check your watch if it is accurate. Tune in the AllIndia Radio (AIR) just before the morning news at 8 a.m. Priorto the start of the News you will hear beep. When you hear thelong beep it mean it is exactly 8 a.m. Check time in your watchmatches with that of AIR if it does then it is ok of not correct the

watch and check it the next day again. If there is a problem theneither get a better watch or correct your time. The clocks inMobile phones are quite accurate.

On the day of the experiment

1. Be at the observing station i.e. where the pole is anchored, byabout 9:30 a.m. and get ready to start the observations byfilling up the form given here.

2. Now exactly at 10 a.m. mark the shadow of the pole. If you are

doing the experiment in the field place one of the marbles orstone or hammer a nail etc. in the ground. If you are on theterrace or on the tiled floor then mark the tip of the shadow

with chalk.


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3. Continue the observations every 10 minutes.

4. But dont worry if you missed one or two observationsbecause of clouds or some other disturbances. Do not put markson your own to make up for it!

Using the observations

The first inspection of the observations should show you when theshadow was shortest and at what time. At this time the shadow wasexactly North-South.Can you tell us why?

Plot your observations on the graph paper and draw a smooth curve.What we mean is that you are not expected to join the points. There

might be some points above and some below this curve. This curve isparabolic. An example is shown on the next page.

The lowest point (orvertex of the parabola) gives us the minimumlength of the shadow, h and the time at which it was minimum t.

Calculations

Make a scale drawing of your pole and least shadow at right angles to eachother. Complete the triangle and measure the angle on top with a

protractor. We denote this number byOne can also use trigonometry tofind it.

The graph above is the data plotted using the observations
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This graph is "trendline" drawn using second order polynomial

How to find out...[Specific to 22ndDecember 2007]

Latitude of the place

On this the latitude of your station would be given by- 23.5 deg. (thedeclination of the sun would be 23.5 deg. Please refer to celestialcoordinate system)

Here h is the height of the verticalpole and x is the length of theshadow. You can find the

angle making a scale drawing on

a graph sheet and measuring theangle or by using trigonometric fun

tan.

tan x/h or tan-(x/h)

Longitude of the place :

Longitude of the place would be thetime difference between the time at which the shadow was minimum at


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you place and the time at which the shadow was minimum at a placewhose longitude we know. See the table below

Latitude(N)

Longitude (E)

Vigyan Prasar,Noida

28o37 77o15 5h 09m

IUCAAMVS,Pune

18o33 73o49 4h 55m

Example: On 22ndDec 07 at Vigyan Prasar, Noida and IUCAA the shadowwould be shortest at 12:19 and 12:33 respectively. This time is also calledtransit time of the sun. Now say at your station the time at which theshadow was minimum was at 12:21. Which is 2 min after the transit timeat Vigyan Prasar or 12 min before the transit time at IUCAA Pune.

[All times are in IST]

Therefore the longitude of your place would be

[5h 09m0h 2m] = 5h 07m or [4h 55m + 0h 12m] = 5h 7m or 76o45.

Measure the Earth experiment

Observation Log

Name and address of the Organization (school, amateur astronomer's clubsetc)

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Name of the principle investigator(s) (PI)

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Height of the vertical pole . . . . . . . . . . . . . . . . . .

(Even though we have fixed the length of the pole to be exactly 200 cmabove the ground but it would pay to check it out and if it is not then writethe exact length.)

Observation table (write length in centimeters)

Time in(IST)

Length ofthe

shadow

Time in(IST)

Length ofthe

shadow

Time in(IST)

Length ofthe

shadow

Time in(IST)

Length ofthe

shadow

10:00 11:00 12:00 13:00

10:10 11:10 12:10 13:10

10:20 11:20 12:20 13:20

10:30 11:30 12:30 13:30

10:40 11:40 12:40 13:40

10:50 11:50 12:50 13:50

Names of the other participants

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Measure the Earth experiment

Calculations and results

x t

it was obtained by - graphical method / using calculator,tick one)

Latitude of the place- 23.5 deg)...........................................

Longitude of the place


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Time at which the shadow was minimum at

Vigyan Prasar, Noida - ............... = .....................(VP time) (your time) (time difference)

Longitude at your place 5h 09m - ..................... = ....................

(VP longitude) (time difference) (yourlongitude in time)

or ....................... x 15 = .................. your longitude in degrees(your long. in time)

OR

Time at which the shadow was minimum at

IUCAA MVS, Pune .................. - ............... = ......................(IUCAA time) (your time) (time difference)

Longitude at your place 4h 55m - ..................... = ....................(IUCAA longitude) (time difference) (your

longitude in time)

or .................... x 15 = .................. your longitude in degrees

(your long. in time )

Doing the experiment

1) One can carry out the experiment (almost) single handed. All one needsto do is to mark the position of the mast. (Here an adjustable mast is beingused.)


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2) One would need some help in measuring the length of the shadow.

3) The floor on which this experiment was carried out has the "graph lines"

made during the cementing of the floor. One can, therefore, measure "x"and "y" and calculate the shadow length.

4) But it could be fun doing in a group helping each other.

We thank Neha (1st year BSc, Fergusson college), Madhuvanti and Mayuri(X std, DAV Public School), Dhruv and Rohan (VI std, Loyola School) And

Jyoti Hiremath for helping out Neha.


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Extra Information about Eratosthenes:

Eratosthenes was born in Cyrene (in modern-day Libya), he worked at

Alexandria and died incapital of Ptolemaic Egypt. He was never married.In 236 BC he was appointed by Ptolemy III Euergetes I as librarian of the

Alexandrian library. He made several important contributions tomathematics and science, and was a good friend to Archimedes. Around 255BC he invented the armillary sphere, which was widely used until theinvention of the Orrery in the 18th century. In 194 BC Eratosthenes became

blind and a year later he supposedlystarved himself to death.

Although Eratosthenes' methodwas well founded, the accuracy ofhis calculation was inherentlylimited. The accuracy ofEratosthenes' measurement wouldhave been reduced by the fact thatSyene is not precisely on the Tropicof Cancer, is not directly south of

Alexandria, and the Sun appears asa disk located at a finite distance

from the Earth instead of as a pointsource of light at an infinitedistance. To put concisely

The two cities are not on thesame meridian (the difference inlongitude is 3o);

Syene is not on the Tropic of Cancer (it is situated 55 km North); The angular difference is not 7o12' but 7o5'. The actual distance between Alexandria and Syene is 729 km, not

800km.

There are other sources of experimental error: the greatest limitation toEratosthenes' method was that, in antiquity, overland distancemeasurements were not reliable, especially for travel along the non-linearNile which was traveled primarily by boat. For overland distancemeasurement, the accuracy rests on the estimated average speed of acaravan of camels. So, the accuracy of Eratosthenes' size of the earth issurprising.
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Eratosthenes' experiment was highly regarded at the time, and his estimate

of the Earths size was accepted for hundreds of years afterwards. Hismethod was used by Posidonius about 150 years later.

Weblinks

Other Eratosthenesexperiment projects:

http://www.k12science.org/noonday/

http://www.physics2005.org/projects/eratosthenes/experiment.html

http://www.algonet.se/~sirius/eaae/aol/market/collabor/erathost/

http://www.phys-astro.sonoma.edu/observatory/eratosthenes/

http://www.youth.net/eratosthenes/

About Eratosthenes:

http://en.wikipedia.org/wiki/Eratosthenes

Contact

Mr. Arvind Paranjpye OR Mr. Samir Dhurde(020-25604601) (020-25604603)([email protected]) ([email protected])

PO Bag #4, IUCAA,Ganeshkhind, Pune -411007

ORVigyan Prasar

Measure the Earth Experiment

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