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The First Step- Measuring the Size
of the Earth Study Material Background While there may have been earlier attempts not yet brought to light, current reading of history gives credit to Eratosthenes (276 BC -195 BC) the Greek, as the first person to actually execute the idea and give the radius of the earth quantitatively. His idea was very simple. Let us follow his idea 1. The earth is spherical 2. The distance to the sun is very large compared t to the size of the earth 3. So the rays of the Sun falling on the earth are all parallel to each other You know that every day the sun rises reaches the highest position at noon or mid day and the sets. You can follow this motion by looking at the direction and the length of the shadows cast. The shortest shadow is cast exactly at noon, when the sun is highest in the sky. Since the earth is a sphere the angle between the vertical and the mid day sun is not the same at different latitudes. Therefore the lengths of the shortest shadow cast by the mid day sun are also different. A and B are two locations on the surface of the earth of radius R. Two sticks (Gnomon) of equal length (AP=BQ) are placed in the two locations. The shadow length AM is shorter than shadow length BN. The direction of the mid -day sun at A and B are MP and NQ. The shorter shadow implies that the sun has a higher altitude at A than at B. The angle APM between the vertical and the direction of the sun at A can be measured by constructing similar triangles on paper or by using trigonometry. The angle BQN between the vertical and the direction of the sun at B, can be similarly measured
How do we calculate the radius of the earth by knowing these angles? Look at the figure below. Let the circle below represent the earth. The vertical at each place is defined as the line joining the centre of the earth to the points A and B The verticals through them are OAP and OBQ respectively. The shortest shadow is cast by the mid day sun. The rays of the sun falling on the earth may be considered parallel, as the sun is very far compared to the size of the earth. From measurement we know the angles APM and BQN and the
difference between them. This is equal to the angle AOB subtended by the arc AB at the centre of the earth. 360 0 is the angle subtended by the whole circle at the centre. Circumference of the earth = C = arc AB x360/ angle ÐAOB. R= C/2pi Thus if we know arc AB and angle, AOB then we can calculate the radius of the earth. We can measure this arc AB from an accurate map for two locations having nearly the same longitude.
ObservationsMaterials needed 2. A plumb line (twine and nut) 4. 40cmx 40cm square board( husk, ply) 12 mm thick 5. A Sheet of white paper, pencil, fine thread, a nut or other suitable weight for plumb line The first item will be supplied by Space on payment/fee by SPACE, New Delhi
Preparation
1.
Paste/ tack white paper on square board. Your shadow stick is now ready for use Marking the shadow 1. Find a site that is flat and open sky where the sun is visible between 11a.m and 1 pm ( A terrace for instance SE to SW segment) 2. Find a flat surface. Place Gnomon with base on the surface. Adjust the base board so that the gnomon is vertical. Check using a plumb line. This is very important 3. Between 11 am and 1 pm mark the tip the shadow formed by the gnomon every 15 minutes. Ensure that the stick is vertical through out the observations. 4. Remove Gnomon from board. 5. Draw a smooth curve through all the observations 6. Measure the shortest distance from the curve to the base of the pencil in millimetres several times. Record the average. This is the length L of the shadow 7. Construct a similar right angle triangle with sides proportional to the Gnomon height H and shadow length L, using a large scale factor. For example draw a horizontal line AM=2L; Draw a line AP perpendicular to it at one end, of length 2H. Join PM. 8. Measure the angle APM. This is the angle between the vertical in your location and the direction of the noon day sun.
Measuring Arc AB 1.Take as large a map of India (Given in the kit) 2. With a piece of fine thread, measure the distance on the map between your location A and partner location B along a longitude line. If there is slight difference in longitude ensure that the arc is measured from your latitude to the latitude of B. Repeat several times and record the average arc length AB. Multiply this by the scale km/mm given on the map. This is the length of arc AB in kilometers. Report results to cooperating centre. Let the length of arc between A and B be arc AB Let the circumference of the earth be = C Let the vertical angle of the noon day sun at A and B be A and B respectively. The difference A - B = d C/360 = arc AB/d C = arc AB x 360/d You have measured the earth!
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