Disclaimer: The earth, according to current mathematical models and observations, is not flat.
ROUND-EARTH MATHEMATICS :
In determining the rotational speed of the earth, we must first assume that the sun is positioned an exceptionally far distance from the earth. If this were not the case, snow, water, and complex biological life would be non-existent. Since this is the case, we must also make the assumption that essentially flat ” windowpanes ” of radiation reach the earth’s surface.
Non-mathematical reasoning, of course, can be used to determine that the earth cannot be flat. If the earth were flat, a hurricane in the Gulf of Mexico would cause rivers and streams to reverse direction, and the flow of water would travel across Canada and melt ice in the North Pole region of the earth. Volcanoes are proof of the existence of tremendous pressure inside the earth’s surface; if such pressure causes soap bubbles to become round over time, why do the same rules of physics not apply to the planet earth? How does a flat earth explain the behavior of a compass needle? If a compass needle ( on a flat surface ) pointed to the center of the surface as opposed to poles on a sphere, how would this region maintain permanent freezing temperatures in the midst of geographic regions that contain permanent deserts? This scenario is, of course, little more than fiction.
Let’s assume that two yardsticks are driven into the ground. One is placed on a flat surface in Columbia, SC. The other yardstick is placed on a flat surface in Little Rock, AR. These two locations can be approximated as being the same distance from the earth’s equator. Additionally, let’s assume that light rays come in at an angle that causes the shadows to be aligned with one another in one direction. On an earth that is flat, the aforementioned sticks would cast shadows that are equal in length ; however, this is not what is observed under real-life circumstances. A person in Columbia, SC will have shadow measurements of a different length compared to measurements made in Little Rock, AR.
If we assume that the aforementioned differences in shadow length can be attributed to curvature on the earth’s surface, mathematics can be used to make assumptions that enable planes, ships, drones, telecommunication systems, guided missiles, and spy satellites alike to function efficiently. Examining the caption here, we can assume that the sun’s rays are equivalent to the ” hypotenuse ” of right triangle with a 90 degree angle between the earth’s surface and our yardsticks.
Another assumption of importance is that the aforementioned yardsticks point towards a common destination within the center of the earth. If yardstick ” A ” is in Columbia, SC, and yardstick ” B ” is in Little Rock, AR, it would be useful to know the angle that exists between the two imaginary lines that leave each of them and meet at the center of the earth.
In order to determine the angle formed at the center of the earth, we must understand the relationship between the different shadows cast from yardstick A and B. For a moment, we can make a cross with our fingers. Assume that the vertical finger is one of the two yardsticks. Additionally, assume that the horizontal finger is a sun ray. Keeping our fingers fixed at 90 degree angles, if we rotate the vertical finger by 45 degrees, the horizontal finger will now be at a 45 degree angle with the earth. Therefore, if there’s any difference in shadow lengths of yardsticks A and B, it is due to A and B having been rotated away from each other due to the earth’s curvature.
There is a direct correlation between the distance between yardstick A and B, the earth curvature that exists between them, and the lengths of the shadows cast from each stick. The difference in the sunlight angles measured from yardstick A and B are directly proportional to the angle of interest in the center of the earth. Fortunately, determining the difference by which sun rays impact yardstick A and B can be done using rather simple and archaic mathematics. The mathematical function of interest is the tangent ( tan ) function described here.
Within any triangle within which a right angle exists, the value of the ratio of the sides that merge to form the right triangle can be determined. The tangent ( tan ) in our case is the ratio of each yard stick’s length ( opposite to theta ) divided by the length of the shadows cast by A and B ( the adjacent sides of theta ). After the measurements are made, a calculator can be used to take the inverse tangent of our measurement ( tan^-1 ), which will give us the angle between the incoming sun ray and the ground at points A and B. We may subsequently subtract the smaller angle from the larger angle. The net result will approximate the angle between the imaginary lines that meet at the center of the earth.
The distance around a circle or sphere is defined to be 360 degrees. If, for example, our angle at the earth’s center is found to be 1 degree, this would mean that the distance between points A and B would be one-360th of the circumference of the earth. ( 1 degree / 360 degrees ) = ( A to B distance / earth’s circumference ). Inverting both sides of the equation gives us ( earth’s circumference / A to B distance ) = ( 360 degrees / 1 degree ). This would quantify the earth’s circumference as being 360 times longer than the distance between point A and B. Different angular measurements obtained by taking measurements with larger or smaller distances between A and B will yield results that necessitate that we multiply our measured distance by some different value. As a consequence, we must divide 360 degrees by our net angular measurement to determine what value we must use to derive the circumference of the earth.
To determine the value of angle A, we’d take the length of yardstick A ( which should be identical in length to yardstick B ) and divide it by the length of the shadow it casts, while our assistant at point B simultaneously calculates tangent values using their measurements for yardstick and shadow length. Once the tangent function is used to solve for the angles of interest, subtract the smaller angle from the larger one. The result obtained is the value of the imaginary angle at the center of the earth. Divide 360 degrees by this number, and multiply the result by the distance between points A and B.
The earth’s circumference is 24,901.461 miles. The distance from Columbia, SC to Little Rock, AR is 729.5 miles. If ( 360 degrees / change from A to B ) = ( earth’s circumference / distance from A to B ), we can determine what change in degree measurements we may obtain in SC vs. AR. Let’s designate the change in angles from location A to B to equal ” x “. ( 360 degrees / x ) = ( 24,901.461 miles / 729.5 miles ). Solving for ” x ” gives us x = ( 360 degrees )( 729.5 miles / 24,901.461 miles ) = a 10.54636915 degree difference between measurements made at points A and B.
Let’s now pretend that we measured the shadow lengths at point A and B without knowing beforehand the length of the earth’s radius and circumference. How may we go about determining these values? Fortunately, this is not difficult to accomplish. We would first use the tangent ( tan ) function to determine the different angles at which sun rays landed upon yardsticks at A and B. If these angular measurements differed by the above value of 10.54636915 degrees, we can assume this to be the angle of separation between two imaginary lines that leave each yardstick and meet at the center of the earth.
Using the tangent ( tan ) function = ( opposite side / adjacent side ), the opposite ( opp ) side of the angle at the earth’s center is our measured distance from Columbia, SC to Little Rock, AR : 729.5 miles. The adjacent side of the angle at the earth’s center is the radius of the earth that we want to determine ( r ). So tan ( 10.54636915 degrees ) = ( 729.5 / r ). Solving for ” r ” gives us r = ( 729.5 miles / ( tan 10.54636915 degrees ) ) = 3,918.329767 miles.
NOTE: The curvature of the earth was approximated as being flat across the 729.5 mile distance from A to B. Therefore, our final calculations will have some error.
The circumference of a sphere or circle is ( 2 )( pi )( r ). Since we now have a value for ” r “, let’s use it to estimate the value of the circumference of the earth. Accordingly, the earth’s circumference is ( 2 )( 3.141592654 )( 3,918.329767 ) = 24,619.59202.
PERCENT ERROR :
The percent error is calculated using the absolute value ( abs ) of the [ accepted value – experimental value ], divided by the excepted value. The outcome is multiplied by 100% to convert the fraction into a percentage. Therefore, ( abs [ 24,901.461 – 24,619.592 ] / ( 24,901.461 ) )( 100% ) = A percent error of 1.1319376 %.
Since it takes 24 hours for the earth to complete one 360 degree rotation, dividing 24,619,59202 miles by 24 hours will give us the approximate speed at which the earth rotates : 1,025.816334 mph ( approximately ). The accepted value of the earth’s rotation is close to 1,000 mph.
RENAISSANCE PHYSICS 