Why are we even having this discussion? It's crazy that there are people who not only believe the Earth is flat like a pizza but also try to prove it. It's like trying to prove you shouldn't add pineapple as a topping on your pizza (which you obviously should). Maybe they just like a challenge. It's not easy to create a flat-Earth model that accommodates all the evidence against it.
I mean, humanity figured this one out a long time ago. The ancient Greeks not only knew the Earth was round, they even measured its diameter. And there are all the actual, real pictures of Earth taken by satellites and astronauts in orbit. But I sort of respect flat-earthers for not just accepting the conclusions of others.
So you want proof? I'm going to describe two experiments that anyone can do to convince themselves the Earth is a sphere.
This one is really simple, but it does depend on location. You need a large body of water that you can visit. I live near Lake Pontchartrain in Louisiana, which is perfect for this experiment. The idea is to see whether the surface of the water is flat or curved. If the Earth is indeed flat, then the water in a large lake or ocean should also be flat. However, if the Earth is spherical, then the water surface should be the same distance from the center of the Earth as it follows the Earth's curvature.
Let's pretend that a large lake is actually flat. You are standing on one side of the lake with a tall building on the other side. What would you see when you look at the building? Here is a simplified diagram:
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Notice that light from both the top and the bottom of the building should be able to reach the observer's eyes. As long as your eyes are even a little bit higher than the water, you should be able to see the whole building. But what if we have a round Earth and a curved water surface? Here's what it would look like:
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Now light from the lower part of the building would be blocked, and you wouldn't be able to see it. In fact, you don't even need a building. If the lake were flat, you would even be able to see the ground on the other side. With a curved lake, you would see water above the lowest point on the building.
Guess what? I have a picture of this exact situation. This photo was taken on the north shore of Lake Pontchartrain looking toward the buildings of New Orleans on the south side. Check it out:
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It's fairly difficult to explain that picture if you think the Earth is flat. (Unless you think I'm part of the conspiracy and photoshopped it.) OK, just for fun—let's figure out how far you would be able to see if you were standing on the edge of a lake. We'll call that distance s. If the Earth is indeed curved, we can use previous experimental results to say that it has a radius (R) of 6.38 million meters. Let's also approximate the height (h) of a human eye at about 1.7 meters above the surface of the Earth. When that person looks at the horizon, it's a point on the surface of the curved Earth. Here's a diagram:
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You can see that we have a right triangle with the hypotenuse equal to the distance from the observer's eyes to the center of the Earth (R + h), with the other two sides being just R and the distance to the horizon (s). Using the Pythagorean theorem, we can solve for s:
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Now we just need to plug in our values for R and h to get a distance of 4,657 meters, or about 2.89 miles. Of course, if you increase your distance from the surface (h), then you can also see farther. But standing on the shore of a lake, it only needs to be 3 miles wide and you won't be able to see the other shore. Yes, it's because the Earth is spherical.
This second experiment is a little tricky to set up, but you don't need a giant lake. You need to get a mass hanging on the end of a string and let that mass swing back and forth—yes, that would be a simple pendulum. However, if you were to carefully let it go (without giving it any circular motion), the pendulum wouldn't just swing back and forth. Instead, it would slowly change the direction that it swings. This is often called a Foucault pendulum (named after Leo Foucault).
Why does this happen? Let's take this to the extreme case so that it will make more sense. Imagine you have a mass hanging from a point exactly at the North Pole (assuming there is still some ice there for you to stand on). As the pendulum swings back and forth, the Earth below it will rotate (because the rotation of the Earth is what causes night and day). Here's an animation of what that would sort of look like (not in real time).
The pendulum just swings back and forth, but it's the Earth that rotates underneath it. This makes the pendulum appear to change directions as it rotates, and it would take half a day for the pendulum to again swing in its original direction (as seen from the North Pole). A full cycle would take one day.
But wait! This doesn't prove the Earth is spherical. Maybe it just shows that the flat Earth rotates like a record on a record player. OK, fair, but how about this? If you take that same pendulum to the South Pole it will also rotate—but in the opposite direction, since you would be standing in Antarctica, completely upside down with respect to the North Pole pendulum.
But you don't have to go to the North or South Pole to do this experiment. You can do it at home. The pendulum will again swing in different directions as the Earth rotates, but it will take longer than half a day to return to its original direction, and the time will depend on your latitude.
First, let's review what latitude is. If you drew a line from the center of the Earth to the equator, and another line from the center of the Earth to your location, those two lines would define an angle, which we'll call θ. That's why latitude is measured in degrees. If you're actually on the equator, the angle would be 0 degrees; the farther north you move, the larger the angle gets, maxing out at 90 degrees at the North Pole. My home in Louisiana is at 30 degrees north.
So the period of time (let's call it T) for one full cycle of your pendulum would be:
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As θ increases, the denominator increases, which means T decreases. There you have it: If you take your swinging mass on a string and travel north, it will take less time to complete an oscillation. You can even record the period and use this to calculate your latitude. Of course, there are easier ways of determining your latitude, but I don't think flat-earthers are looking for easy answers.
