Thanks to Brilliant for helping support this episode. Hey Crazies. What if we dug a hole all the way through the Earth. How long do you think it take someone to fall all the way through? About 40 Minutes. That’s it. Just 40 minutes and it doesn’t even matter how you dig the hole. This tunnel takes 40 minutes. That tunnel takes 40 minutes. All the tunnels take 40 minutes! But, how did I get that number? This episode was made possible by generous supporters on Patreon. Unless you’re making really sensitive measurements or you’re near a neutron star or a black hole, gravity is accurately described by Newton’s laws. But even that can be challenging. If you try to solve this problem as is, you’re going to run into trouble. See, we’re used to gravity or weight being pretty constant. My clone has a weight that’s pulling him toward the ground, but that weight is the same up here, or down here, or over here. If he falls, that weight stays the same. It’s constant. Unfortunately, most people have spent their entirely lives near the surface of the Earth and that’s created a bit of a bias. My clone’s weight might be 165 pounds on the ground, but it’s 140 pounds in an average low-Earth orbit, and effectively zero in deep space. Weight is actually different depending on where you. Weight is just another word for the force of gravity. It depends on how far apart things are. It’s just that the Earth is so big and so massive that we don’t notice the differences, at least when we’re near it. My clone’s weight is only approximately constant. If we drop him from low-Earth orbit, we’ll have to be a little more careful. In a case like that, we have to consider how far the clone is from the Earth. When a force changes with position, even Newton’s laws can be notoriously difficult to solve. We tend to get non-linear differential equations like this one and that’s assuming the Earth doesn’t move, which is only approximately true. Seriously, I spent, like, four hours trying to solve that thing symbolically before I gave up. But a computer can give us an answer as long as we plug in specific numbers. Assuming no air resistance, it takes my clone 6.3 minutes to reach the ground. If you assume a constant weight, you get 5.8 minutes, which is clearly wrong. The math gets even harder when both objects are allowed to move. It’s called the two-body problem where “body” is just another word for “object.” Although, if we’re using clones, “body” is a fine word. Let’s say I put two of my clones in the middle of deep space about 5 feet apart. They’ll each have a force of gravity toward the other. Both of them have mass, about 75 kg worth, so they will attract each other. How long will it take that gravity to bring them together? Not as long as you might think. The computer says about 5.8 hours, which is apparently just enough time for them to run out of oxygen. The thing is though, using a computer is kind of a brute force method. There’s nothing particularly elegant about it. It’ll work on just about anything. Orbits, balls on ramps, single pendulums, double pendulums, masses on springs. It’ll even work on many-body problems as long as you’ve got the processing power. Boring! We want a solution that’s a little more elegant something outside the box. In physics, sometimes it’s helpful to imagine a challenging problem as a completely different problem. Consider our two clones again. They might be falling toward each other, but let’s imagine for a minute that they aren’t. Let’s imagine that they’re orbiting a common center of mass instead. Believe it or not, orbits are way easier to solve than free fall paths. These elliptical orbits obey very simple patterns. We know each clone will be opposite each other at all times. We know the farther they are from each other the slower they move and vice versa. We know their common center of mass will be a focus of both ellipses and the foci of an ellipse are always on its major axis. Half of that major axis is called the semi-major axis, typically labeled “a.” We also know their motion is periodic. My clones will always return to their original starting places after a certain amount of time. That time is called the period and is given by this simple equation. If we know the semi-major axis of each clone’s path and we know the mass of each clone, we can find their orbital period. What does that have to do with falling though? Everything! An orbit is the same as falling. The International Space Station is falling toward the Earth. It’s just moving so fast sideways that it continuously misses. The Earth curves away from it as it falls. It’s the same with my clones. They’re also falling toward each other. They’re just moving so fast sideways that they miss. But what if they didn’t miss? What if the ellipses were so long and skinny that the clones ran into each other somewhere? That would happen around here when they’re closest to their center of mass. If we make the ellipses really skinny, this is pretty much a free fall problem. Actually, it’s exactly a free fall problem. In fact, we already know how to find the time. The sum of the semi-major axes is just half the distance between them. This equation tells us the period is 11.6 hours, but the clones run into each other about halfway through their orbital cycle or after half the period. That’s 5.8 hours, exactly what the computer got earlier. Bingo bango! This trick will work for any free fall problem. Want to know how long it would take for the Moon to fall into the Earth? The sum of the semi-major axes is just half the distance to the Moon. The time? 4.8 days! Did you drill a hole all the way through the Earth and fall in? The sum of the semi-major axes is just half the distance across the Earth, otherwise known as the radius of the Earth. The time? 42 minutes! Of course, that last one is assuming the Earth is perfectly uniform, which it isn’t. Henry from Minute Physics went through the trouble in one of his videos if you’re interested. Spoiler Alert: It’s still around 40 minutes. There’s another aspect of this tunnel that I happen to find a little more interesting. See, gravity doesn’t magically come from the center of the Earth. If we’re outside the Earth, we can pretend like it does. We imagine the Earth is a tiny point mass and then measure everything from there. In reality though, a separate gravity comes from every tiny particle that makes up the Earth. Remember, everything with mass attracts everything else with mass at all times. A falling person is being attracted toward every particle in the Earth. When they’re outside the Earth, all of those particles are underneath them. But, once they’re inside, some of those particles are above them. Some of the Earth is pulling them upward, which makes them lighter. You weigh less underground. Your weight only depends on the part of the Earth that’s closer to the center than you. Let’s say one of my clones is in a deep hole. His weight comes from the mass inside this sphere. These other two sections of the Earth cancel each other’s effects. The part above him pulls him up just as much as this part pulls him down and that’s true no matter how deep the hole is. If there’s a tunnel all the way through the Earth and my clone falls into it, it’s as if the Earth is getting less massive as he falls. He would still speed up as expected, but his weight would decrease on the way down. As he passes through the core, his weight would go to zero. He would be weightless. After that, the weight changes direction, which will slow him down. Eventually, he’ll come gently to rest at the other end. And, as we said before, the whole trip takes about 40 minutes. But what happens if we don’t catch him? He’ll fall down again. Eventually, he’ll reach his original starting point another 40ish minutes later. If we never catch him, he starts to look like a block bouncing on a spring. It’s the same motion. I don’t mean kind of the same. I mean exactly the same. The forces that govern both motions are some constant stuff times a distance, so it’s only natural that their motions would be the same. We could have imagined this tunnel problem as a spring problem instead and 42 minutes would still have been our answer. Reimagining is like the ultimate physics trick! But the coolest thing about this is that it doesn’t even matter where the hole goes. When we write the equation like this, it makes it seem like you have to travel the entire diameter of the Earth. But all that matters here is the density of the Earth. We can drill the hole wherever we want. The math will always predict a 42-minute travel time. Actually, if the tunnel doesn’t go directly between the poles, you have to consider the rotation of the Earth. Dude. Why do you have to be such a buzz kill all the time? Ugh. He’s right. Since the Earth is spinning, the Coriolis force would have to be factored in for most tunnels. Unfortunately, we can’t drill a tunnel between the poles. Creating a tunnel like this requires there be land on both sides of the Earth, but there’s no land at the north pole. We live on a planet that’s mostly covered in water. Fun Fact: If you take a map of the Earth and superimpose it on the same map rotated 180 degrees, you’ll see all the places you can make an Earth sandwich. Because no one wants soggy bread on their sandwich. Am I right? So what’s the crazy physics trick that makes gravity easy? It’s the ability to imagine one type of problem as a completely different problem, an easier problem. Instead of a direct gravitational attraction, imagine gravitational orbits and then just make the orbits very eccentric and take half the period. Instead of a person falling through an Earth tunnel, imagine they’re bouncing on a spring and take half the period. It’s the easiest way to calculate travel times in gravitational systems. So, are you an out of the box thinker? Let us know in the comments. Thanks for liking and sharing this video. Don’t forget to subscribe if you’d like to keep up with us. And until next time, remember, it’s ok to be a little crazy. If you’re naturally curious or want to build your problem-solving skills, then get Brilliant Premium to learn something new every day. Brilliant has over 60 interactive courses in math, science, and computer science. If you liked this video, you might like their course on gravitational physics, where you can learn all about Newtonian gravity. When you’re ready, try their quiz that walks you through the two-body problem or their quiz on gravitational pendulums, which should look familiar. If you’re finding them a bit challenging, you can always brush up on the math with their new interactive calculus course, where visual and physical intuition is used to present the major pillars of calculus. Brilliant puzzles you, surprises you, and expands your understanding of the modern world. If this sounds like a service you’d like to use, go to brilliant dot org slash Science Asylum today. The first 200 subscribers will get 20 percent off an annual subscription. The Sun uses quantum tunneling for nuclear fusion and nuclear fusion is how the Sun creates light, so is this the light at the end of the tunnel? You’ve won my comment section today, Master Therion. Nice one. And, to everyone else, thanks for watching!
