Mental experiments
with gravity

Michele Diego

This is the power of the thought experiment: it is able to idealize a phenomenon, even making it unrealistic, but with the aim of extracting the essence from it.

In college, I often heard of the distinction between two categories of physicists: the intuitive and the analytic. In this trenchant subdivision of the scientific world, those belonging to the first category understand concepts through drawings, figures, and analogies with other physical systems, while the latter is guided mainly by mathematical theorems and formal proofs. I used to belong to the second category, so much so that at a certain point I decided that everything that needed an image to be explained was not worthy of consideration. I also discovered, with some admiration, that there was someone even more uncompromising who had tried to publish mathematical books without any spoken word, that is, composed only of mathematical symbols and equations.
Needless to say, my desire for formal strictness has shattered over the years. And today, I am the first to affirm that a scientific article needs, first of all, clear figures from which the message of the article transpires. But today I want to go back to the glorious past for the sake of this article of La Livella, discussing with you, armed only with words, two thought experiments that deal with gravity.
The creator of the first experiment is none other than Sir Isaac Newton. His purpose is to show that orbiting the Earth (or any other planet) in a certain way means falling, but too fast to touch the ground. In the second thought experiment, we will question what would happen if we could tunnel through the entire Earth (from the North pole to the South pole for example, or from Spain to New Zealand) and jump into it.

Let’s start with Newton’s experiment, here revisited in a modern-fashioned way. We are on a very high mountain, like Mount Everest for one, with no obstacles around us. We can imagine a large plain at the centre of which rises the mountain on which we are standing. With a slingshot, we throw a cue ball straight in front of us. What happens to the ball? While it is in flight, the force of gravity acts on its centre of mass and deflects its trajectory downward. By doing so, after a few dozen meters, it will end up falling to the ground. If we increase the force with which we pull back the ammo, the range of the shot also increases accordingly.
Nothing extravagant so far. But what would happen if we started throwing the ball really far, reaching tens or hundreds of kilometers of range? We should now change the description of our system. The Earth’s curvature, in fact, would begin to matter. The image to be visualized would no longer be that of a large plain in the center of which a mountain is positioned, and in which the force of gravity drags objects downwards. Instead, it would become that of a sphere (the Earth), from which a mountain rises, and there we are above it, armed with our slingshot. Gravity, in this description, no longer points downwards, but towards the center of the Earth. ‘Throwing straight’ now means that the cue ball follows the Earth’s curvature. By continuing to increase the force with which we launch the cue ball, we’ll make it reach ever greater distances, sending it to places on Earth that we cannot see. Sooner or later, we’ll be able to drop the ball at the exact antipode of the world. Continuing further, sooner or later it would hit us in the back of the head. At that point, it will have completed its entire rotation around the Earth: we will have managed to put it into orbit. If we were able to dodge it, it would continue to orbit the Earth, like a satellite at very low altitude.
This thought experiment, as obvious as it may seem today, contains one of the fundamental pivotal points of Newton’s theory: the unification of celestial mechanics and earth’s gravity. The force that holds the Moon in orbit around the Earth is the same force that causes objects to fall to the ground. This idea is described in the work De mundi systemate, and represents a historical stage within Newton’s thought, a much more reliable event compared to the legendary apple which, falling on his head, earned him his famous ‘Eureka’.

The second thought experiment concerns gravity as well, but this time, instead of orbiting the Earth, we’ll go through it. Let’s imagine we are digging a tunnel from one side of the globe to the other; a well of some kind, with the same length as the Earth’s diameter, hence passing through the centre of the Earth. It could extend from the North pole to the South pole, from Spain to New Zealand, or from Brazil to the Philippines. For this experiment, Italy is not a great choice: our tunnel would end up leading into the Pacific Ocean.
Anyway, once this hypothetical tunnel is dug, what would happen if we jumped into it? Gravity tends to cause objects to fall towards the centre of the Earth, which clearly implies that we would begin our fall towards the centre of the Earth. As our fall continues, our speed increases, just like the speed of any object falling to the ground increases. It takes about twenty minutes to reach the centre of the Earth. Here, however, things change. At the exact central point, representing a point of symmetry of the globe where there is equal mass in all directions around us, the force of gravity is zeroed. No forces act on us, in the exact instant in which we are in the centre of the planet, and therefore, due to the famous law F = m a (force equals mass times acceleration), there is no acceleration. However, even if there is no acceleration, our speed is the maximum achievable in this system, because we have accumulated the entire velocity of the fall up to that point. By definition of a nonzero speed, therefore, we will continue our advance, moving now away from the center of the Earth and starting to head towards the exit of the tunnel, opposite to the one we entered from. However, as we continue through the tunnel, our speed is now decreasing. Being attracted to the centre of the Earth, as we move away from it we begin to slow down, just like an object thrown upwards slows down until it stops and begins to descend. Our speed will be zero at the exact moment we reach the exit of the tunnel and we see the Earth’s crust (or the Pacific Ocean, if we have launched from Italy). But what happens at that point? The force of gravity will continue to attract us towards the centre, reversing the motion and accelerating us towards the Earth’s core. In doing so, we will go back to the Earth’s centre and return to the entrance of the tunnel, from where we started. And go on like this, like a human yo-yo eternally doomed to fall into a tunnel across the world.

Fortunately, no one can start throwing pellets from a mountain with a slingshot that go around the world, much less dig tunnels from one side of the Earth to the other and then throw someone inside. Moreover, to be a stickler, in our two thought experiments we considered negligible some effects that would substantially modify the description we have provided: the air resistance, for example, which slows the motion of objects in flight. But it is normal and indeed essential that this be so: in order to understand the essence of a phenomenon, scientists must be able to eliminate those variables that intervene indirectly and that can divert attention from the heart of the problem that is trying to tackle. This is the power of the thought experiment: it is able to idealize a phenomenon, even making it unrealistic, but with the aim of extracting the essence from it. Today, thought experiments, in some ways, have been replaced by computer simulations, in which with one click you can eliminate air resistance, but also halve the weight of the Earth or increase it a thousandfold. With these completely ideal attempts, we are able to detach ourselves from reality to enter a world in which the physical contributions of a complex dynamic can be analyzed one by one, allowing us to understand their real individual importance. It is a dive into an abstract world, a ‘mathematical’ world in a certain sense, but it is not an irrational and fantastic dream: when we re-emerge, we will know something more about the world around us.


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