Journey to the Centre of the Earth

The world seems to be getting a smaller and smaller as transport becomes more and more efficient. Although a plane could take us to the other side of the world in under forty-eight hours, the cost is the main deterrent. As a physicist I am obliged to think of other options (however theoretical).

Why not take a direct route through the centre of the Earth? Even though I’ll have disintegrated whilst I’m digging a couple of kilometres below the Earth’s surface, it’s a nice thought experiment anyway.

‘Okay, I’ll see you down under,’ I say to Harvey while he boards his flight and I’m preparing to jump into my ready-made hole.

Contrary to popular belief, you are not able to dig from the USA through the centre of the Earth to China – assuming you don’t deviate from a perfectly straight path, of course. For one, both countries lie in the northern hemisphere, and that should be a massive giveaway. In fact there is no location in any of the fifty states that would enable you to arrive on land. (Okay, there may be some dodgy islands that few people have visited, but you probably wouldn’t want to end up there anyway.)

I came across Antipodes Map, a particularly riveting website that spits out the exact location on the other side of the Earth to the one you enter. As a resident of the UK, I would end up a somewhat thousand kilometres off the coast of New Zealand, so I’d probably want a better place to dig my hole.

We can tell that a route directly through the centre of the Earth is certainly a shorter distance (approximately 12,700 km) compared with the route as the crow flies (approximately 20,000 km). However, we can’t be certain that it will take a shorter time, because the only force accelerating our fall is the force of gravity.

If we model the Earth as a perfect sphere, the centre of mass of the Earth can be assumed to be at the centre of the Earth. This means that, when we stand on the ground, even though there is mass to the left of us and to the right of us, we can assume the gravitational force of the Earth is pulling us perfectly downwards.

Newton’s law of gravitation states that force of gravity F decreases inversely with the square of the distance r from the centre of the field:

newtons-law-of-gravitation

m1 and m2 are the two masses in question, and G is the universal gravitational constant.

This equation shows that if you were to double your distance from the centre of the Earth, the gravitational force pulling you down would quarter. If you were to triple your distance from the centre of the Earth, the gravitational force pulling you down would be  nine times weaker.

Unfortunately, this relationship doesn’t apply once you’ve fallen below the surface of the planet, because now there is an increasing amount of mass above you that pulls you upwards – we have to establish a new relationship between force and distance below the surface of the Earth.

Maths and experimental evidence has demonstrated that there is no gravity inside a hollow sphere. There is no gravity at the centre, because there is an equal amount of mass surrounding you in all directions and you are being equally attracted in all directions. If you were to shift closer to one side of the shell, then the mass still cancels out, since now there is more mass further away from you and less mass closer to you.

Gravity in a Hollow Sphere.png

This means that none of the’shell’ of mass beyond the distance you are from the centre will contribute to the gravitational force. Essentially, as you fall, the force is acting as if the layers of the Earth above you  are continually being shaved off.

gravity-within-earth

Using this, we can approximate the relationship between force and distance while below the surface of the Earth – the new mass of the ‘shaved’ Earth is the product of its volume (4/3)πR3 and its average density ρ, and we can substitute this mass into the gravitational force equation we saw earlier.

Sadly the different layers of the Earth have different densities – the crust is the least dense layer, being made of rock, whereas the inner core is packed with solid iron. To make the maths simpler, we’ll just assume that the Earth has an average uniform density (we’ll find that it doesn’t make too much of a different on the result).

We also need to assume no air resistance, because otherwise we would reach terminal velocity and we wouldn’t be travelling fast enough past the centre of the Earth to reach the other side, and we would spend eternity doomed to oscillate about the centre of the Earth. Without air resistance, we can only just reach the other side.

With some mathematical manipulation, we can evaluate how the force varies below the surface of the Earth:gravitational-force-within-earth

This equation indicates that force decreases linearly with distance as you approach the centre, and equals zero at the centre.

After some classical mechanics calculations, we end up with a time period of a brisk 42 minutes. If we take into consideration the varying density within the Earth, the time period decreases to approximately 38 minutes, because more of the mass of the Earth is contained within the core.

It turns out gravity is pretty powerful, as you would be reaching velocities of up to 30,000 km h-1. Beat that, Boeing.

Even after writing this post, I could jump into the hole and still be waiting almost two days for Harvey to arrive. Maybe I’ll just have a snooze.

Yanhao

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