The Hole Through the Earth Problem with some practicalities and simplifications.

This is an interesting thought experiment and involves being dropped through a hole drilled between the North and South Poles. (If you take any other journey, you would bash into the sides as the Earth rotated.)

Let's assume, for a minute, that the hole is completely empty and ignoring the incredibly high temperatures and pressures down there (probably 5000 degrees and about 5 million times atmospheric pressure).

We start to fall down the hole; accelerating under 'normal gravity'. Newton tells us that only the bits of the Earth that are nearer to the centre than we are would have any effect on us. The bits nearer the surface than we are have no net effect; their attractions would all cancel out.

This force on us is, in fact, proportional to our distance from the centre. When half way down, we would weigh half as much as at the surface. This means that we exhibit what is known as Simple Harmonic Motion (just like a pendulum or a weight on a spring). We would oscillate backwards and forwards, from pole to pole for ever, if it weren't for friction.

The period of this motion is fairly easy to calculate. It's about 90 minutes for the return journey; exactly the same as the time for a satellite in a low Earth orbit to go round the Earth once.

To be a bit practical; what would be the effect of the hole being open to the air? Well, we know that, as we go up in the air, the atmospheric pressure halves every 5.5km. As we went down the hole, the reverse would happen; every 5.5 km downwards, the pressure would double. After only 110km (only 2% of the way down) the pressure would have doubled 20 times! That corresponds to a million times atmospheric pressure. The air molecules would have long since been squeezed together and this simple calculation runs out of steam. But, in any case, the atmosphere would become a highly compressed liquid and we would never get through,

We could do this experiment though, if we went out into space and found ourselves a nice, spherical asteroid or tiny planetoid of the same density as the Earth - say it was a couple of hundred km across and made of Granite If our planetoid were in an orbit near to Earth, it would be warm enough so that any atmosphere would have boiled away so the hole would be completely empty all the way through.

It would work.

But here is the interesting bit. Because we are dealing with simple harmonic motion, the return trip would still take 90 minutes and this would also be the orbit time of a tiny satellite in orbit just above its surface. It must also be true for bits of dust orbiting close to rocks. I wonder if astronauts have ever observed it outside their window. .

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