People studying science and mathematics, love to calculate things and they simply adore calculating for “assumed situations”. Scientists study earth and its structure on a regular basis and a lot is known about our very own Mother Earth as of now. This time around, the maths community has a very interesting thing for you.
Assume that you were to fall through our Earth from North Pole to South Pole, then how long would it take exactly? Well well… don’t look at us that way… we are not ‘actually’ making you fall through that distance, we are asking you to ‘assume’! So instead of making you fall through that space actually, we will make you fall mathematically with certain assumptions.
- It was assumed that the Earth has got a hole through it and it goes from north pole to the south.
- There is no air in the hole otherwise you would reach terminal velocity quite quickly and you will move too slowly.
- It was assumed that the Earth is perfectly spherical and has the same density everywhere (This is to note that the gravitational attraction from any spherically symmetric object is the same as if all its mass were concentrated at the centre of the object)
- The parts closer to the sphere attract more and the farther ones attract less but in whole the gravitational pull averages out)
- So inside the Earth, any parts that are farther away from the centre than you are; basically cancel out each other so that they create a smaller sphere trimming out the outer layers.
- Assuming the earth also behaves like this shaved sphere, we come to the assumption that the shaved earth’s mass is simply proportional to its volume, which is proportional to its radius cubed and we safely assume that all the mass is concentrated at a single point in the middle.
- The gravitational attraction between two objects is proportional to their masses but inversely proportional to the distance between them (squared)
- Calculating on the basis of all the assumptions and mathematical formulae, we arrive at the result- 42 minutes. Yes this is the time you will take to fall through the Earth from North Pole to South Pole.
The calculations done above are based on the assumption that the earth has same density throughout, but if you chart the more detailed density profile then you will come to the more accurate falling-through-the-earth time, which is 38 minutes and 6 seconds from pole to pole. Apart from the mathematics, the physics behind this fall is similar to that of a moving pendulum.
This means, as you fall, the mass beneath you decreases more quickly as the average gravitational pull on you from that mass increases. But the mass decreases more quickly, so as you approach the centre of the Earth, you start going faster and faster but the force pulling you towards the centre getting smaller. If you are interested and are a maths and physics enthusiast, then you can refer the video below which gives detailed calculations and formulae. Have fun calculating!
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Author: Technology and Beyond
