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Gravity train

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Sophie.A
670567.  Sun Feb 14, 2010 3:48 pm Reply with quote

In the last episode of QI, it was mentioned that if you have a tunnel through the centre of the Earth, an object dropped from one end would emerge at the other in 42 minutes 12 seconds. I did the calculation myself (using just A-level physics and maths) and got 42 minutes 10 seconds instead.

In fact, assuming that the Earth is spherical in shape and its density is uniform, I found the exact formula to be



where ρ is the density of the Earth and G is the universal gravitational constant.

Suppose the object has mass m and is at distance r from the centre of the Earth (and falling towards the centre of the Earth). The gravitational force on it is due to the spherical mass of Earth of radius r (the rest of the Earth forming a spherical shell around the object and contributing no net gravitational force on it). This mass is given by (4/3)πr³ρ (mass = volume × density). By Newton’s law of gravitation, the force on the object is given by G × (4/3)πr³ρ × m / r² = (4/3)πρGmr, where G is the universal gravitational constant. Combining this with Newton’s second law of motion and cancelling the m should give



(the minus sign being due to the fact that r is decreasing with time t). This is the second-order differential equation of simple harmonic motion. The period of the motion, T, is given by 2π/ω, where ω² = (4/3)πρG. And the time for the object to get from one end of the tunnel to the other is half the period motion – which should work out to be the formula mentioned above:



The beauty of this formula is that it does not depend on the mass of the falling object. Nor does it depend on the mass or the radius of the Earth separately – only the density of Earth matters. The values of ρ and G are readily available from Wikipedia; substituting



should give a value of approximately 2530 seconds – which is 42 minutes 10 seconds.


Last edited by Sophie.A on Wed Feb 24, 2010 7:39 am; edited 2 times in total

 
AlmondFacialBar
670569.  Sun Feb 14, 2010 3:51 pm Reply with quote

but the earth is not a perfect sphere, as we all know from season e. would that account for the extra two seconds? i could well imagine that near the equator you'd experience a slightly increased gravitational pull to the side.

:-)

AlmondFacialBar

 
Sophie.A
670580.  Sun Feb 14, 2010 3:59 pm Reply with quote

AlmondFacialBar wrote:
but the earth is not a perfect sphere, as we all know from season e. would that account for the extra two seconds?

Yes, and it could also be due to the fact that the Earth’s density is not uniform.

Still, I think the formula



is a close enough approximation. It shows that tunnelling time increases if the density is less. This would happen on the Moon, which has as lower mean density than Earth – 3346 kilograms per cubic metre according to Wikipedia. Indeed, substituting this value into the formula gives the lunar tunnelling time as approximately 54 minutes 8 seconds. I believe this was also mentioned on the show.

Note also that the tunnel does not have to be straight – it can be in two straight halves, each connected to the centre of the Earth. Thus, in order to get from one place on earth to anywhere else in exactly 42 minutes 10 seconds, you only have to travel in a straight tunnel to the centre of the Earth, and then in another straight tunnel from the centre of the Earth to your destination point.

A straight tunnel between two points on the Earth not passing through the Earth’s centre may be shorter, but it would actually take you LONGER than 42 minutes 10 seconds to get from one end to the other. For example, if you are on latitude L° (N or S) and you have a straight tunnel to the diametrically opposite point on the same latitude, the time required to get from one end of the tunnel to the other would be



which increases as L increases, tending to infinity towards the Poles.

 
Posital
670599.  Sun Feb 14, 2010 4:37 pm Reply with quote

I only assume that one of newton's assumptions was a perfect sphere...

Someone should tell Isaac that he was two seconds out...

 
mckeonj
670627.  Sun Feb 14, 2010 5:45 pm Reply with quote

Probably the time it took for the apple to fall from the tree........

 
thegrandwazoo
670636.  Sun Feb 14, 2010 6:07 pm Reply with quote

So, when are these tunnels getting built? I am trying to plan next years holidays. I bet you would still have to book in 4 hours early and find your luggage had been sent down the wrong tunnel at the core interchange. Hopefully the wrong sort of snow wouldn't affect it though.

 
Ion Zone
670644.  Sun Feb 14, 2010 6:29 pm Reply with quote

On the moon they would be quite handy, if powered, on Earth, even assuming you could do it cheaply, by the time you did you would have excavated enough material to make a country of significant size, and, by the time you finish, taking a scram-jet to Australia would be faster (you could probably walk all the way if you dug to Australia, and add a small third moon - the tunnel would have to be curved) and cheaper - not to mention tunnel safety, it wouldn't be fun if part of the tunnel was crushed or shifted out of alignment with the rest.

This gives me an interesting idea (not, technically, a gravity lift, I might add)....

 
Mort
670709.  Sun Feb 14, 2010 10:21 pm Reply with quote

Heh, this is an awesome post. I think I'm in love with Sophie.A. Great calculations. It is so easy to visualise that the gravity train can operate through the centre of the Earth, but the fact that off-diagonal gravity trains are exactly the same is decidedly harder to understand conceptually - but these calculations show it to be true.

 
eggshaped
670808.  Mon Feb 15, 2010 7:07 am Reply with quote

The person in our version of events stands in awe of Sophie A's quibbling for exactly two seconds before jumping into the gravity train.

 
Mort
670955.  Mon Feb 15, 2010 11:10 am Reply with quote

Here's a great video, though it does mention the alleged 42 minutes and 12 seconds, not 10 seconds - I think we need to launch an independent inquiry into the veracity of this 42 minute and 12 seconds claim and whether Tony Blair has lied to us about the legality of a Gravity Train.

http://www.youtube.com/watch?v=wrQjMslFutc

It comes complete with the obligatory Penguin Cafe Orchestra 'generic science music!' track at the end.

 
Leith
671025.  Mon Feb 15, 2010 2:33 pm Reply with quote

In case anyone's curious, here are a couple of papers with gravity train calculations for comparison.

This one gives the 42m 12s result:
Gravity Train Solution - Alexandre Eremenko, Purdue University

..while this one comes up with ~42m 14.4s (in addition to describing a quicker route with a curved tunnel):
Brachistochrone inside the Earth: The Gravity Train - Amanda Maxham, University of Nevada, Las Vegas

I haven't spotted where the extra few seconds come from yet, but I'd guess it's in the precision of the approximations used for various dimensions of the Earth.

See also:
QI Channel Tunnel thread

 
britishsm
671158.  Mon Feb 15, 2010 6:49 pm Reply with quote

AlmondFacialBar wrote:
but the earth is not a perfect sphere, as we all know from season e. would that account for the extra two seconds? i could well imagine that near the equator you'd experience a slightly increased gravitational pull to the side.

:-)

AlmondFacialBar


Ooh Ooh SIR !! .. Oblate Spheroid Sir ?? :D

B.

 
bobwilson
671185.  Mon Feb 15, 2010 10:40 pm Reply with quote

Ban Sophie A for being far too clever by half

(ooops wrong thread)

 
soup
671253.  Tue Feb 16, 2010 4:40 am Reply with quote

Sophie.A wrote:



which increases as L increases, tending to infinity towards the Poles.


Eh? I may be being thick (probably am) but as L increases would the value for

not decrease?

 
Alfred E Neuman
671261.  Tue Feb 16, 2010 5:07 am Reply with quote

I reckon that, if you have to make any assumptions at all, getting two answers that differ by 0.08% is a good result.

 

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