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A question about the earths curve.


rick89
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Hello all.

 

If a very long bridge was built, say from England to Australia, it would be correct to say that the bridge would follow the earths curve and therefore not be level.

 

If, however, a bridge that was this long was built and was perfectly level, would I be correct in thinking that it would seem to gradually slope 'upwards' up to a point where it would be impossible to climb?

 

Regards

Ricky

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Its impossible but if a bridge which is so long and it is not curved,the bridge would be out of earth's surface and it would take you to the more and more height from the earth.if you keep travelling on it you would be out of earth's gravity.

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Hello all.

 

If a very long bridge was built, say from England to Australia, it would be correct to say that the bridge would follow the earths curve and therefore not be level.

 

If, however, a bridge that was this long was built and was perfectly level, would I be correct in thinking that it would seem to gradually slope 'upwards' up to a point where it would be impossible to climb?

 

Regards

Ricky

"level" gains a certain amount of ambiguity when used in reference to a curved surface. What do you mean by level?

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By level I mean flat, not curved, straight.

 

The bridge would go into space, but for someone walking along the bridge, which would be flat, they would eventually have trouble due to gravity and it would eventually become impossible without the use of a ladder. Would this be correct?

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By level I mean flat, not curved, straight.

 

The bridge would go into space, but for someone walking along the bridge, which would be flat, they would eventually have trouble due to gravity and it would eventually become impossible without the use of a ladder. Would this be correct?

 

yes the angle would approach 90 degrees as the length approaches infinity

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True, but as the distance approaches infinity the effect of gravity approaches zero.

Which happens first?

Also, if you built it on top of the north pole would the fact that it's spinning tend to throw you out along it (assuming it had handrails)?

 

Let's look at it this way:

 

Assume that you built it on the Equator in a west East direction. When the "bridge" reaches a length of 41,820 km, you would be a geosynchronous orbit altitude and would be in orbit. This would be the point where you would feel zero g. At this point, the Bridge would be 81.32 degrees from level. So the question becomes: Just how steep an incline can you climb ( assuming a flat surface)? Sure gravity is less, but as a result so is the friction between your feet and the bridge due your lessened weight. My guess is that sometime before reaching the zero -g point you will no longer be able to climb the increased slope.

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Let's also not forget altitude sickness. At 150 miles, you would be walking on a slope of about 2°, and you will have attained an altitude of 15,000 feet, which is the maximum allowed in aircraft without pressurized oxygen. You could also suffer altitude sickness below that altitude. Fly to La Paz Bolivia and see for yourself.

 

At 850 miles, you'd be walking at about 12°, and you'd be 90 miles in altitude where the low-orbit stuff "floats by" at 17,000 mph. Cool!

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yes the angle would approach 90 degrees as the length approaches infinity

 

Would the angle depend on each individuals perspective?

 

 

For example, if I was at the beginning of the bridge and you were at a point on the bridge where it was starting to get difficult to climb and I had a very powerful telescope and was looking straight forward, I would be able to see you and beyond. It would look to me like the bridge was flat. If you, however, looked straight forward you would see the bridge surface because of the gradient. To see beyond the bridge you would need to look up.

 

 

Surely what I would see through the telescope wouldn't make sense? Unless i'm missing something here!

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If the bridge followed the curvature of the ocean, then wherever you put a level on the surface, it would say the bridge was level. Conclusive proof that the earth really is flat.

 

The only way to build a bridge in a straight line between England and Australia, is to drill a hole through the centre of the earth, and such a bridge would be vertical everywhere except the centre where the level would read level.

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One aspect not considered is centrifugal force. If a straight bridge was built at a pole this would increase with distance. At something over 2 million miles anyone at the end of the bridge would be travelling at the speed of light (but I'm not too sure what this would be relative to). By comparison, if the bridge was built at the equator the end of the bridge would only describe a circle the same diameter as the earth however long the bridge was, but this lower speed would still throw you off the bridge at approximately 90 degrees to the bridge surface if the bridge was very long.

Edited by TonyMcC
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By comparison, if the bridge was built at the equator the end of the bridge would only describe a circle the same diameter as the earth ...

 

That is just flat wrong. A tangent to the earth's surface can have an end point arbitrarily far from the center of the earth. Unless the tangent is paralell to the axis of the earth the radial distance from the center of the earth will increase without bound as the length of the tangent segment increases.

 

Your conclusion holds if and only if the briddge were to run perfectly north and south.

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One aspect not considered is centrifugal force. If a straight bridge was built at a pole this would increase with distance. At something over 2 million miles anyone at the end of the bridge would be travelling at the speed of light (but I'm not too sure what this would be relative to). By comparison, if the bridge was built at the equator the end of the bridge would only describe a circle the same diameter as the earth however long the bridge was, but this lower speed would still throw you off the bridge at approximately 90 degrees to the bridge surface if the bridge was very long.

Not considered by some, but it's the whole point of the post I wrote.

However, notwithstanding the fact that I asked about a bridge that was built on top of the North pole it was answered by saying

"Assume that you built it on the Equator in a west East direction".

 

Imagine it's built on the North pole, straight out, tangential to the earth.

Stand on top of it and start walking.

The whole earth is "below" you, so gravity will always tend to pull you "down" onto the bridge rather than away from it.

Ignoring questions of oxygen and temperature I can walk up a slope provided that I can get enough grip on it. (in the limit, I can climb a vertical ladder)

I did mention a handrail so, as long as I don't get too tired, I can drag myself up using that.

 

Am I right in thinking that, if I go far enough, I get thrown outwards by centrifugal force? (As an aside, does that happen at the point where I reach the height of a geosynchronous orbit?)

 

The further I get from the centre of the earth, the lower my weight gets.

The further out I am, the faster my (tangential) rotational speed gets and the bigger my radial acceleration gets.

At some point surely, one overtakes the other and I "fall" outwards.

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That is just flat wrong. A tangent to the earth's surface can have an end point arbitrarily far from the center of the earth. Unless the tangent is paralell to the axis of the earth the radial distance from the center of the earth will increase without bound as the length of the tangent segment increases.

 

Your conclusion holds if and only if the briddge were to run perfectly north and south.

 

Yes, of course you are correct, but I am assuming a bridge built at the equator (i.e. at a tangent to the surface at the furthest distance from the axis of rotation) would be in alignment with the earth's axis of rotation. Also, I believe in agreement with John Cuthber, I am assuming a similar bridge built at the pole where the axis of rotation "emerges" would provide maximum centrifugal force. At least as long as its length to one side exceeded the radius of the earth. .

 

 

Also, if you built it on top of the north pole would the fact that it's spinning tend to throw you out along it (assuming it had handrails)?

 

Sorry, I didn't spot this point when I posted my later entry (#11)

___________________________________________________________________________________________________________________

 

As long as we are discussing impossibly long structures the question of what a bridge might look like from an observer on the earth arises . If the end was a couple of million miles from the earth and built straight I assume it would look distinctly curved since light would take about 10 seconds to reach earth from the far end (Edit - just realised how little the earth rotates in 10 secs - curve would be there but very slight). What happens if the end reaches the speed of light (assuming it holds together!)?Since you stand on the end of the bridge and rotate with it it seems to me that the other end is not moving at all relevant to you. Or are things just getting too ridiculous ?

Edited by TonyMcC
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Yes, of course you are correct, but I am assuming a bridge built at the equator (i.e. at a tangent to the surface at the furthest distance from the axis of rotation) would be in alignment with the earth's axis of rotation.

 

As I said that works if and only if the bridge is built at the equator in a perfectly north-south direction. That is an extremely restrictive and unrealistic assumption.

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As I said that works if and only if the bridge is built at the equator in a perfectly north-south direction. That is an extremely restrictive and unrealistic assumption.

 

Be fair Dr. the whole darn thing has been pretty unrealistic - lol

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Be fair Dr. the whole darn thing has been pretty unrealistic - lol

 

To be fair to you, I realise that what I said did not properly describe what I meant!! The end result of what I concluded should make what I meant quite clear. But, of course that isn't really good enough. ( especially for a mathematician?) :(

Edited by TonyMcC
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At ground level the steepest you can climb is about 30°. Even if you build it at the pole the centrifugal force will not be strong enough to help you climb before you reach the 30°. The gravity won't be much less either so I think you will lose grip to go any further at about 30-35°.

 

This is me at the monza oval at ~30° banking:

P1010878.JPG

 

Edited by mertol
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"At ground level the steepest you can climb is about 30°"

What?

http://en.wikipedia.org/wiki/Ladder#Safety

suggests more than twice that is safe and, as I pointed out, it's not difficult to climb a vertical ladder.

Even if you say that the ground lacks the good grip you get with a ladder, your own picture shows that a 30 degree slope isn't a problem to stay on without sliding off.

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suggests more than twice that is safe and, as I pointed out, it's not difficult to climb a vertical ladder.

Even if you say that the ground lacks the good grip you get with a ladder, your own picture shows that a 30 degree slope isn't a problem to stay on without sliding off.

 

Of course you can climb any angle (including vertical or an overhang) if you have rungs or handholds. It was very clear from the context that everyone was talking about a relatively smooth surface.

Once you get past 30-45 degrees somewhere it gets a little difficult. If you start sliding or rolling, it can be hard to stop. Depending on footwear/surface I can stand still on a surface up to around 50 degrees, or slip on a surface of around 20 degrees, so 30 is a reasonable limit before you'd call your bridge a ladder/staircase/climbing wall.

 

Re centrifugal force, your bridge would have to be roughly the diameter of the earth before this would become overly important. (Before then it would just add to the impression that the ground is sloping or make you feel lighter, although the direction of the 'slope' would depend on where your bridge was).

 

At one Earth radius (or one Earth's diameter if it's symmetric), the bridge is going to feel like it's sloped at 45 degrees and be almost half Earth's radius from the ground. Gravity will be half what it is at the surface (distance is sqrt(2)* earth's radius, gravity is prop. to 1/r^2). Depending on your footwear/surface it's going to start getting difficult to keep moving.

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To be fair to you, I realise that what I said did not properly describe what I meant!! The end result of what I concluded should make what I meant quite clear. But, of course that isn't really good enough. ( especially for a mathematician?) :(

 

I strongly doubt that even you know what you meant.

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Well, according to WIKI's article on friction "Coefficients of friction range from near zero to greater than one – under good conditions, a tire on concrete may have a coefficient of friction of 1.7" and if we assume that your shoes have soles made from the same rubber as the tyre and that the bridge is concrete then the limiting angle is about 69 degrees.

Higher values of the coefficient are not impossible.

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Well, according to WIKI's article on friction "Coefficients of friction range from near zero to greater than one – under good conditions, a tire on concrete may have a coefficient of friction of 1.7" and if we assume that your shoes have soles made from the same rubber as the tyre and that the bridge is concrete then the limiting angle is about 69 degrees.

Higher values of the coefficient are not impossible.

 

Really now? What about the dynamic friction if you slip? What if you start falling and no longer have all your weight on the shoes?

You go for a nice long stroll on a 70 degree roof, and I'll come to see you in hospital if you survive.

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