Mobius

Yes I have seen that one before where the mistake made is the divide by 0. It is a good one though. I can't remember which mathematician did this but I'm sure someone will enlighten us. He proved that if 1=2 you could prove anyone was the pope. i.e. Fact 1: John and the pope are 2. Fact 2: 1=2 therefore we can deduce that John and the pope are 1.

As I stated we could not observe a black hole merging except the gravitational waves released. This is a good article (albeit from 3 years ago...) http://www.spaceref.com/news/viewpr.html?pid=8945

If two supermassive black holes were to merge you would get a larger supermassive balck hole. This event could surely only be detected by the large gravitaional waves it would produce but I don't think this is possible at the moment.

I'm just wondering if anyone knows any decent books (not magazines) for puzzles. I am not interested in sudoku, word searches, crosswords etc... I prefer maths, geometry, logic and science puzzles. Especially ones that are challenging (but not impossible i.e. PhD standard!!!!) Websites are good, but I'd prefer books... Thanks

When two galaxies collide the actual probabilities of stars, black holes etc.. colliding is slim. The galaxies completely change due to gravitational interactions.

Pascal's trinagle is not just good at finding coeficients of polynomials. If you add up each row the numbers increase by powers of two... If you split the rows into diagonals and add them they give you the Fibonacci sequence. There are loads more examples where the triangle shows the relationship between numbers and elementary number patterns.

Although I put up on the other post that the evidence for naked singularities is slim. It is theoretically possible. As a naked singularity by definition is not "hidden" in a black hole, if one did exist it COULD be observed.

Ok here we go... Subtracting 1 from the original terms gives... [math]\sum_{3}^{\infty}({1\over sin({(n-2)\times 180)\over 2n})}-1)[/math] Not sure of how to get the limit of the series but I can find the limit of the expression as n approaches [math]\infty[/math] multiplying out the sin expression:- [math]\lim (n\rightarrow \infty)({1\over sin({180n-360\over 2n})}-1)[/math] [math]\Rightarrow \lim (n\rightarrow \infty)({1\over sin({180-{360\over n} \over 2})}-1)[/math] Now as n approaches infinity the 360/n term approaches 0. [math]\Rightarrow ({1\over sin(90)}-1)[/math] [math]\Rightarrow 1-1=0[/math] Therefore the limit series does indeed converge. Is this the idea are have I made another mistake?

Naked singularity is one without an event horizon right? I'm not sure on the details on naked singularities but it was a computer simulation that showed they could exist right. Is there any proper theory behind them and of course they have not been observed.

Yeah typo: I meant the second eqn. converges to 8.7. Not to worry I will put some thought into your points over the next few days as I am not sure how to convert it to a series (I just took 1 from the two n's in the equation). The first equation is the limit (from n =4 to infinity) of the sum of the equations (just didn't come out that way). Cheers for the help...

I think I see your problem. y-y1=m(x-x1) You have chosen the point (0,2/3) therefore x1 = 0 y1=2/3 That will give the right answer

Ah yes Is see they ARE the same force now, it has been some time since I thought about satellites and circular motion in that way. It is far easier to mess around with the equations without thinking what is really going on. I think that goes for most of physics

Talk about moving off the topic a bit. Maybe Obnoxious you should state what you mean by singularity... Mathematical singularity :-where maths breaks down e.g. 1/0 Gravitational singularity:- Infinite spatial curvature (same maths as above) (Black holes and big bang being examples). Technological singularity:- A future time we cannot comprehend. I presume you mean Gravitational singularity. The problem as I stated is that equations end up similar to 1/0. the maths behind general realtivity break down as mass is compressed to a single point. It is obvious that the current theory of gravity is inadequate to describe what happens to objects, space or time at these extremely small, dense, high temperature situations.

Ok I'm with you on the gravitational effect on the walls of a rotating space station, that is not in a gravitational field, is due to centripetal force. However I am unsure of what you mean by the fact that the two forces are not balancing the gravitational force of the earth. All satellites of the earth experience the earths gravitational field. Do you mean that they are effectively in free fall and do not feel the gravitational force? So it falls around the earth's surface as opposed to on it. I honestly thought that there was a "balancing" effect. As is the case: the nearer the satellite is to the earth, the faster it must go. Even high school students must equate the formulas for centripetal force with gravitational force in order to work out their height. Just explain the fact that the two forces don't cancel. Is it because it is the fact that you are moving in a circle in a freefall fashion and therefore the only force experienced must be centripetal, keeping us in that circle. However you must account for the fact that if we did not have this centripetal force we would plunnit towards the ground due the the earths gravity!

Not so good at my limits it seems... On your logic is this the series my product changes to?:- [math]\lim \limits {1\over sin({(n-3)\times 180\over 2(n-1)})}[/math] Surely this is an infinite sequence as the angle is always less then 180 degrees therefore the sign of the angle is always positive and between 0 and 1 (it would be quickly approaching 1) and therefore 1/this number will be greater than 1, so the infinte series diverges! The computer seems to agree with this also. Is there no way to work out the original limit of?:- [math]\prod \limits_{3}^{\infty}{1\over sin({(n-2)\times 180\over 2n})}[/math] The computer says it does diverge to 8.7 As this problem is based on increasing regular polygons it stands to reason that the distance between consecutive circles gets shorter and shorter and should converge.

Ooops sorry in the attachement it has from n=1 to inf, it should be from n=3 to inf... [math]R=\prod \limits _{3}^{\infty }{1\over sin({(n-2)\times 180\over 2n})}[/math]

The series is attached. series.doc

Finding limits of infinite series is possible but I don't know how to calculate the limit of a multiplicative series. I put the problem I have on a website including the background to the problem, would be interested in any insights or alternative ways of doing the sum.... http://www.geocities.com/mobiusmaths/infpoly.htm

The "artificial gravity" from a spinning object is when the centripetal force balances the gravitational force. Einstein stated that the two forces were equivalent. Satellites stay in orbit by balancing their centripetal force with the gravitational force from the earth.

The problem with a singularity is the infinite density associated with it.

Gravity is still causing problems in physics today. It is completely different from the three other natural forces (electromagnetic, strong and weak nuclear forces) due to weakness. Gravitational forces for small masses is negligible capared to the other forces. It is thought that at an extremely high temperature that all forces were the same i.e. close after the big bang. Although the strength of the force is weak its range is infinte compared to the short range of the others. The graviton theory is based on an "exchange particle". Forces work by exchanging particles (electromagnetic forces exchange photons etc...). This "exchange particle" means that objects feel the force by exchanging gravitons. Gravitons and their wave equivalent (gravitaitional waves) have not yet been discovered or confirmed (as far as I know). The problem with Newton's picture of gravity was that it did not include time. So according to Newton Gravity was instantaneous. If the sun disappeared right now we would feel the effects right now and career on a tangent out of the solar system. This is not the case we would not feel the effects for another 8 mins. Einstein showed that gravitational effects travel at the speed of light. As already mentioned General relativity sees gravity as curved space and objects travel the path of least resitance within that curved space. And indeed if gravity gets very strong it can curve space into a singularity i.e. a black hole. As I said gravity is still causing problems as it cannot be combined with quantum mechanics. The problem is not only the extreme difference in the nature of the magnitude of the force but also the "smoothness" of gravity. Gravity casues a curve in space that should be smooth. Quantum space is very different as it is a see of virtual particles and destroys this smoothness. Most modern theories are trying to rectify the situation. M theory (a combination of 5 different string theories) is one of the best models to combine gravity with the other forces and to fully understand what this mysterious force that keeps us to this earth. Or you could say it's just magic!!!!