Pi. A Universal Constant?

incredibly clever stuff :)

Thanks Maisie... I'm not sure that it is clever yet but I'm hoping it might end up being ;o)

My system is to post something and wait for people to read and think about it for a while...
and then come back to it again - on the principle that, as in quantum theory, looking at something forces it to become better defined :O)

I intend to get around to examining all the little rules we have discovered from playing with our virtual pendulum.
One of the most obvious is that the velocity of the pendulum is related to it distance from the centre of its swing. In other words, the velocity of the bob is proportional to the radius of the swing.

We have also noticed that four times the gravity results in twice the velocity (for any given radius). Now since gravity is an acceleration we can say that the velocity equals half the acceleration.

We have noticed that as the radius (L) reduces the swing period time (T) reduces too - proportionally to the square root of the radius. Yet, the speed of the bob actually reduces too! This shows us the added inertia associated with increasing curvature.

We’ve discovered how to keep a constant speed of the bob with increasing radius by proportionally reducing the acceleration due to gravity.

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Okay we’ll look at the other things we’ve learned later when I can spare the time but for now I want to discuss the possibility that Newton was WRONG.

For the sake of argument let’s say that Newton took Kepler’s findings, his laws and measurements, and came up with a way to not only improve the accuracy of the predictive power but also to tie it all together with his laws of Gravitation.
No great argument yet :O)

Now let’s say that Newton made a grave mistake and assumed that all the planets orbits were simply reliant on the gravitational force of the sun - which followed the simple inverse square rule.

Now this seems to fit in perfectly with Kepler’s findings but we now know that it doesn’t.
To fit with Kepler’s laws then it would mean that there was no added inertia from increasing curvature… and yet we have not only found that there definitely is, but we have also measured it.

If Newton were alive today he would look at the weightlessness experienced by orbiting astronauts and realise that Einstein was correct, that there is no effect of gravity on bodies which are free falling because they all fall together and hence there is no gravitational force from the sun acting on the Earth and so it does not effect the tides.

Gravity curves space and things that move through space curve with it.

The planets follow the curve of space and it is their inertia that determines how far from the sun they are, and hence their velocity, not the acceleration due to gravity that Newton thought.

Hence Kepler’s laws are based on the conservation of momentum and not on a elastic gravitational string that ties them to the sun.

Newton's law of gravitation works well enough in our solar system because it was based on local observations and made to fit...
but it is not a Universal Law and inventing dark matter and dark energy to try and make it fit elsewhere is the last refuge of a desperate bunch of failed sci-fi writers trying to hang onto to their jobs :O)

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Thinking about the crucial difference between pendulums, swung weights and orbits it is in the direction of their desire :O)
Pendulums and swung weights desire to move away from the centre and gravitational orbits are drawn toward it.
Of course all have zero mean acceleration but only the pendulum has zero mean inertia.

Now the result of this is that only the gravitational orbit actually has the objects velocity increasing with decreasing radius...
The main reason for this is that when the force is acting outward it tends to move faster with increasing radius because the ‘connecting force’ acts rigidly and so the object is swung ever faster as its radius increases.

With orbit’s the ‘connecting force’ is inward and not rigid and so the object tends to slow with increasing radius because it has decreasing restraints and the only way it can orbit is by having a lower inertia to match.

In other words the velocity of an orbiting body relies solely on the resistance it faces in moving, nearer to, or further from, its sun.

The larger the radius (the lessening curve) the lesser the inertia of curving but the lesser the reason to want to curve.

So, the curve of the orbit ‘selects’ a specific velocity and inertia.
If the inertia and the velocity are not balanced then they will seesaw and cause an eccentric orbit or the orbit will not be stable.

Thinking a little more about orbiting bodies - and that they are a balance between what pulls them down and what pushes them up - I think it is probably time to start considering what pulls them up and what pushes then down :O)

Which ever theory you choose to support it should have a reasonable explanation as to why the orbiting body does not fall.
The simplest explanation is that there is no reason why it should - that its curve fits exactly with the curve of gravity at that particular distance from its sun.

However, for both Newton and the Einstein the tendency for a body to want to follow a straight line must be fundamental. It’s just that for Einstein an orbit IS a straight line whereas for Newton the orbit is a curve which results from a constant battle between the straight line and the force of gravity that seeks to bend it.

My point being that if you have an outward force then you must have a balancing inward force and since they are balancing then Einstein’s is the simplest explanation - that, space is curved and left to themselves bodies will follow that curve. So why do they fall? Because they don’t follow the curve.

If you want the simplest explanation you should imagine that the gravitational field is rotating and that anything that rotates relatively slowly, to it, will tend to be deflected inwards toward the centre of spin and anything that rotates too fast tends to be deflected outwards...

For some reason this reminded me of Fleming‘s Rules and set me thinking about Maxwell‘s ‘curl’ which made me wonder...
has anyone thought to try and explain gravity using Heaviside's versions of Maxwell's equations?

"The most interesting thing about orbits is what prevents the orbiting body from falling - the curve.
It seems that they are a balance between what pulls them down and what pushes them up...
yet, I think it is probably time to start considering what pulls them up and what pushes then down :O)"...
If we replace 'up' with 'out' and 'down' with 'in' then the following does exactly that - it explains what pushes things in and what pulls them out.

Not feeling well enough to put my mind to gravity I was sat out in the sunshine when I noticed a couple of courting butterflies.
I was amazed to see what a wonderful demonstration of Kepler’s second law these two lovers described.
They were orbiting each other faster as the grew closer, slowing as they moved apart.
Then they flew around me in an almost perfect circle while flying around each other!

Now when I say these lovers were a wonderful demonstration of Kepler’s law I mean before Newton added his theory of gravity… I say that because, of course, it could not have been gravity that was holding the two together.

If I was speculating I would argue that there must be an ether and that it is spinning in the ether that draws them together, the faster they spin around each other the more they are drawn together but the harder it gets to move closer because the curve 'pushes' them out.

So, to me that implies that gravity is a not one force but at least two.
A strong local force caused by the effect of a body’s mass and a secondary force caused by the spin it generates - with the mass effect probably reducing much faster than Newton guessed and the spin taking over as the main ‘gravitational’ force as the radius increases.

Now this suggest to me that where there is gravity which connects two bodies there must always be spin and that spin will be proportional to the masses of the two bodies and their distance apart.

With the butterflies they were of equal mass and so they circled each other at the same speed…

Kepler suggests that had one been heavier that the other the lighter one would move faster than the heavy one as though they were spinning on opposite sides of a fulcrum, like a spinning seesaw, where the seesaw was balanced by the heavier one being closer to the middle.

So what is the seesaw, what draws them together and balances their speed?
It certainly can’t be gravity... So, it must be a function of their spin!

I can see it may be something to do with the inertia of curving that supplies the resistance to them actually colliding but I can’t see what draws them together but it must be something pushing inward rather than pulling - could it be that their spin causes a pressure gradient which pushes inward...

Perhaps it is like a tornado or a water spout - somehow their spin causes them to be drawn together but again however fast they spin they can’t beat the curve.

I’ll give it more thought.

Great. I managed the whole essay. When all is said and done more was said than done.

"If Newton were alive today he would look at the weightlessness experienced by orbiting astronauts and realise that Einstein was correct, that there is no acceleration due to gravity on bodies which are free falling". Makes one think doesn't it?

And certainly original, "Anyway, the point I was trying to make, before I wandered off on a tangent, was that a pendulum lets you investigate the effect of ever decreasing curvature by the simple means of lengthening the pendulum".

Thanks for reading Nolan!
I applaud your determination :O)

AS you say a lot more is said than done because it is difficult to find anything that could prove Newton wrong since his theory of gravitation was so good it lasted until Einstein.

The laugh is that people pretend that there is only a small difference between Einstein and Newton and this only effects very high speed calculations.

The truth is that Einstein has a completely different concept of gravity and the reason we pretend that it is similar to Newton is all the intellectual legacy of Newton's reign which still works wonderfully well in our Solar System, and, of course, the fact that Einstein’s equations are too tough to work with and that most people don’t really understand his theory despite being ‘experts’ ;O)

I wrote a joke piece suggesting that the Earth’s gravity might be a result of the Earth acting like a spaceship accelerating at the speed we attribute to gravity only to find later that Einstein may actually have believed that!

Personally I think he should have kept the ether as that would have solved a lot of problems that will probably bring the theory down quite soon…
the laugh is they will probably default to Newton again even though they must know that his Universal Gravitation can’t be right.

People will throw their E=MC2 T-shirts in the rubbish bin onlt to regret it a few years later when it emerges that Einstein was right and it was MICHELSON and MORLEY who were wrong because they failed to realise that the ether spins with the Earth, or more properly the Earth spins with the ether :O)

It seems that it is always the way that great men are always misunderstood by those that follow and like Darwin, Marx and countless others their work becomes something they can no longer identify with.
As Marx might have said “I don’t know what I am but I know I’m not a Marxist.”
Einstein might have said “I don’t know what I am but I know I’m not a Relativist.”

Mind you, have you ever tried to get your head around the explanation of tides from an relativistic viewpoint? The Newtonian explanation isn’t any more reasonable but it is a heck of a lot simpler and, of course, it has had a lot longer for people to come up with ingenious solutions to plaster over the more obvious cracks!

For the record I’ve never been a great fan of any of the dark things that have invaded our view of the Universe from Dark Energy and Dark mass through to Black Holes and since I’m not a great fan of time dilation either then I can’t see how light could be trapped within an event horizon since gravity could never spin fast enough :O)

If you think about it frozen time makes no sense if it is somehow allowing radiation to escape - mind you, it would have to be along the axis of spin and so I can see that it could be theoretically possible.

So I don’t see how our galaxy could ever be gobbled up from within and because of its spin it would deflect anything large which was spinning in the same direction and anything spinning in the opposite direction would likely curve around.

So, that’s pretty much punctured all the sci-fi horror stories that physicists have been apt to roll out as their party-piece to end with a flourishing “… but don’t worry it won’t happen for a few million years.”

Persistence and determination. I shall attempt the comments once I've summoned the courage.
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Well, I managed to get it in on time although it was a bit rushed... Of course I could easily be wrong but even then I should get points for originality :O)

So, here it is. The problems with gravity stems from using mass instead of weight :O)
Gravity has a dual function in that it accelerates and it also increases weight :O)

A pendulum is only effected by the acceleration of mass and not the by the varying weight and so it gives results that are independent of weight and solely dependent on the acceleration.

However, when gravitational equations are used they almost always use mass as an acceleration and not as weight and yet quite often weight is involved too because weight is the downward force of mass accelerated by gravity.
Hence as things fall their weight increases, even though it may not be apparent because they are falling, and this will have a marked effect on how well they curve!

Hence could it be that the inverse square rule is mostly an artefact of this simple confusion between when to use mass and when to use weight.

Oops, checking again I think I got mixed up between R and P squared and that g should be divided by the square of R...

But, serious now, a pendulum allows you to vary gravity from essentially zero upto almost what ever the local gravity is...

Imagine a Pendulum with an infinitely long string with a heavy weight at the bottom.
Essentially that weight is weightless to you but not massless.

It is weightless because it is isolated from the acceleration due to gravity, simply because moving it from side to side has virtually no effect on its height because its curve is so relatively shallow.

As you shorten the length of the string then the amount of fall, the height it can reach, gradually increases.
So we have a virtual means to test the effect of radius on a rotating body because a rotating body sees the same curve at the same radius and hence sees a similar proportional amount of acceleration.

So why doesn't the results from a pendulum agree proportionately with the results from using Newton's equations?

With a pendulum the square of the period is proportional to the radius...
With orbits the square of the period is proportional to the cube of the radius!

The only way to make the pendulum period fit the orbital period is to have the gravity (g) proportionally decrease with the increasing radius such that g = 4Pi squared / radius squared

Ah, but of course, the pendulum calculates the effect on the period due to increasing radius but only the effect due to the decreasing acceleration of gravity...
the added g/(r squared) allows for the decreasing weight and so it fits with Kepler's observations.

Interesting that the decreasing weight changes P squared from equalling R to becoming R cubed!

I have worked out your evil scheme Dastardly! You are writing the Science Lectures for Children & trying them out on us!!!!!!

actually if you are - you need to slow up and explain a few of the complex references you throw at us - some of us don't identify with them at all. Can children?

i bet some of them can, stretchier minds!

No fooling you Maisie ;O)

Actually, it is more of a record of what I'm thinking in the hope that others may catch my drift and help me out...

Well, I managed to get it in on time although it was a bit rushed.
Of course I could easily be wrong but even then I should get points for originality :O)

So, here it is. The problems with gravity stems from using mass instead of weight :O)
Gravity has a dual function in that it accelerates and it also increases weight :O)

A pendulum is only effected by the acceleration of mass and not the by the varying weight and so it gives results that are independent of weight and solely dependent on the acceleration.

However, when gravitational equations are used they almost always use mass as an acceleration and not as weight and yet quite often weight is involved too.

Hence could it be that the inverse square rule is mostly an artefact of this simple confusion between when to use mass and when to use weight.

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Since F=ma then do a thought experiment on a planet with a very low gravity.

You could throw things into the air that you could not even lift on Earth but since the mass has remained the same then you must have suddenly developed super strength :O)

The interesting thing is that if you threw these heavy things forwards then you would, more than likely, throw yourself backward more than you threw them forward because of the proportionality of mass.

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But, serious now, a pendulum allows you to vary gravity from essentially zero upto almost what ever the local gravity is...

Imagine a Pendulum with an infinitely long string with a heavy weight at the bottom.
Essentially that weight is weightless to you but not massless.
It is weightless because it is isolated from the acceleration due to gravity, simply because moving sideways has virtually no effect on its height.

As you shorten the length of the string then the amount of fall, the height it can reach, gradually increases.
So we have a virtual means to test the effect of radius on a rotating body because a rotating body sees the same curve at the same radius and hence sees a proportional amount of acceleration.

So why doesn't the results from a pendulum agree proportionately with the results from using Newton's equations?

With a pendulum the square of the period is proportional to the radius...
With orbits the square of the period is proportional to the cube of the radius!

The only way to make the pendulum period fit the orbital period is to have the gravity (g) proportionally decrease with the increasing radius such that g = 4Pi squared / radius squared
NB This gives the Orbital Period at any given distance from the sun (ie. the planet's year at that radius) and so its square is the cube of R.

Ah, but of course, the pendulum calculates the effect on the period due to increasing radius but only the effect due to the decreasing acceleration of gravity... the added g/(r squared) allows for the decreasing weight and so it fits with Kepler's observations.
In fact the equation reduces to P = sqrt(R cubed) but now we know why :O)

Interesting that the decreasing weight changes P squared from equalling R to becoming R cubed!

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Thinking about the crucial difference between pendulums, swung weights and orbits - it is in the direction of their desire :O)

Pendulums and swung weights desire to move away from the centre and bodies in gravitational orbits appear to be drawn toward it.

Now the result of this is that only the gravitational orbit actually has the objects velocity increasing with decreasing radius...

The main reason for this is that when the force is acting outward it tends to move faster with increasing radius because the ‘connecting force’ acts rigidly and so the object is swung ever faster as its radius increases.

With orbit’s the ‘connecting force’ is inward and not rigid and so the object tends to slow with increasing radius because it has decreasing restraints and the only way it can orbit is by having a lower inertia to match.

In other words the velocity of an orbiting body relies solely on the resistance it faces in moving, nearer to, or further from, its sun.

The larger the radius (the lessening curve) the lesser the inertia of curving but also the lesser the gravitational reason to want to curve.

So, the curve of the orbit ‘selects’ a specific velocity and inertia.
If the body's inertia and its velocity are not balanced then they will seesaw and cause an eccentric orbit or the orbit will not be stable.

The most interesting thing about orbits is what prevents the orbiting body from falling - the curve.
It seems that they are a balance between what pulls them down and what pushes them up...
yet, I think it is probably time to start considering what pulls them up and what pushes then down :O)

Not feeling well enough to put my mind to gravity I was sat out in the sunshine when I noticed a couple of courting butterflies.

I was amazed to see what a wonderful demonstration of Kepler’s law these two lovers described.
They were orbiting each other faster as the grew closer, slowing as they moved apart.
Then they flew around me in an almost perfect circle while flying around each other!

Now when I say these lovers were a wonderful demonstration of Kepler’s law I mean before Newton added his theory of gravity… I say that because, of course, it could not have been gravity that was holding the two together.

If I was speculating I would argue that there must be an ether and that it is spinning in the ether that draws them together, the faster they spin around each other the more they are drawn together but the harder it gets to move closer because the curve 'pushes' them out.

So, to me that implies that gravity is a not one force but at least two.
A strong local force caused by the effect of a body’s mass and a secondary force caused by the spin it generates - with the mass effect probably reducing much faster than Newton guessed and the spin taking over as the main ‘gravitational’ force as the radius increases.

Now this suggest to me that where there is gravity which connects two bodies there must always be spin and that spin will be proportional to the masses of the two bodies and their distance apart.

With the butterflies they were of equal mass and so they circled each other at the same speed…

Kepler suggests that had one been heavier that the other the lighter one would move faster than the heavy one as though they were spinning on opposite sides of a fulcrum, like a spinning seesaw, where the seesaw was balanced by the heavier one being closer to the middle.

So what is the seesaw, what draws them together and balances their speed?
It certainly can’t be gravity... So, it must be a function of their spin!

I can see it may be something to do with the inertia of curving that supplies the resistance to them actually colliding but I can’t see what draws them together but it must be something pushing inward rather than pulling - could it be that their spin causes a pressure gradient which pushes inward...

Perhaps it is like a tornado or a water spout - somehow their spin causes them to be drawn together but again however fast they spin they can’t beat the curve.

How fast could a body theoretically orbit?

Well, since the orbital speed increases with decreasing radius then at some point, the speed might start to approach the speed of light…
Now since circumference c = 2Pi * Radius even if the body remained at the same orbital speed with the decreasing radius it would still mean that its angular velocity, and hence its orbits per second, increased...
So, as a minimum we get orbits O approaches infinity as the radius R approaches zero.
Now since an orbit with the radius of C/2Pi would give an orbital velocity of the speed of light - using identical time periods - then C/2Pi is our limit
In other words if the radius was 186,000 miles/2Pi and the body was orbiting once a second, or, say, if the radius was 0.186000 miles/2Pi and the body was orbiting a million times a second.

Mind you, I would suspect that the latter would be much harder to approach than the former because I feel that the increased curving would mean it needed far more energy to achieve...

Can this be similar to Einstein’s assertion that the inertia of a body approaches infinity as its speed approaches the speed of light? :O)
I've wondered before if it could be the inertia of curving that limits speed because as the Universe is curved anything approaching the speed of light must pass through a lot of curved space.

We know that nothing can travel faster than light so we know that the inertia of curving will approach infinity as the radius approaches zero so there must be a theoretical minimum length for the radius... perhaps it is related to the reduced Planck constant.

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Which ever theory you choose to support it should have a reasonable explanation as to why the orbiting body does not fall.
The simplest explanation is that there is no reason why it should - that its curve fits exactly with the curve of gravity at that particular distance from the sun.

However, for both Newton and the Einstein the tendency for a body to want to follow a straight line must be fundamental. It’s just that for Einstein an orbit IS a straight line whereas for Newton the orbit is a curve which results from a constant battle between the straight line and the force of gravity that seeks to bend it.

My point being that if you have an outward force then you must have a balancing inward force and since they are balancing then Einstein’s is the simplest explanation - that, space is curved and left to themselves bodies will follow that curve. So why do they fall?
Because they don’t follow the curve.

If you want the simplest explanation you should imagine that the gravitational field is rotating and that anything that rotates relatively slowly, to it, at that locality, will tend to be deflected inwards toward the centre of spin and anything that rotates too fast will tend to be deflected outwards...

For some reason this reminded me of Fleming‘s Rules and set me thinking about Maxwell‘s ‘curl’ which made me wondered if anyone has thought to try and explain gravity using Heaviside's versions of Maxwell's equations.

I found some of Maxwell’s thoughts regarding gravity and it seemed to me that he was making the same assumption that Newton makes that gravity actually adds energy to a system as the bodies that compose it draw apart.

Being a great believer in balance this doesn’t work for me. When a spinning skater draws in his arms to spin faster what do we argue that he has stolen energy from? Surely energy is conserved.
Perhaps using Lagrangian mechanics might help to make the balance more obvious.

For Newtonia mechanics we could use the Lagrangian L = T - V
Where T is kinetic energy, (1/2)mv2, and V is potential energy, so we could then use V = mgh for a local area :
where mgh = mass, gravity constant, height.

However, for the moment we simply wish to calculate the Orbital Velocity when the ‘centrifugal force’ balances the gravitational attraction...
Orbital Velocity = sqrt (g/r) where, g = 887.2 and r = distance from Sun.

Okay, since we know that this equation works but it only works as a proportionality.
What I need is a similar equation that gives the orbital velocity for any radius which also gives
similar values to the equation above.

Hopefully I can do this so that it is but it may be that I have to add at least one more variable… I already have an equation that gives a constant orbital speed for any radius so it may be that all I have to do is tack on something that accounts for the added speed of falling / inertia of curving.

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If you look at the values for the planets of our Solar System, then you can see that as bodies move toward the sun their Orbital Velocity increases, as obviously it must, because they are moving toward the centre of a spinning field.
Unless their direction is directly down then their desire to fall must decreases as they move faster because of the growing resistance from the increasing inetia of curving...
So their energy is given to the curve which must add, at least infinitesimally, to the spin.
Of course the reverse is true on the way back :O)

If Maxwell needed lots of energy to make his field equations work then how much is stored in a spinning galaxy - no need for Black Holes.

I’ve pointed out earlier that things fall because they move, it is just that, relatively speaking, we don’t see the extent of that movement.

Conversely, if you look at a geosynchronous satellite then it doesn’t seem to be moving at all... but if so, why doesn’t it fall?

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As some of you will know I have been searching for some method or means of calculating how much ‘effort’ is involved in curving… the answer has been staring me in the face for years and I’ve just been too dim to notice.
The pendulum.

First, though, I to try and make it clear what I’m hoping to discover:
Imagine two points on the circumference of a circle, say C1 and C2, whose distance apart we will assign an arbitrary value of Unity (1)...
What is the maximum and minimum amount that they can curve?
Well, that’s simple, the maximum curve would be if they were the diameter of the circle and hence curved through 180 degrees...

The minimum, is harder to define but it would be when they were part of an infinitely large circle and hence the curve would be infinitesimal and so their ’curve’ would be almost a straight line in terms of the whole.

Perhaps a better way to illustrate that is to imagine you construct a polygon with an infinite number of sides. However many sides you give it you can always add one more and it will be an infinitesimally better approximation to a true circle - but will it ever be perfect?
This explains why Pi is a transcendental number - you can always add another side to the approximation :O)

Anyway, the point I was trying to make, before I wandered off on a tangent, was that a pendulum lets you investigate the effect of ever decreasing curvature by the simple means of lengthening the pendulum.
Obviously as the pendulum lengthens then C1 and C2 become ever closer, even though they are always the same distance apart :O)

In other words, for a diameter of one they would start out at almost Pi/2 and shrink with the growing radius until the are almost down to unity - when the bit of circumference they describe is almost exactly the same length as their distance apart. Clear as mud then :O)

A much simpler way to explain it is by imagining that you set a very small limit on the actual angle that the pendulum is allowed to swing through. Essentially don’t lift it too high before you let it start :O)

Obviously, the longer the pendulum the larger the arc it describes but, the point is the less actual ‘falling’ that it does.
Yeah, well, I think I’ll give up trying to explain what I mean and just get on with it.

Now, I’ve argued before that gravity is the tension between the curved and the straight.
With the pendulum we have a chance to calculate the effect of that tension.

Okay the time period for a complete pendulum oscillation, there and back again, is : T = 2Pi * sqrt(L/g)
where T = the time of complete swing period (there and back)
L= length of the pendulum and g = the local acceleration of gravity.

Obviously the pendulum will swing fastest when it is ‘falling’ the most in relation to the distance it moves horizontally, ie when it is at its shortest. As we lengthen the pendulum it has to travel in an ever greater arc to ‘fall’ the same distance.

Playing with the Hyper physics simple pendulum simulator I discovered that if first you set the force of gravity (g) to 39.4784 [about 4 times Earth's gravity of 9.81 m/s per second - to be more precise it is 4*(Pi squared)] then it is exactly the gravity needed for -
L = 1 metre when T = 1 second in the Sim!

The simple pendulum simulator is here -
http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html

Now you will quickly notice that once you have set g=39.4784 that this causes the length L (in metres) to be equal to time T (in seconds) squared. So if the Earth’s gravity was about 4 times what it is then the time it takes for a pendulum’s swing period would always be the square root of the pendulum’s length.
But why is that?

It is simply because when g=(2Pi) squared its square root (2Pi) cancels with the multiplying 2Pi and hence T = 2Pi * sqrt(L/g) becomes T = sqrt(L).

A little bit of thought reveals that the need for g to be about 4 times that of the acceleration due to gravity on Earth is simply that the equation is for two swings - there and back - and so the result is twice the time of one swing.

Now since the equation is set for a circumference
by the use of 2Pi all we need to do is loose the 2 and we will get half the circumference or 1 swing.
Since we can't adjust the formula used by the Sim we can simply divide the resulting Time by two when we use g, or leave it set at 39.4784 and use the Time as calcultated... and wonder if it is simply coincidence that 1 metre is 39.37 inches which is actually closer to 4*g(Earth) than 39.4784... makes me wonder why the metre and the second don't quite work with 9.81 - although it is close giving 1.003 seconds, (pi squared) 9.8696 gives 1.0000002 seconds!

So, now we have a very simple way of seeing the result of increasing the length of the pendulum…
The pendulum period slows down with the inverse square of the length.
So where have we come across this before? Yes, in orbits.
But why does a pendulum mimic an orbit?
Well, in a way it is an orbit… or at least part of one...

It certainly demonstrates that the slowing of orbits doesn’t necessarily mean that the gravity is reducing only that the increasing radius means that even though it describes a greater distance, it actually falls down less and less in the same time and so, in a way, it mimics reducing gravity.

In other words the greater the radius the less the pendulum can fall because the difference between its curve and the curve of gravity is less pronounced.

If you imagine the radius to be very large you can see that it would be almost horizontal and the gravitational force would essentially all be straight down…

Having realised that orbits are necessarily slower with increased radius how can we know how much of a planet’s velocity is caused by the strength of the gravitational field and how much by the fact that it cannot utilise as much as it would like to because it isn’t allowed to fall enough?

In other words does a gravitational field reduce with the square of the distance or is it that, like a pendulum, the planet’s curve controls how much of the strength of the gravitational field it can use?
That it can only ‘fall’ enough not to fall at all :O)

If you would like to know a little more about the similarity between pendulums and orbits or would like a simple way to calculate how long a planet would take to orbit the sun from almost anywhere in the Solar System then check out the final post to -
http://www.abctales.com/story/mangone/matter-curvature

I still haven't gotten around to calculating exactly how much extra inertia is caused by curving because we have only been using a simple pendulum sim and yet, as you will have noticed if you followed the link and used the sim to calculate the planetary years, it is still a very powerful tool and it may be that it does include the data I'm looking for if I can figure out how to extract it.

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looking at it again I realise that I have fallen into the same trap that some people accuse Newton of falling into... assuming that spinning a ball weight around on a piece of string is analogous to an orbiting body - no it isn't!
Well, certainly not if it is physical string.

The relationship between an orbital planet's velocity and the distance from the sun, as expressed by its year, is related to the square root of the cube of the year - but for pendulums and swung weights it is related to the square.

In other words; the velocity of a pendulum swung by gravity increases with the distance from its centre whereas the velocity of an orbiting body increases the nearer to the centre it gets.
The confusing aspect is that the period of both increases with the distance from the centre.

So, that's the bad news. The good news is that realising the differences has made me realise that in principle doubling the length of the pendulum's 'string' should double the speed.

However, by experiment you will find that the speed increases with the square of the length if g=Pi squared. So what is holding it back?

Well, at least some of the slowing factor must be inertia and we will look at that shortly, bearing in mind that a pendulum has the same period over a reasonably large angular range which suggests that the extra acceleration is being balanced by the extra distance that the bob must travel.

Looking at this from the opposite end it demonstrates that it becomes increasing more difficult to revolve as the radius reduces...
This fact begs the question why do gravitational orbits increase in velocity as the radius reduces despite the increasing resistance?

I've been too preoccupied with health concerns to get much time to do more than think a bit more on this subject but it has occurred to me that the slowing effect might actually already be incorporated into the orbital period and that might be why it is Radius to the power of 3/2 that gives the Orbital Period.
That is, without the slowing it would give the expected Radius squared (assuming that the slowing effect is square root of the Radius) that would give R to the power (2 - 0.5) = 1.5 (3/2).

Feeling a bit better now so a quick check reveals that if g (the local acceleration due to gravity) were to follow the same inverse square rule as it does for planets then the radius
(length of the pendulum) = time in seconds.

That is :
if we use g = 4*(Pi squared) then we get r = 1, t = 1
if we double r then we must halve g so -
g = 2*(Pi squared) give r = 2, t = 2
halving again g = Pi squared gives r = 4, t = 4...

In other words if we adjust the gravitational field strength g to follow the inverse square law, the field halves as r, the distance, doubles...
then r = t whatever length the radius.

In, yet more, other words the speed of the bob remains constant if the acceleration falls proportionally with the increasing radius.

So, using the Sim r = t when g is 4/r Pi squared.
gives you r = t when t is the time for a full double swing period (there and back)...

for a single swing r = t when g = Pi squared/r

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So, has that given us any clues to why a pendulum can take the same time to do increasingly greater swings?
Well, let's consider an earlier conclusion...
'It certainly demonstrates that the slowing of orbits doesn’t necessarily mean that the gravity is reducing only that the increasing radius means that even though it describes a greater distance, it actually falls down less and less in the same time and so, in a way, it mimics reducing gravity.'

However with a pendulum if it is allowed a greater arc then the angle of fall is steeper which means the pendulum gets a higher acceleration but the radius remains the same - and just as a steeper hill makes a bike accelerate faster if the rider doesn't use the brakes - so the pendulum goes faster... Consequently it has further to go because its increased inertia causes it to swing higher on the upward swing.

In other words since the changing acceleration is identical to increasing gravity then time t would decrease but the distance travelled by the larger arc balances it out and so time t remains almost perfectly constant upto 5 or 6 degrees and within 1 percent upto 20 degrees.

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I intend to get around to examining all the little rules we have discovered from playing with our virtual pendulum.
One of the most obvious is that the velocity of the pendulum is related to it distance from the centre of its swing. In other words, the velocity of the bob is proportional to the radius of the swing.

We have also noticed that four times the gravity results in twice the velocity (for any given radius). Now since gravity is an acceleration we can say that the velocity equals half the acceleration.

We have noticed that as the radius (L) reduces the swing period time (T) reduces too - proportionally to the square root of the radius. Yet, the speed of the bob actually reduces too! This shows us the added inertia associated with increasing curvature.

We’ve discovered how to keep a constant speed of the bob with increasing radius by proportionally reducing the acceleration due to gravity.

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Okay we’ll look at the other things we’ve learned later when I can spare the time but for now I want to discuss the possibility that Newton was WRONG.

For the sake of argument let’s say that Newton took Kepler’s findings, his laws and measurements, and came up with a way to not only improve the accuracy of the predictive power but also to tie it all together with his laws of Gravitation. No great argument yet :O)

Now let’s say that Newton made a grave mistake and assumed that all the planets orbits were simply reliant on the gravitational force of the sun - which followed the simple inverse square rule.

Now this seems to fit in perfectly with Kepler’s findings but we now know that it doesn’t.
To fit with Kepler’s laws then it would mean that there was no added inertia from increasing curvature… and yet we have not only found that there definitely is, but we have also measured it.

If Newton were alive today he would look at the weightlessness experienced by orbiting astronauts and realise that Einstein was correct, that there is no acceleration due to gravity on bodies which are free falling.
Hence little or none of the sun's gravitational force acts on the Earth... however, I'm sure it might have a less direct effect, say, sunshine, that could account for the small effect that are attributed to the sun's gravity in a Newtonian based explaination of tides.

Gravity curves space and things that move through space curve with it.
The planets follow the curve of space and it is their inertia that determines how far from the sun they are, and hence their velocity, not the action of gravity from a distance that Newton used (but admitted, as did Einstein, that he didn't like the action at a distance with no known way for it to propogate through space)

Could it be that Kepler’s laws are based on the conservation of momentum and not on a elastic gravitational string that ties them to the sun?

Newton's law of gravitation works well enough in our solar system because it was based on local observations and made to fit...
but it is not a Universal Law and inventing dark matter and dark energy to try and make it fit elsewhere is the last refuge of a desperate bunch of failed sci-fi writers trying to hang onto to their jobs :O)

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As you probably know you can find the orbital period of things that orbit the sun if you know their distance from the sun relative to Earth’s distance from the sun.
Put simply, you either find the distance of the planet from the sun in AU’s on the Internet or you convert it yourself from the distance to the sun in kilometres divided by the Earth’s distance from the sun which is 1AU or 149597890 Km. Now the radius of the sun is 696,000 Km so we divide by 1AU and we get 0.0046524720 AU. Now all we need to do is to multiply it by its square root and we should get the orbital period expressed as a fraction of an Earth year… which turns out to be quite small at 0.000317340. However we have to multiply it by the number of seconds in a year to convert to seconds. So 365.25 * 0.000317340 = 0.115908659 days *24 = 2.7818 hours * 3600 = 10014.48 secs.

Okay now we are ready to use the simple pendulum sim at…
http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html

Now 696,000 Kilometres is 696 million meters so we can enter that in the second L box, the metre box.
Then enter 10014 in the T box and click on the acceleration due to gravity just above ‘g =‘ and you’ll get an acceleration of 274 m/sec squared.

A quick visit to - http://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_...
Where you will find that the equatorial gravity = 274 m/sec squared!

Oddly, the rotation period is 25.38 days not the 0.1159 days predicted by Kepler’s 3rd law…
But, to be fair that’s for orbiting planets and not rotating gases - so why does it work?

I’m not sure but 25.38 days = 2192832 seconds… dividing by 10014 seconds gives 218.97
The sun’s orbital speed is given as 220 Km/s…

Cor! Longest comment on record! Gold Star! *

Thanks maisie... I'll glue it onto my dunce cap :O)

"You're definitely no dunce!" /me remarks fluttering her eyelashes furiously and leaving mascara everywhere... "That's my middle name..."

Thanks again Maisie... and consider me fluttered :O)

At this point I have to concede defeat. However I do have some final remarks:

A curious way of looking at things, “The larger the radius (the lessening curve) the lesser the inertia of curving but also the lesser the gravitational reason to want to curve”.

I liked these ones too, some other novel ideas. The claim “Einstein was right and it was MICHELSON and MORLEY who were wrong because they failed to realise that the ether spins with the Earth, or more properly the Earth spins with the ether”. Could be an alternative? An other fundamental explanation for the famous experiment?

This has to be a very hard problem: “How fast could a body theoretically orbit?” And yes, “the velocity of the bob is proportional to the radius of the swing” - catchy!

*

Your numerical calculations in themselves are flawed e.g. there are actually about 31557600 seconds in a year.

You have been repeating sentences “Gravity curves space and things that move through space curve with it”.
“If Newton were alive today he would look at the weightlessness experienced by orbiting astronauts and realise that Einstein was correct, that there is no acceleration due to gravity on bodies which are free falling”.
“However, for both Newton and the Einstein the tendency for a body to want to follow a straight line must be fundamental. It’s just that for Einstein an orbit IS a straight line whereas for Newton the orbit is a curve which results from a constant battle between the straight line and the force of gravity that seeks to bend it”.

Perhaps one could try just to interchange the word orders e.g. “Gravity does not accelerate free-falling objects” and “before going off on the tangent I barely touched the point I wished to make... ”

*

Congratulations Mangone. I believe you have successfully devised and implemented a self-sustaining self-propagating self-perpetuating algorithm. The tone, the confidence and authority and delivery style in general are perfect for a respectable undergraduate lecture. A pleasant droning well suited to a catnap between breaks.

I do also have to commend you for apparently not inferring concrete conclusions- i.e. anything that could possibly be directly verified.

This is now my last say except I still would like some material on string theory of which I haven't managed to make any sense of whatsoever, ever, and now it appears to have gone out of fashion. Knot theory is of course a different story altogether.

Cheers. Nolan

( Once I remarked to one of my brother's colleagues that to me it seemed “Research in pure mathematics is an instance of intellectual masturbation- because it means nothing to anyone save yourself” on which he replied “And what is wrong with that? It is the 21st century you know” ).

Thank you for the incredible determination it must have needed to wade through all the stuff above Nolan!

I have finally reached a point where lots of interesting experiments are possible, but mainly from other places than Earth, so I’ve essentially maintained the untestabilty status that is so important in the effort to make it as hard as possible to disprove :O)

I would be interested to see if a pendulum with a length that exactly matches the Earth’s acceleration of gravity at the place where it is to be used (about 9.8 meters) has super swing properties, that is can keep time at far greater swing angles.

I’m also interested in checking to see if anything that rotates at 1 radian per second (or maybe 2) has equal acceleration to its radius.

The experiments off planet would be to check that the angular momentum is the same everywhere and not effected by the curve of the Universe - which I suspect it must be.

All my efforts have been in simple 2D and so have ignored at least one dimension.

As it is beyond my simple skill to model a two dimensional sphere (except as a circle) and the Universe must resemble a sphere so then all the 2D curves will be distorted approximations - but then again, that seems to sum up most of modern science ;O)

Thanks again Nolan. I’ll try and return the favour but I’m not very knowledgeable about the poetry of love - but then, few are.
Cheers!

I was wondering why there hadn't been anything new from your good self for quite some time. I should've known you were down here sparring with Nolan and dragging that poor chap Pi, after everything he's been through, back on to the main stage for several thousand encores. Well, I hope you'll let him put his feet up now so he can write another travel book.

The good thing about Pi is you can eat it when you're done - I'm just hopimg it won't be humble.

Pottering with Pi is a pleasant way to cheer up after reading a wonderfully written tale about a poor kid whose lost his mum and living rough...
and you dare say I don't treat my characters well ;O)