Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: jeffreyH on 23/09/2013 08:49:21
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I have noted a similarity in plots between the gravitational field decrease with distance and increasing velocities. Does this imply that velocity has its own inverse square law?
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No that is not the case. They are essentially different things. and it will allow me to explain in simple terms some really important physical facts that are not well understood
You see the inverse square law for the decrease in the forces gravity and electrostatic fields form point sources is due to the fact that we live in a largely three dimensional universe and the area over which the field is spread increases as the square of the distance from the source so the force decreases inversely as the square. if we lived in a universe with a different number of dimensions (or even approximately so) the fall off of the force would be different.
We do experience and can show that the effect of this reduced dimensionality is real and this will lead to the second part of this answer but I will extend the ideas of reduced dimensionality first.
The gravitational or electrostatic force from an infinite (or approximately an infinite) line source falls off just as the inverse of the distance and the gravitational or electrostatic force from an infinite plane source is constant this is because the field can only expand into one or zero dimensions as you move away from the source.
We experience this on the earth's surface because to us gravity is essentially a plane source of force and to a first approximation is constant over the few miles of vertical dstance within which we all live
This also works the other way if there were four spatial dimensions, forces would decrease as the inverse cube of distance and there would be no such thing as a stable orbit!
Now back to the distance travelled by an object accelerating in a constant (note well, constant as we experience on the earth's surface)gravitational field
a constant force produces a constant acceleration (as long as things don't move too fast of course) so for a given time a body has moved a particular distance and in another similar time period it will move that distance again ie twice the distance but in that second period it has also acelerated enough to cover twice the distance in the same time well do it does both and the distance travelled goes up as the square of the time (this of course is a direct square law not an inverse one!)
Thinking a bit more about this it means that the distance fallen a body free falling towards a gravitating body a long way away where the force increases as the square of the distance travels much further than a simple square law would imply because the acceleration is increasing as the body approaches the source this is why comets in long elliptical orbits take such a long time to come back but only spend a brief period close to the sun.
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No that is not the case. They are essentially different things. and it will allow me to explain in simple terms some really important physical facts that are not well understood
You see the inverse square law for the decrease in the forces gravity and electrostatic fields form point sources is due to the fact that we live in a largely three dimensional universe and the area over which the field is spread increases as the square of the distance from the source so the force decreases inversely as the square. if we lived in a universe with a different number of dimensions (or even approximately so) the fall off of the force would be different.
We do experience and can show that the effect of this reduced dimensionality is real and this will lead to the second part of this answer but I will extend the ideas of reduced dimensionality first.
The gravitational or electrostatic force from an infinite (or approximately an infinite) line source falls off just as the inverse of the distance and the gravitational or electrostatic force from an infinite plane source is constant this is because the field can only expand into one or zero dimensions as you move away from the source.
We experience this on the earth's surface because to us gravity is essentially a plane source of force and to a first approximation is constant over the few miles of vertical dstance within which we all live
This also works the other way if there were four spatial dimensions, forces would decrease as the inverse cube of distance and there would be no such thing as a stable orbit!
Now back to the distance travelled by an object accelerating in a constant (note well, constant as we experience on the earth's surface)gravitational field
a constant force produces a constant acceleration (as long as things don't move too fast of course) so for a given time a body has moved a particular distance and in another similar time period it will move that distance again ie twice the distance but in that second period it has also acelerated enough to cover twice the distance in the same time well do it does both and the distance travelled goes up as the square of the time (this of course is a direct square law not an inverse one!)
Thinking a bit more about this it means that the distance fallen a body free falling towards a gravitating body a long way away where the force increases as the square of the distance travels much further than a simple square law would imply because the acceleration is increasing as the body approaches the source this is why comets in long elliptical orbits take such a long time to come back but only spend a brief period close to the sun.
Sorry I totally messed this question up. I meant to include the time dilation and length contraction elements with relativistic gamma/beta at velocity. I have my stupid head on today. Now that does look like a mirror of the gravity curve. However, as you rightly pointed out, not inverse square. Hence why it appears to mirror relativistic velocities.
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I have noted a similarity in plots between the gravitational field decrease with distance and increasing velocities. Does this imply that velocity has its own inverse square law?
No. For instance; a particle moving through a gravitational field speeds up as if starts to fall but as it gets closer to the planet time dilation makes the speed start to decrease until, if it's a black hole, it nearly comes to a stop. But as the particle moves outside the field the speed slows down but may never stop and becomes constant if the speed is greater than the escape velocity.
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I have noted a similarity in plots between the gravitational field decrease with distance and increasing velocities. Does this imply that velocity has its own inverse square law?
No. For instance; a particle moving through a gravitational field speeds up as if starts to fall but as it gets closer to the planet time dilation makes the speed start to decrease until, if it's a black hole, it nearly comes to a stop. But as the particle moves outside the field the speed slows down but may never stop and becomes constant if the speed is greater than the escape velocity.
Thanks trying to get my head round a few things. A question. For a super massive black hole could the time dilation effect as it drops off away from the mass be a reason for the galaxy rotation profiles that are observed or is the effect too weak?
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I have done some plots of gravity decrease for various mass sizes. Do these look realistic? I have attached the graph. The x-axis is distance from mass (r) the y-axis is gravitational field strength.
For anyone who already read this I got the axes the wrong way round in the above post. I have corrected this. Sorry for the confusion.
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Thanks trying to get my head round a few things. A question. For a super massive black hole could the time dilation effect as it drops off away from the mass be a reason for the galaxy rotation profiles that are observed or is the effect too weak?
You're talking about the effects due to dark energy. No. It can't. If there was a chance that would have been the first thing that they'd think of. These people are very good at what they do. Remember that these people are scientists and are very smart indeed!
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I have noted a similarity in plots between the gravitational field decrease with distance and increasing velocities. Does this imply that velocity has its own inverse square law?
No. For instance; a particle moving through a gravitational field speeds up as if starts to fall but as it gets closer to the planet time dilation makes the speed start to decrease until, if it's a black hole, it nearly comes to a stop. But as the particle moves outside the field the speed slows down but may never stop and becomes constant if the speed is greater than the escape velocity.
Thanks trying to get my head round a few things. A question. For a super massive black hole could the time dilation effect as it drops off away from the mass be a reason for the galaxy rotation profiles that are observed or is the effect too weak?
What I may do next is investigate the relationship between escape velocity and altitude. I believe I am correct in assuming that we could never actually reach escape velocity on earth because the energy required is too great. We actually force our way out into space at a slower rate.
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Not an energy limitation, but a matter of practical aerodynamics. Wikipedia puts it nicely
Because of the atmosphere it is not useful and hardly possible to give an object near the surface of the Earth a speed of 11.2 km/s (40,320 km/h), as these speeds are too far in the hypersonic regime for most practical propulsion systems and would cause most objects to burn up due to aerodynamic heating or be torn apart by atmospheric drag. For an actual escape orbit a spacecraft is first placed in low Earth orbit (160–2,000 km) and then accelerated to the escape velocity at that altitude, which is a little less — about 10.9 km/s. The required change in speed, however, is far less because from a low Earth orbit the spacecraft already has a speed of approximately 8 km/s (28,800 km/h).
You could in principle build a gun that would fire a bullet at escape speed (Jules Verne's calculations are adequate) but atmospheric ablation would make it a bit pointless (literally!) and you'd have quite a problem preventing any human cargo from being squished by the acceleration. 3 - 6 g is tolerable, maybe 15g for unmanned payloads, so practical rockets take their time to leave us.
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Not an energy limitation, but a matter of practical aerodynamics. Wikipedia puts it nicely
Because of the atmosphere it is not useful and hardly possible to give an object near the surface of the Earth a speed of 11.2 km/s (40,320 km/h), as these speeds are too far in the hypersonic regime for most practical propulsion systems and would cause most objects to burn up due to aerodynamic heating or be torn apart by atmospheric drag. For an actual escape orbit a spacecraft is first placed in low Earth orbit (160–2,000 km) and then accelerated to the escape velocity at that altitude, which is a little less — about 10.9 km/s. The required change in speed, however, is far less because from a low Earth orbit the spacecraft already has a speed of approximately 8 km/s (28,800 km/h).
You could in principle build a gun that would fire a bullet at escape speed (Jules Verne's calculations are adequate) but atmospheric ablation would make it a bit pointless (literally!) and you'd have quite a problem preventing any human cargo from being squished by the acceleration. 3 - 6 g is tolerable, maybe 15g for unmanned payloads, so practical rockets take their time to leave us.
This brings up another interesting issue. If spacecraft were ever to have some means of generating their own artificial gravity would this in turn pose the g force problem when travelling in space.