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Nir Dahan Inner circle Munich, Germany 1390 Posts |
Most of us who took physics know that Newton's law of Gravitation is expressed by:
F = G * m1 * m2 / r^2 F is the force between two masses m1 and m2 G is the gravitational constant r is the distance between the two center of masses is this formulation really correct or is there something wrong with it??? |
Jonathan Townsend Eternal Order Ossining, NY 27297 Posts |
That's pretty good for two point masses.
Do we need to bring in the stress energy tensor and the twice contracted Riemann tensor today?
...to all the coins I've dropped here
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Nir Dahan Inner circle Munich, Germany 1390 Posts |
I fyou think this is not applicable in certain cases please state why.
dark matter/dark energy theories not accepted :-] |
Jonathan Townsend Eternal Order Ossining, NY 27297 Posts |
Confused by that one. In the early 1900s folks measuring the deflection of light as it passes by the sun found Einstein's equations gave more accurate predictions than Newton's.
Though for everyday occasions, including space travel to date, looks like Newton's equation (or the Laplace/Lagrange work that followed) seems to work just fine.
...to all the coins I've dropped here
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Nir Dahan Inner circle Munich, Germany 1390 Posts |
I can show you a VERY simple case where the above formula (as stated) will not work.
and I am not talking relativistic cases... |
TomasB Inner circle Sweden 1144 Posts |
A sphere inside a hollow sphere, for example. According to that formula the force would be infinite when the mass centres overlap.
/Tomas |
Nir Dahan Inner circle Munich, Germany 1390 Posts |
Absolutelty true Tomas!
another easy example is a right angle ruler and a point mass. we have to integrate over all points in those cases and get the sum of all forces between two points. Nir |
TomasB Inner circle Sweden 1144 Posts |
Would it be correct to say that the formula is valid if the convex hulls of the objects don't intersect?
/Tomas |
Nir Dahan Inner circle Munich, Germany 1390 Posts |
Quote:
On 2006-05-26 17:00, TomasB wrote: oops, I guess I don't understand you. could you maybe draw a simple picture or two? |
Jonathan Townsend Eternal Order Ossining, NY 27297 Posts |
Ah, I forgot about that sphere with someting inside problem. we saw it in first term physics. the basic fun was to see what gravity would be like inside sphere.
...to all the coins I've dropped here
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TomasB Inner circle Sweden 1144 Posts |
Nir, I'll try to attach a picture. The black blobs are objects and the red is the convex hull for each object.
I assume that the formula is valid for the left figure, but what about when the convex hulls start to intersect? Is that when the formula isn't valid any more? Or is it valid all the way up to when one of the convex hulls surround the mass centre of the other object? Wonders, /Tomas Click here to view attached image. |
Jonathan Townsend Eternal Order Ossining, NY 27297 Posts |
Wondering if the problem starts when the center of mass moves inside the "hull" of other object.
...to all the coins I've dropped here
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Nir Dahan Inner circle Munich, Germany 1390 Posts |
Quote:
On 2006-06-01 07:13, TomasB wrote: Tomas, I am not even sure it is valid for the left figure. The formula is valid for point like objects. or some special cases (like round planets, I am sure you can also find other private cases) otherwise one would have to integrate over all points in space and calculate the sum of all forces. to simplify your example. imagine 2 bananes close to one another. one like so -> ( the other like that -> ). assume the top of one banana is touching the bottom of the other. naturally the gravitational force there is stronger then between the other tips. the center of mass is somewhere close to the middle but outside of the banana towards the curvature side. in a very similar way to what you drew. I think there is no way out of this but to integrate over all points and calculate the total force. of course when the bodies are distant enough we can assume point like bodies... |
TomasB Inner circle Sweden 1144 Posts |
You are very right, Nir. I found that Newton himself showed that the integration over a spherical body is the same as considering it a point mass, but it is not general for other shapes.
/Tomas |
Nir Dahan Inner circle Munich, Germany 1390 Posts |
Quote:
On 2006-06-02 14:48, TomasB wrote: True only if you are not within the body itself. in that case one has to sum up only the spherical layers below him towards the center - the attraction of the layers above just cancel out. another puzzle comes to mind based on this principle. I will try to post it later... |
The Magic Cafe Forum Index » » Puzzle me this... » » Newton's gravitational law (another one from the book) (0 Likes) |
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