Question on the medium gravitational force through a body [Archive]

Markus

Dec20-04, 10:05 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Perhaps an absolutely trivial exercise, but for some reason I don\'t\ncome up with this simple problem when I have to prove it analytically.\n\nImmagine a an extended (regular or irregular) body immersed in a\ngravitational force field (not necessarily that produced by a\npointmass, but more generally of any kind). I have to show that the\nmedium gravitational force exerted on the body turns out to be the\nsame force we find at its center of mass. That\'s quite intuitive, but\nI have no certainty, and I read that somewhere those people working\nwith tidal forces have shown this. "It has been shown", they tell, but\nthen don\'t furnish any reference, and I couldn\'t find any.\n\nHas somebody a reference or a link that proves this? Otherwise can\nsomeone outline briefly the proof?\n\nThanks so far.\n\nMarkus.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Perhaps an absolutely trivial exercise, but for some reason I don't
come up with this simple problem when I have to prove it analytically.

Immagine a an extended (regular or irregular) body immersed in a
gravitational force field (not necessarily that produced by a
pointmass, but more generally of any kind). I have to show that the
medium gravitational force exerted on the body turns out to be the
same force we find at its center of mass. That's quite intuitive, but
I have no certainty, and I read that somewhere those people working
with tidal forces have shown this. "It has been shown", they tell, but
then don't furnish any reference, and I couldn't find any.

Has somebody a reference or a link that proves this? Otherwise can
someone outline briefly the proof?

Thanks so far.

Markus.

Uncle Al

Dec22-04, 05:55 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Markus wrote:\n>\n> Perhaps an absolutely trivial exercise, but for some reason I don\'t\n> come up with this simple problem when I have to prove it analytically.\n>\n> Immagine a an extended (regular or irregular) body immersed in a\n> gravitational force field (not necessarily that produced by a\n> pointmass, but more generally of any kind). I have to show that the\n> medium gravitational force exerted on the body turns out to be the\n> same force we find at its center of mass. That\'s quite intuitive, but\n> I have no certainty, and I read that somewhere those people working\n> with tidal forces have shown this. "It has been shown", they tell, but\n> then don\'t furnish any reference, and I couldn\'t find any.\n>\n> Has somebody a reference or a link that proves this? Otherwise can\n> someone outline briefly the proof?\n\nOne imagines summing the gravitational interaction with each point\nwithin the extended mass will give you an overall total identical to\nthe classical expectation that (less tidal forces) all the mass could\nhave been cocentrated at the center of mass to give identical results.\n\nSounds like job for Green\'s function.\n\n--\nUncle Al\nhttp://www.mazepath.com/uncleal/\n(Toxic URL! Unsafe for children and most mammals)\nhttp://www.mazepath.com/uncleal/qz.pdf\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Markus wrote:
>
> Perhaps an absolutely trivial exercise, but for some reason I don't
> come up with this simple problem when I have to prove it analytically.
>
> Immagine a an extended (regular or irregular) body immersed in a
> gravitational force field (not necessarily that produced by a
> pointmass, but more generally of any kind). I have to show that the
> medium gravitational force exerted on the body turns out to be the
> same force we find at its center of mass. That's quite intuitive, but
> I have no certainty, and I read that somewhere those people working
> with tidal forces have shown this. "It has been shown", they tell, but
> then don't furnish any reference, and I couldn't find any.
>
> Has somebody a reference or a link that proves this? Otherwise can
> someone outline briefly the proof?

One imagines summing the gravitational interaction with each point
within the extended mass will give you an overall total identical to
the classical expectation that (less tidal forces) all the mass could
have been cocentrated at the center of mass to give identical results.

Sounds like job for Green's function.

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf

Kralorelan

Dec22-04, 06:00 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>If I understand what you are to show... it\'s certainly not true, which\nmakes me wonder if the problem is rather odd or if I misunderstand or\nif it\'s been lost in translation:\n"Immagine a an extended (regular or irregular) body immersed in a\ngravitational force field (not necessarily that produced by a\npointmass, but more generally of any kind). I have to show that the\nmedium gravitational force exerted on the body turns out to be the same\nforce we find at its center of mass."\n\nThe force exerted by a (Newtonian) gravity field on an object is not\nthe force that would be exerted on a point mass at the center of mass,\nwhich is what I think you\'re saying you have to show. Rather, there is\na point called the "center of gravity" where the force exerted on the\nobject is the same as the force on that point. To dispel the\ncenter-of-mass idea, consider a very long dumbbell with heavy weights\non the end and a light rod connecting them held vertically in the\ngravity field of a spherical mass. Let the rod be several times longer\nthan the distance from the center of the sphere to the center of the\nlower weight on the dumbbell. The force is a bit over the weight of\nthat dumbbell; if both weights were moved to the center of mass it\nwould clearly be much less.\n\nNow, if you want to prove the existence of the center of gravity,\nwhich is what I suspect you really want to do, then what you\'re\nprobably looking for is the (integral) Mean Value Theorem for\n3-vectors. "A continuous function attains its average at some point".\nYou can look that up at http://www.mathreference.com/ca-int,mvt.html.\nNote that gravity fields are assumed to be continuous while density may\nnot be.\n\nNote that in a uniform gravity field, the center of mass and the\ncenter of gravity coincide, but in general, they do not.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>If I understand what you are to show... it's certainly not true, which
makes me wonder if the problem is rather odd or if I misunderstand or
if it's been lost in translation:
"Immagine a an extended (regular or irregular) body immersed in a
gravitational force field (not necessarily that produced by a
pointmass, but more generally of any kind). I have to show that the
medium gravitational force exerted on the body turns out to be the same
force we find at its center of mass."

The force exerted by a (Newtonian) gravity field on an object is not
the force that would be exerted on a point mass at the center of mass,
which is what I think you're saying you have to show. Rather, there is
a point called the "center of gravity" where the force exerted on the
object is the same as the force on that point. To dispel the
center-of-mass idea, consider a very long dumbbell with heavy weights
on the end and a light rod connecting them held vertically in the
gravity field of a spherical mass. Let the rod be several times longer
than the distance from the center of the sphere to the center of the
lower weight on the dumbbell. The force is a bit over the weight of
that dumbbell; if both weights were moved to the center of mass it
would clearly be much less.

Now, if you want to prove the existence of the center of gravity,
which is what I suspect you really want to do, then what you're
probably looking for is the (integral) Mean Value Theorem for
3-vectors. "A continuous function attains its average at some point".
You can look that up at http://www.mathreference.com/ca-\int,mvt.html.
Note that gravity fields are assumed to be continuous while density may
not be.

Note that in a uniform gravity field, the center of mass and the
center of gravity coincide, but in general, they do not.