20100617, 11:51  #1 
"Lucan"
Dec 2006
England
194A_{16} Posts 
Gravity Force of a Spherical Shell
As we know,
A) the field inside is zero B) outside it is the same as if the mass was located at the centre. The "grownup" way of explaining this is symmetry and Gauss' Flux Theorem (easy to understand and derive without calculus). However, it is easy to see that the field inside is zero: a chord through a point P inside the sphere has the same angle of incidence at each end. Consider a narrow cone with apex P. The mass in the cone goes as distance^2 compensating for the inverse square force law. So the fields due to the masses at opposite ends of the cone cancel at P. Can B be explained as elegantly? Hint: ccorn is temporalily ineligible for this puzzle David Last fiddled with by davieddy on 20100617 at 12:08 
20100617, 12:02  #2 
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2×5,557 Posts 

20100617, 13:39  #3  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2·5,557 Posts 
Quote:
I argue that it is not. The CC manifests itself as a discrepancy from a strict inverse square law, something which the classical proofs require. Paul Last fiddled with by xilman on 20100618 at 08:00 Reason: Added "nonzero" 

20100617, 20:17  #4 
"Lucan"
Dec 2006
England
2×3×13×83 Posts 

20100618, 02:03  #5  
"Richard B. Woods"
Aug 2002
Wisconsin USA
1111000001100_{2} Posts 
Quote:
So I won't answer (though realizing I've just given others a hint). Last fiddled with by cheesehead on 20100618 at 02:07 

20100618, 03:41  #6  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:


20100618, 04:50  #7 
"Richard B. Woods"
Aug 2002
Wisconsin USA
7692_{10} Posts 

20100618, 05:20  #8  
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Quote:
Of course Newton is famous for gravity, calculus and was presumably intimately familiar with Greek geometry. The problem is a good exercize in integration: take your pick how you set the problem up. Surface area between planes h apart = 2 pi R h is tempting. Or rings at angle theta from either the centre or the point outside where you are calculating the field. I am after something neater! David Last fiddled with by davieddy on 20100618 at 05:24 

20100623, 16:43  #9  
Apr 2010
151 Posts 
Quote:


20100624, 16:49  #10 
Dec 2008
Sunny Northern California
3×19 Posts 
Extremely precisely. But not entirely rigorously according to modern standards. Newton had already invented the integral calculus at the time he published the Principia, but would not publish it for some time yet. So he was forced, in order to be understood, to couch his arguments in longwinded geometric reasoning which makes his proofs difficult reading for anyone but experts. (Check it out... it's actually really interesting, but dense, reading.) And I'm sure it wouldn't stand up to modern scrutiny as rigorous mathematics.

20100624, 21:23  #11  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
not altogether unrigorous, and is IMO elegance itself. I meant "Can you tell me precisely how Newton did it?" David 

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