In the Hollow Center of a Large Mass

Date: 12/23/95 at 2:46:21
From: Anonymous
Subject: physics and calculus

In physics lecture I was told that if a large mass, such as the earth, 
was hollow and one was to be in the hollow part, then you would be 
weightless.  I hypothesised that unless you were in the exact center 
you would be drawn towards the closest side; but my teacher (who 
refused to go into detail) said that Newton had proved that my 
hypothesis was wrong by using three-dimensional integral calculus.  
I would very much like to see this proof.  Please send it to me. 

Date: 5/31/96 at 11:17:39
From: Doctor Ceeks
Subject: Re: physics and calculus

Hi,

Your teacher is correct... an object inside a uniform hollow shell 
would have no net gravitational force on it.

Intuitively, the reason is that although the parts of the sphere closest 
to you would produce a greater force on the object, there is more 
mass on the other side to counteract.

Imagine a cone with vertex the object. By cone, I mean something 
that roughly looks like two party hats stuck together at their apexes 
so that their axes form a single line. (Precisely, I mean something 
that looks like the solutions to x^2+y^2 = az^2, where a is constant.)

Such a cone will intersect the sphere in two pieces, one a distance 
roughly d from the object, the other a distance roughly d' from the  
object. The gravitational forces will point roughly in opposite  
directions and equal (with suitable units) m/d^2 and m'/d'^2 where 
m and m' are the masses of the two pieces a distance d and d', 
respectively, from the object.

The point is that m (with suitable units) is  equal to d^2 and m' is 
equal to d'^2.

This argument is not rigorous. To be made rigorous, careful 
attention must be paid to the orders of magnitude of the errors made 
in the rough approximations. This can be done using the techniques 
of Calculus. Since I am not going to include these computations in 
my answer, this answer must be considered incomplete. However, if 
you know Calculus, try to analyze the problem using spherical 
coordinates.

-Doctor Ceeks,  The Math Forum